Harry F. Olson - Elements of Acoustical Engineering

Harry F. Olson - Elements of Acoustical Engineering
G} u . J,..,
05.2 (zI i" ;'
C,L
ACOUSTICAL
ENGINEERING
HARRY F. OLSON, PH.D.
Director, Acoustical and Electromechanical
Research Laboratory, RCA Labora,tories,
Princeton, New Jersey
UNVERSIDADTECNOLOGICADECHiLE
B ISUOTECA
SEDE PEREZ ROSALES
D. VAN NOSTRAND COMPANY, INC.
PRINCETON, NEW JERSEY
LONDON
TORONTO
NEW YORK
1
D. VAN NOSTRAND COMPANY, INC.
120 Alexander St., Princeton, New Jersey (Principal office)
24 West 40 Street, New York 18, New York
D. VAN NOSTRAND COMPANY, LTD.
358, Kensington High Street, London, W.14, England
D. VAN NOSTRAND COMPANY (Canada) LTD.
25 Hollinger Road, Toronto 16, Canada
COPYRIGHT ©1957, BY
D. VAN NOSTRAND COMPANY, INC.
Library of Congress Catalogue Card No. 57-8143
Published simultaneously in Canada by
D. VAN NOSTRAND COMPANY (Canada), LTD.
No reproduction in any form of this book, in whole or in
part (crcept for brief quotation in critical articles or reviews).
may be made without written authorization from the publishers.
This book is based on an earlier work entitled Elements
of Acoustical Engineering, by Harry F. Olson, copy­
right 1940, 1947 by D. Van Nostrand Company, Inc.
First Published May 1957
Reprinted August 1960
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
The first edition of this book, published in 1940, was the subject matter
of thirty lectures prepared for presentation at Columbia University. It was
an exposition of the fundamental principles used in modern acoustics and
a description of existing acoustical instruments and systems.
Many and varied advances were made in acoustical engineering in the
seven years following the issuance of the first edition. The second edition
of the book, published in 1947, covered the advances in acoustics which
were made in the period between the first and second editions. Since the
publication of the second edition, the developments in acoustics have been
on an ever greater scale than in the period between the first and second edi­
tions. Today, the science of acoustics includes the generation, transmission,
reception, absorption, conversion, detection, reproduction and control of
sound. An important division of acoustical engineering is sound repro­
duction as exemplified by the telephone, radio, phonograph, sound motion
picture and television. These sound reproducing systems are universally
employed in all variations of modern living. The impact of the reproduc­
tion of sound by these systems upon the dissemination of information, art
and culture has been tremendous.
The ultimate useful destination of all informative sound, direct or repro­
duced, is the human ear. In this connection, great strides have been made
in obtaining knowledge on the characteristics and action of the human hear­
ing machine. Measurements play an essential part in the advancement of
any scientific field. Instruments have been developed and standards have
been established for the measurement of the fundamental quantities in
acoustics. The applications of acoustics in the field of music have led to a
better understanding of the stuff of which music is made. This knowledge
has been applied to the development of new musical instruments employing
the latest electronic and acoustical principles.
Accelerated by the requirements in W orId War II, tremendous advances
were made in underwater sound. The developments in underwater sound
have resulted in systems for detection and accurate location of underwater
craft and obstacles over great distances, depth sounders and other acoustic
applications in undersea communication. The industrial applications of
ultrasonics have unfolded a new field in acoustics. Some of the important
ultrasonic developments include the cleaning of machine parts, drilling and
flaw detection. The science of architectural acoustics has advanced to the
point where auditoriums, studios and rooms can be designed to obtain ex­
cellent acoustics under severe artistic conditions. With ever increasing iniii
11
IV
PREFACE
dustrial expansion comes an increase in noise. Work is now under way
actively to control noise by the use of a variety of acoustic countermeasures.
The preceding brief description of the present status of acoustics shows
that it plays a very important part in our modern civilization. Furthermore,
the fundamentals and applications of the science of acoustics are so well
formulated and substantiated that a large area of the field of acoustics has
attained an engineering status. In preparing new material and in revising
existing material in the third edition, the same principles were followed as in
the first and second editions. Particular efforts have been directed towards
the development of analogies between electrical, mechanical and acoustical
systems because engineers have found that the reduction of a vibrating
system to the analogous electrical network is a valuable tool in the analysis
of vibrating systems. Each chapter has been brought up to date and ampli­
fied. Two new chapters on Complete Sound Reproducing Systems and
.Yfeans for the Communication of Information have been added. As in the
first and second editions most of the illustrations contain several parts so
that a complete theme is depicted in a single illustration.
The author wishes to express his appreciation to Miss Patricia Duman
for her work in typing the manuscript and to his wife Lorene E. Olson
for assistance in compiling and correcting the manuscript.
HARRY
March,1957
F.
OLSON
CONTENTS
PAGE
CHAPTER
1.
SOUND WAVES
1.1
1.2
1.3
INTRODUCTION
ACOUSTICAL WAVE EQUATION . . . . . . . . • . . . . . . . . . . . . . . . . . .
A.
B.
C.
D.
E.
1.4
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • .
SOUND WAVES ... . . . . . . . . . . . . . . . . . . . . . . . •.• . • . . . . . • . . . .
Equation of Continuity ............................
Equation of Motion ...............................
Compressibility of a Gas ..........................
Condensation ....................................
D'Alembertian Wave Equation ... .... .............
PLANE SOUND W A V E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Particle Velocity in a Plane Sound Wave...........
B. Pressure in a Plane Sound Wave . .. ...............
C. Particle Amplitude in a Plane Sound Wave .........
1.5
SPHERICAL SOUND WAVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Pressure in a Spherical Sound Wave ...............
B. Particle Velocity in a Spherical Sound Wave.......
C. Phase Angle Between the Pressure and Particle Veloc­
ity in a Spherical Sound Wave....................
D. Ratio of the Absolute Magnitudes of the Particle Ve­
locity and the Pressure in a Spherical Sound Wave. ..
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
STATIONARY SOUND WAVES.............................
SOUND ENERGY DENSITY
SOUND INTENSITY
DECIBELS
. . . . . . ...•. . . . . . . . . . . . . . . . . . . . .
........................•.........••.
(BELs) ......................................
DOPPLER EFFECT . • . . • . • . . . . . . . . . . . . .• . . . . • . . . .•.••... •.
REFRACTION AND DIFFRACTION • . . . . . . . . . . . . . . • . . . . . . . . . .
ACOUSTIEAL RECIPROCITY THEOREM .. . . ... . . . . . . . . . . . • ...
ACOUSTICAL PRINCIPLE OF SIMILARITY . . . . . . . . . . . . . . . . . . .
LONGITUDINAL WAVES IN A ROD..... .. . . . . . . . . . . . . . . . . . .
TORSIONAL WAVES IN A ROD........ . .. . . . . . . . . . . • . . . . ..
CYLINDRICAL SOUND WAVES
............................
1
2
4
4
5
5
6
6
10
10
10
10
11
12
13
13
14
14
15
15
15
17
17
24
26
26
28
28
II. ACOUSTICAL RADIATING SYSTEMS
2.1
INTRODUCTION
2.2
SIMPLE POINT SOURCE ... • .. • . . . . . . . . . . . . . • . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . .
A. Point Source Radiating into an Infinite Medium. Solid
Angle of 471' Steradians ............................
B. Point Source Radiating into a Semi-Jnfinite Medium.
Solid Angle of 271' Steradians ......................
v
30
30
30
31
vi
CONTE:'-JTS
CHAPTER
PAGE
C. Point Source Radiating into a Solid Angle of 71" Stera­
dians ............................................
D. Point Source Radiating into a Solid Angle of ?T12 Ste­
radians ......................................... .
E. Application of the Simple Source ................. .
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
DOUBLE SOURCE (DOUBLET SOURCE)
...•••••....•..•....•
SERIES OF POINT SOURCES ••.•..•.••••.•.•••••..•.••••••
STRAIGHT LINE SOURCE ..•.••.••.••••....••..•.••.•.••.
BEAM TILTING BY PHASE SHIFTING ...•••.•.•..•.•••...•
TAPERED STRAIGHT LINE SOURCE ••••••••••••••••••••••••
NONUNIFORM STRAIGHT LINE SOURCE ••.•••••••..••••••.
END FIRED LINE SOURCE •....••••....•.••....•••••.....
SUPER DIRECTIVITY SOURCE .••••••....•.••••..••••••....
CURVED LINE SOURCE (ARC OF A CIRCLE) ..••.••...••••.•
CIRCULAR RING SOURCE ..•••••...••.•....••••••..••••••
PLANE CIRCULAR-PISTON SOURCE . • . . . . . . • • . . . . • • . • . . . . . •
NONUNIFORM PLANE CIRCULAR SURFACE SOURCE •.......•
PLANE CIRCULAR-PISTON SOURCE SET IN THE END OF AN
INFINITE PIPE •.•••••.•.••••••.••••••.••.••••...••••••.
PLANE CIRCULAR-PISTON SOURCE IN FREE SPACE •.•••.•.•
PLANE SQUARE SURFACE SOURCE ••....••••....•••••.... ,
PLANE RECTANGULAR SURFACE SOURCE .....••••...•.•....
HORN SOURCE
•.•.••..•.•••••••..••••.••..•••.•..•.••..
A. Exponential Horns .............................. .
B. Conical Horns .................................. .
C. Parabolic Horns ................................. .
2.20
2.21
III.
CURVED SURFACE SOURCE ..•.•.•••....••.••••.••••.••.•.
CONE SURFACE SOURCE . . . . . . . . . • . . . . . • • • . . . . . . . . • . . . . • .
31
31
31
32
35
36
36
37
38
38
39
40
43
43
44
45
45
45
46
47
47
48
48
50
53
MECHANICAL VIBRATING SYSTEMS
3.1
3.2
3.3
INTRODUCTION
A.
B.
C.
D.
E.
F.
G.
3.4
.•....••.....•.••••....••........•.•...•.
STRINGS ..•••......••.•....•••.•...•••.•.•.••••.....••
TRANSVERSE VIBRATION OF BARS ..•••...•.••••...••••.•.
Bar Clamped at One End ..........................
Bar Free at Both Ends ...........................
Bar Clamped at Both Ends ........................
Bar Supported at Both Ends .......................
Bar Clamped at One End and Supported at the Other
Bar Supported at One End and Free at the Other. . ..
Tapered Cantilever Bars ..........................
STRETCHED MEMBRANES
.•.••.•...••.•....•.••....••.•..
A. Circular Membrane .. ~ . . . . . . . . . . . . . . . . . . . . . . . . . . ..
B. Square Membrane ................................
C. Rectangular Membrane ............................
3.5
CIRCULAR PLATES ••••..••••....•.••.•....•.••....••....
A.
B.
C.
D.
Circular
Circular
Circular
Circular
Clamped Plate ...........................
Free Plate ...............................
Plate Supported at the Center .............
Plate Supported at the Outside ... . . . . . . . . ..
56
56
57
58
59
60
60
60
60
60
61
62
63
63
63
64
66
66
66
vii
CONTENTS
CHAPTER
PAGE
3.6
3.7
3.8
LONGITUDINAL VIBRATION OF BARS
TORSIONAL VIBRATION OF BARS • • • • . . . . • • . . • . . . . . . . . . • • .
OPEN AND CLOSED PIPES • • • . . . • . . . . . . . . . • • . . . • . . • . . • . • . •
66
68
69
IV. DYNAMICAL ANALOGIES
4.1
4.2
4.3
4.4
INTRODUCTION
DEFINITIONS
ELEMENTS
.•.••........•.••..•••.•.•............••.
..•..•...•....••••••••••....••....•.•..•..
.............•••••.•.•.••........•....•.....
RESISTANCE
•.....•.•..........•...•.•........••••••••.
A. Electrical Resistance ..............................
B. Mechanical Rectilineal Resistance ..................
C. Mechanical Rotational Resistance ..................
D. Acoustical Resistance .............................
4.5
INDUCTANCE, MASS, MOMENT OF INERTIA, INERTANCE .•...
A. Inductance .......................................
B. Mass ............................................
C. Moment of Inertia ................................
D. Inertance ........................................
4.6
ELECTRICAL CAPACITANCE, RECTILINEAL COMPLIANCE, ROTA­
TIONAL COMPLIANCE, ACOUSTICAL CAPACITANCE •••.••..••
A. Electrical
B. Rectilineal
C. Rotational
D. Acoustical
4.7
71
73
77
78
78
78
78
79
79
79
79
80
80
Capacitance
Compliance
Compliance
Capacitance
............................
............................
............................
............................
81
81
81
82
82
REPRESENTATION OF ELECTRICAL, MECHANICAL RECTILINEAL,
MECHANICAL ROTATIONAL AND ACOUSTICAL ELEMENTS.....
83
V. ACOUSTICAL ELEMENTS
5.1
5.2
5.3
INTRODUCTION
504
ACOUSTICAL IMPEDANCE OF A NARROW SLIT ...•••.••••...
5.5
5.6
5.7
5.8
ACOUSTICAL
•..••.••.•...•.•••....••.....••••.•..•...
RESISTANCE
....•.•.••••..•.•...••••.••••••
ACOUSTICAL IMPEDANCE OF A TUBE OF SMALL DIAMETER. .•
ACOUSTICAL RESISTANCE OF SILK CLOTH •...•.••.•••.••••
INERTANCE
...........•••.••••••.••..••.•••••.•••••....
ACOUSTICAL CAPACITANCE
MECHANICAL
AND
..•........•......•....•.•....
ACOUSTICAL
IMPEDANCE
LOAD
UPON
A
VIBRATING PISTON . . • . . • • . . . . . . . . . . . • • . . • . . . . . . . . • . . . • .
5.9
MECHANICAL
AND
ACOUSTICAL
PULSATING SPHERE
5.10
5.11
UPON
SPHERE
MECHANICAL
AND
MECHANICAL
AND
MECHANICAL
AND
.••••.•....•..••••..••.•...•••••••
ACOUSTICAL
IMPEDANCE
LOAD
UPON
93
.....•....••..•.••.•••••.•..••.•.•
ACOUSTICAL
IMPEDANCE
lOAD
UPON
IMPEDANCE
LOAD
UPON
MECHANICAL AND
ACOUSTICAL
IMPEDANCE
LOAD
UPON
95
A
••••••••••••..•.•......•....••••••••..
ACOUSTICAL
94
A
96
A
VIBRATING PISTON IN THE END OF AN INFINITE TUBE ••••
5.14
92
A
•...•••••..•.•••..............••...•
OSCILLATING
VIBRATING STRIP
5.13
LOAD
MECHANICAL AND ACOUSTICAL IMPEDANCE LOAD UPON AN
PULSATING CYLINDER
5.12
IMPEDANCE
88
88
88
89
90
91
91
97
A
VIBRATING PISTON IN FREE SPACE ••..••..••.......••••.
99
.,
CONTENTS
VJ11
CHAPTER
PAGE
5.15
ACOUSTICAL IMPEDANCE OF A CIRCULAR ORIFICE IN A WALL
5.16
OF INFINITESIMAL THICKNESS ••••..........••••.••.....
ACOUSTICAL IMPEDANCE OF AN OPEN PIPE WITH LARGE
FLANGES
5.17
5.18
5.19
5.20
5.21
5.22
5.23
5.24
HORNS
••.••.•..••..••••.•...••.........••••••••.....
•••••••.••.••••••••••••••••...•.••••••••••••••••
FUNDAMENTAL HORN EQUATION
INFINITE CYLINDRICAL HORN
INFINITE PARABOLIC HORN
INFINITE CONICAL HORN
..•••••••.•........•••.•
(INFINITE PIPE)
.......•..
.......•••••••••.•......•••••
•••..•••.••••••.•••.•••••••••••
INFINITE EXPONENTIAL HORN
•.•••..•....•••••••••••...
INFINITE HYPERBOLIC HORN ••••••...••....••.•••••••...
THROAT ACOUSTICAL IMPEDANCE CHARACTERISTIC OF INFI­
99
99
100
100
101
102
102
103
104
NITE PARABOLIC, CONICAL, EXPONENTIAL, HYPERBOLIC AND
CYLINDRICAL HORNS
5.25
5.26
5.27
5.28
5.29
5.30
5.31
5.32
5.33
•••.•.........••••••••........••..•
FINITE CYLINDRICAL HORN
.......•..•....•........•.•.•
FINITE CONICAL HORN ••..•......••.••.•.•.....•..•••..
FINITE EXPONENTIAL HORN
.•.••.••.•••••••.•••••••••••
THROAT ACOUSTICAL IMPEDANCE CHARACTERISTICS OF
FI­
NITE EXPONENTIAL HORNS
•••••.........•.•••••••••....
EXPONENTIAL CONNECTORS
.••••.••........•.•••••••••.•
A
HORN CONSISTING OF MANIFOLD EXPONENTIAL SECTIONS
CLOSED PIPE WITH A FLANGE . . . . . • • . • . • • • . . . . . . . . . . • • . •
SOUND TRANSMISSION IN TUBES • . . • • . . . • . • . . . . . . . . • . . • •
TRANSMISSION FROM ONE PIPE TO ANOTHER PIPE OF DIF­
FERENT CROSS-SECTIONAL AREA . . . . . . • . . • . . . . . . . . . . • • • . • .
5.34
5.35
5;36
5.37
5.38
104
105
106
108
TRANSMISSION THROUGH THREE PIPES .....•••••••••....
TRANSMISSION FROM ONE MEDIUM TO ANOTHER MEDIUM..
TRANSMISSION THROUGH THREE MEDIA ....•••••.•••..•.
TUBES LINED WITH ABSORBING MATERIAL •••••••••••...•
110
112
114
115
116
117
119
120
121
121
RESPONSE OF A VIBRATING SYSTEM OF ONE DEGREE OF FREE­
DOM
..•............•.•..........•....•..•...........••
122
VI. DIRECT RADIATOR LOUDSPEAKERS
6.1
6.2
6.3
6.4
6.5
6.6
6.7
INTRODUCTION
••........•••••••.•.....•...•••.••..••...
SINGLE-COIL, SINGLE-CONE LOUDSPEAKER . . . . • . • • . • • . . . . .
MULTIPLE, SINGLE-CONE, SINGLE-COIL LOUDSPEAKER •••...
SINGLE-COIL, DOUBLE-CONE LOUDSPEAKER •..•••.•••••....
DOUBLE-COIL. SINGLE-CONE LOUDSPEAKER
•.........••.••
DOUBLE-COIL. DOUBLE-CONE LOUDSPEAKER ••.....•..••••.
124
125
137
141
143
143
MECHANICAL NETWORKS FOR CONTROLLING THE HIGH FRE­
147
A. Conventional Single-Coil Loudspeaker .. . . . . . . . . . . . .. 147
B. Loudspeaker with a Compliance Shunting the Cone
Mechanical Impedance ............................ 148
C. Loudspeaker with a Compliance Shunting; a Compli­
QUENCY RESPONSE OF A LOUDSPEAKER ••••••.•...•••.••••
ance and Mass in Parallel. Connected in Series with
the Cone Mechanical Impedance ................... 148
D. Loudspeaker with a "T" Type Filter Connecting the
Voice Coil Mass and the Cone Mechanical Impedance 148
ix
CONTENTS
CHAPTER
6.8
PAGE
LOUDSPEAKER BAFFLES
A.
B.
C.
D.
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
Irregular Baffle ........................ ... ........
Large Baffle, Different Resonant Frequencies .. . ... . .
Low Resonant Frequency, Different Baffle Sizes ... . .
Different Resonant Frequencies and Different Baffle
Sizes ........................... .... .... . ... . ... .
LOUDSPEAKER
r
6.27
6.28
6.29
6.30
6.31
6.32
..•..•••.•••••.•• • • • • •• ••.•.• .• •
MECHANISMS
FOR
SMALL
SPACE
REQUIRE­
•...• . ••. • •.. . . . . . •• •.•..•.• •• .• • •••••••••••••.•
FEEDBACK ApPLIED TO A LOUDSPEAKER
• ... • •... • ••••••.•
CABINET CONFIGURATION
... . .. • . • • . •• .•.•. • • • ...••..•.•
LOUDSPEAKER
ARRANGEMENT
MOUNTING
IN
THE
167
168
169
CABINET
169
171
LOUDSPEAKER LOCATIONS IN PHONOGRAPHS . • . . . . . . • . . . . • 173
LOUDSPEAKER LOCATIONS IN RADIO RECEIVERS • .. •• • • •. . •• 176
LOUDSPEAKER LOCATIONS IN COMBINATION INSTRUMENTS .. 177
CONCENTRATED SOURCE LOUDSPEAKER .•.. • • • .•. • ..•.•.. • . 178
TRANSIENT RESPONSE . . . . . . . . . •. ... . . . . .. . . .. . . • . . .•.•. 178
DISTORTION . . . . . . . . . . . . . • . . . . . . . . . . . .•.• . . . • . . . . . . . . . • . 183
A. Distortion Due to Nonlinear Cone System .......... 183
B. Nonlinear Suspension System .. ... . ................ 184
C. Distortion Characteristics of a Nonlinear Suspension
System ......... . ... ... .. .. . . ................... 186
D. Response Frequency Characteristics of a Direct Radi­
ator Loudspeaker with a Nonlinear Suspension System 188
E. Distortion Due to Inhomogeneity of the Air Gap Flux 188
F. Frequency Modulation Distortion ................... 190
G. Air Nonlinear Distortion . . ....... . ................ 190
DIAPHRAGMS, SUSPENSIONS, AND VOICE COILS ••.••.••...• 192
HIGH FREQUENCY SOUND DISTRIBUTOR •• • •• . . • . • . . . . • • . . 197
FIELD STRUCTURES . .. .. .•• .••.•. • ••••• • .•.. • .•.••...•• 198
ELECTROSTATIC LOUDSPEAKER . . •. .•..•. •••. .. •. .. • • . . . . . 205
SOUND POWER EMITTED BY A LOUDSPEAKER • . . . • . . . . . •••. 210
LOUDSPEAKER DIRECTIVITY INDEX ..•.• •• .••.. • .•.••.•• •• 211
WALL
6.20
6.21
6.22
6.23
6.24
6.25
6.26
149
149
149
150
152
152
A. Low Resonant Frequency, Different Cabinet Sizes .... 153
B. Different Resonant Frequencies and Different Cabinet
Sizes ....................... . .... . . . ..... . ....... 154
C. Effect of the Depth of the Cabinet .. .. ...... . .. . ... 155
BACK ENCLOSED CABINET LOUDSPEAKER . • • • • ••••• . • ••..•• 155
COMPOUND DIRECT RADIATOR LOUDSPEAKER •..••• • ..•.•.• 157
ACOUSTICAL PHASE INVERTER LOUDSPEAKER . •• .•••..•.••. 159
DRONE CONE PHASE INVERTER .•..•••••••.• ••• •...•••••• 161
ACOUSTICAL LABYRINTH LOUDSPEAKER .••..•.••••.•.•.... 162
COMBINATION HORN AND DIRECT RADIATOR LOUDSPEAKER .. 163
CABINET LOUDSPEAKERS
MENTS
6.17
6.18
6.19
• • •• • • . . . • . . . • .•• ••• ... • . ••• • • • ••
. • . . . . . . . . . • • • . • . • . . .. •...• • • • . •...• •• •••..•••••
LOUDSPEAKER LOCATIONS IN TELEVISION RECEIVERS • • .••..
VII. HORN LOUDSPEAKERS
7.1
7.2
INTRODUCTION
EFFICIENCY
•• •. • . ••• ••• • • • • • . • • . . . . . • . . • • • • • . . . • • • •
. • ... • . • ... •• ••••.••. . •. . .. •• •.•• ••••• •.•••
212
212
CONTEKTS
x
CHAPTER
PAGE
A. The Relation between the Voice Coil Mass, the Load
Mechanical Resistance and the Initial Efficiency ......
B. The Effect of the Mass of the Vibrating System upon
the Efficiency ....................................
C. The Effect of the Air Chamber upon the Efficiency . . ..
D. The Effect of the Generator Electrical Impedance and
the Mechanical Impedance at the Throat of the Horn
upon the Efficiency ...............................
E. The Effect of the Voice Coil Temperature upon the Ef­
ficiency ..........................................
F. The Effect of the Sound Radiation from the Unloaded
Side of the Diaphragm upon the Efficiency ..........
7.3
7.4
7.5
7.6
213
216
218
220
221
221
223
A. Distortion Due to Air Overload in the Horn ......... 223
B. Distortion Due to Variation in Volume of the Air
Chamber .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 225
C. Distortion Due to the Diaphragm Suspension System .. 226
D. Distortion Due to a Nonuniform Magnetic Field in the
Air Gap ......................................... 227
E. Subharmonic Distortion .......................... 228
F. Power Handling Capacity and the Voice Coil Tempera­
ture ............................................. 228
G. Power Handling Capacity and the Amplitude of the
Diaphragm .................. . . . . . . . . . . . . . . . . . . .. 229
HORN LOUDSPEAKER SYSTEMS •••.•.•••.••••••••••••••.• 230
A. Single Horn, Single Channel System .............. 230
B. Multiple Horn, Multiple Channel System ............ 233
C. Compound Horn Loudspeaker ...................... 237
D. Multiple Horn, Single Channel System .............. 238
E. Horn Loudspeaker for Personal Radio Receivers .... 239
F. Folded Horns .................................... 240
G. Horn Loudspeaker Mechanisms .................... 242
H. Diaphragms and Voice Coils ...................... 242
1. Field Structures .................................. 243
J. Horn Walls. Vibration and Absorption ............ 243
THROTTLED AIR FLOW LOUDSPEAKER •••...••••••.••...••. 243
IONOPHONE LOUDSPEAKER
244
DISTORTION
•.•••••••••••••••.••••••••••••••••.••••••••
VIII. MICROPHONES
8.1
INTRODUCTION
8.2
PRESSURE MICROPHONES ••••.••••.•......•.••.•••••..•..
A. Carbon Microphones ............................. .
1. Single Button Carbon Microphone ............ .
2. Double Button Carbon Microphone ........... .
B. Condenser Microphone (Electrostatic Microphone) ..
C. Piezoelectric (Crystal) Microphones ............... .
1. Direct Actuated Crystal Microphone .......... .
2. Diaphragm Actuated Crystal Microphone ...... .
246
246
246
246
251
253
257
259
260
CONTENTS
Xl
CHAPTER
PAGE
3. Diaphragm Actuated Barium Titanate Micro­
phone ...... . ................................
D. Moving Conductor Microphones ...................
l. Moving Coil Microphone (Dynamic Microphone)
2. Inductor Microphone (Straight Line Conductor)
3. Ribbon Microphone ..........................
4. Probe Microphone .................. . ...... . .
5. Comparison of Electrodynamic Microphones. . . ..
E. Magnetic Microphone .............................
F. Electronic Microphone ............................
8.3
VELOCITY
MICROPHONES,
FIRST
ORDER
GRADIENT
260
260
260
263
264
267
268
270
273
MICRO­
275
A. Pressure Gradient Microphone ..................... 275
B. Velocity Microphone .......................... . ... 279
8.4 UNIDIRECTIONAL MICROPHONES . . . . . . . . . .. . . . . • . . . . . . . . . 291
A. Combination Unidirectional Microphones ............ 291
l. The Response of the Unidirectional Microphone as
a Function of the Distance and the Frequency . . .. 292
2. Efficiency of Energy Response to Random Sounds
of the Unidirectional Microphone as a Function of
the Relative Sensitivities of the Bidirectional and
Nondirectional Microphones .................. 293
3. Efficiency of Energy Response to Random Sounds
of a Unidirectional Microphone as a Function of
the Phase Angle Between the Two Units ........ 295
4. Distortion of the Directional Pattern in the Uni­
directional Microphone ....................... 297
B. Single Element Unidirectional Microphones ......... 297
l. Phase Shifting Unidirectional Microphone .... . 297
2. Polydirectional Microphone ................... 298
3. Uniaxial Microphone .. . .................. . ... 303
4. Uniphase Dynamic Microphone ................ 305
5. Variable-Distance Unidirectional Microphon~ .. 307
6. Directional Condenser Microphone ............. 307
7. Dipole Microphone ........ . ................ .. 308
8. Differential Microphone. Lip Microphone ...... 310
8.5 HIGHER ORDER GRADIENT MICROPHONES • • . . . . . . • • . . . . • . . . 311
A. Second Order Gradient Microphones ............... 311
B. Gradient Microphones of Any Order . ... ........... 311
C. Noise Discrimination of Gradient Microphones ....... 312
D. Higher Order Unidirectional Grau: ~.lt Microphones .. 315
E. Second Order Gradient Uniaxial Microphone ........ 316
8.6 W AVE TYPE MICROPHONES • . . . . . . . . . • . . . . . . . . • . . . . . .. ... 319
A. Parabolic Reflector ............................... 320
B. Lens Microphone .... .. . . ......................... 321
C. Large Surface Microphone ........................ 321
D. Line Microphones .................. . ............. 322
l. Line Microphone: Useful Directivity on the Line
Axis. Simple Line .. . ........................ 323
PHONES
..• • • . • . . • . . . • . • • • • . . . . • • • . . . . . . . . . . . . . . . . . . . .
CO NT E:-'.JT S
xu
CHAPTER
PAGE
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15
8.16
2. Line Microphone: Useful Directivity on the Line
Axis. Line with Progressive Delay ............ 324
3. Line Microphone: Useful Directivity on the Line
Axis. Two Lines and a Pressure Gradient Ele­
ment ....................................... 325
4. Ultradirectional Microphone .................. 327
THROAT MICROPHONE .•..••••.••.••••.•..•••••....••••• 329
LAPEL, LAVALIER AND BOOM MICROPHONES ••..•••.•.•.••• 329
HOT-WIRE MICROPHONE • . . . . • • . • . . . . • . • • . . . . . . . • • • . . . • 330
RADIO MICROPHONE .•••.•.....•••....•.•.•...•.•••..... 331
DIRECTIONAL EFFICIENCY OF A SOUND COLLECTING SYSTEM 331
WIND EXCITATION AND SCREENING OF MICROPHONES •••••• 332
NONLINEAR DISTORTION IN MICROPHONES • . . . . . . . . • • . . . . . 333
TRANSIENT RESPONSE OF MICROPHONES ...••.•.•....••••. 334
NOISE IN A SOUND PICKUP SYSTEM ••••....••••.••.•••••• 335
A. Ambient Noise in the Studio ...................... 335
B. Noise Due to Thermal Agitation of the Air Molecules 335
C. Noise Due to Thermal Agitation of the Atoms in the
Vibrating System ................................ 336
D. Noise Due to Thermal Agitation of the Electrons in the
Conductor ....................................... 336
E. Noise Due to Barkhausen Effect in the Transformer .. 337
F. Noise in the Vacuum Tube ........................ 337
G. Noise Due to Thermal Agitation of the Electrons in the
Plate Resistor .................................... 337
H. Example of Noise in a Sound Pickup System ...... ,. 337
SHAPES OF MICROPHONES •.•.•.•.•••..•••...•.•••..••..• 339
IX. MISCELLANEOUS TRANSDUCERS
9.1
9.2
INTRODUCTION
A.
B.
C.
D.
9.3
•.•••••.•.•••••....••••.•...•.••....•..•.
TELEPHONE RECEIVERS
.......••......••.•....••......••
Magnetic Telephone Receiver ......................
Crystal Telephone Receiver ........................
Dynamic Telephone Receiver ......................
Inductor Telephone Receiver ......................
PHONOGRAPHS
.• • • • • • • • • . • • • • . • • • • • • • • . • • • • • • • . . . • • • ••
A. Recording Systems ...............................
1. Recorders ...................................
2. Lateral Cutter ...............................
3. Vertical Cutter ...............................
4. Recording Characteristics .....................
5. Heated Stylus ...............................
B. Reproducing Systems .............................
1. Record Player ...............................
a. Electrical Record Player ....... . . . . . . . . . ..
b. Mechanical Phonograph .................
2. Phonograph Pickups .........................
a. Crystal Pickup ........................•.
340
340
340
346
347
348
350
351
351
351
355
355
357
357
357
357
358
359
359
CONTENTS
Xlll
CHAPTER
PAGE
b.
c.
d.
e.
f.
g.
h.
Ceramic Turnover Pickup ................
Magnetic Pickup ........................
Dynamic Pickup .........................
Frequency Modulation Pickup ............
Electronic Pickup .......................
Variable Resistance Pickup ...............
Feedback Pickup ........................
1. Compliance of Pickups ...................
j. Tone Arm Resonance ....................
3. Distortion in Record Reproduction .............
4. Record Noise ................................
C. Selection of Rotational Speed and Record Diameter ...
D. Commercial Disk Phonograph Records ..............
9.4
9.5
9.6
9.7
VIBRATION PICKUP
ELECTRICAL MEGAPHONE
9.9
••••.••......•.•.•.••••.......•
MAGNETIC TAPE SOUND RECORDING AND REPRODUCING SyS-
384
390
A. Frequency Conversion System ..................... 390
B. Frequency Compression System .................... 391
C. Time Compression System ........................ 392
TEM
9.8
••..••••••••..•.•••.•••.....••••....
SOUND-POWERED PHONES . • . . . . . • . . . . . . . . • . . . . . . . • . . . . . .
••....•..•...•.•••...•.•..••.•..•....•••••••...•.•
MAGNETIC TAPE CONVERSION SYSTEMS
.•..•...•...•.•.••
SOUND MOTION PICTURE RECORDING AND REPRODUCING SySTEM
•.••...•.••..•.•••••...•••....•••.•.••••••...•.•••
A. Film and Sound Track ............................
B. Recording System ................................
1. Variable Area ...............................
2. Variable Density .............................
3. Recording Film Transport ....................
C. Reproducing System ..............................
1. Optical Electronic Reproducer ................
2. Reproducing Film Transport ..................
9.10
362
364
367
369
371
371
372
373
373
373
377
378
379
380
382
382
393
394
395
395
397
397
399
399
400
MOTION PICTURE MAGNETIC TAPE SOUND RECORDING AND
SYSTEM •.•.•..•.•..•..••.•....••••••.••. 400
A. Magnetic Tape .................................. 401
B. Recording Tape Transport ........................ 402
C. Reproducing Tape Transport ...................... 403
VOLUME LIMITERS, COMPRESSORS AND EXPANDERS • . . • . . . . 403
SYNTHETIC REVERBERATION •••..•••••••••••••••••••.•..• 404
HEARING AIDS .•.••..•.•••.•.•••..•••••.••....••.••••• 406
SIRENS . . . • • • . • . • . • . . • . • . . . . . . . . • • . . . . • . • • • . . . • . . . . • • • 409
SEISMIC DETECTORS •...•.•..•.•.••••.••••••..•..•.••... 409
STETHOSCOPES ••.••..••.....••...••.•••.••••.••.•..•..• 410
EAR DEFENDERS .•.••.••••••..•.•..•..•••••.••••••.•••• 414
REPRODUCING
9.11
9.12
9.13
9.14
9.15
9.16
9.17
9.18
ELECTRONIC
SOUND
AND
VIBRATION
REDUCERS
AND
AB­
415
A. Free-Field Zone-Type Sound Reducer ............... 415
B. Free-Field Electronic Sound Absorber ............... 415
SORBERS
,
,0
..•.•..•.•...••..•..••.•.••.••.••...•••••••••..
CONTENTS
XIV
PAGE
CHAPTER
C. Headphone-Type Noise Reducer .................... 417
9.19
D. Electronic Vibration Reducer ...................... 418
. • . • • • • . • . . . . • . . • . . . . • • • • . • • 420
NOISE REDUCTION CIRCUITS
X. MEASUREMENTS
10.1 INTRODUCTION . • . • . . . . . • • . . • . . . . . . • . . • . . . . . . . . . . . • • • • • •
10.2 CALIBRATION OF MICROPHONES . . • . . • • . . . . . . . . . • • • . . . • . • •
A. Response Frequency Characteristic .................
1. Pressure Response ...........................
a. Pistonphone .............................
b. Thermophone ...........................
c. Electrostatic Actuator ....................
d. Reciprocity .............................
2. Field Response ..............................
a. Rayleigh Disk ...........................
b. Reciprocity .............................
3. Secondary Calibration ........................
4. Artificial Voice ..............................
5. Artificial Throat .............................
B. Directional Characteristic .........................
C. Nonlinear Distortion Characteristic .................
D. Phase Distortion Characteristic ....................
E. Electrical Impedance Frequency Characteristic ......
F. Transient Response Characteristic ..................
G. Measurement of Wind Response . . . . . . . . . . . . . . . . . . ..
10.3 TESTING OF LOUDSPEAKERS •..••••••••..•••.••••.•••.•••
A. Response Frequency Characteristic .................
1. Pressure Response ...........................
2. Apparatus for Measuring the Sound Pressure Fre­
quency Relationship of a Sound Source ........
3. Calibration of the Sound Measuring Equipment ..
4. Free Field Sound Room ......................
5. Outdoor Response ............................
6. Small and Partially Deadened Room ... . . . . . . . ..
7. Arrangement of Loudspeakers for Test .........
8. Living Room Measurements ...................
9. Theater Measurements .......................
10. Automobile Measurements ....................
B. Directional Characteristic .........................
C. Nonlinear Distortion Characteristic .................
D. Efficiency Frequency Characteristic ................
1. Direct Determination of Radiated Power .......
2. Indirect Determination of Radiated Power ......
E. Phase Distortion Characteristic ....................
F. Electrical Impedance Frequency Characteristic ......
G. Transient Response Characteristic ..................
H. Subjective Measurements .........................
1. Loudspeaker Environment ....................
2. Loudspeaker Housing, Placement and Mounting .
423
423
423
423
424
425
426
427
428
429
430
433
433
433
434
435
436
437
437
437
438
438
438
439
444
445
450
450
451
451
451
451
452
452
460
461
463
464
464
465
466
466
466
•
CONTENTS
xv
PAGE
CHAPTER
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
10.4
Signal Sound Level ..........................
Ambient Noise Level .........................
Signal or Program Material ...................
Reference Systems .......................... .
Relative Loudness Efficiency ............. . ....
Relative Directivity ................. . ........
Frequency Range ............................
Power Handling Capacity ....... . .... . . . . . . . ..
Response Frequency Contour ..................
Nonlinear Distortion .. . ................... .. .
Transient Response ..........................
General Aspects .............................
TESTING OF TELEPHONE RECEIVERS •..•••..•....•••..••..
A. Subjective Measurements ..................... . ...
B. Objective Measurements ..........................
1. Artificial Ear ................................
2. Artificial Mastoid ............................
10.5
TESTING OF PHONOGRAPHS .••. •.. . . . . . . • . . . . . . . • • • . . . . . .
A. Measurement of the Response of a Phonograph Record
by the Optical Method ............................
B. Testing of Phonograph Pickups ....................
C. Testing of Mechanical Phonographs ................
D. Measurement of Mechanical Noise Produced by a
Phonograph Pickup ..............................
10.6
10.7
10.8
10.9
1O.l0
1O.l1
10.l2
10.l3
1O.l4
10.l5
10.l6
1O.l7
1O.l8
1O.l9
10.20
10.21
XI.
MEASUREMENT OF WOWS . . . . . . . . • . . . • . . . • . . . . . . • . . . • • . .
MEASUREMENT OF ACOUSTICAL IMPEDANCE .••.•.....•..••
MEASUREMENT OF MECHANICAL IMPEDANCE ••.•.•.••.•••.
MEASUREMENT OF POROSITY
MEASUREMENT
OF
D.C.
.•••• • .••••..... • ••.••..••••
ACOUSTICAL
RESISTANCE
MEASUREMENT OF ABSORPTION COEFFICIENT . . . . . . . . • . . . . .
MEASUREMENT OF NOISE • • . . . . . . . . . . . • . • . • • • . . . . • . . . • • •
MEASUREMENT OF THE COMPONENTS IN A COMPLEX WAVE
MEASUREMENT OF TRANSMISSION COEFFICIENT . • . . • • . . . . . .
ARTICULATION
•...••.•• , •••••••••..•..•..•...•..••.••••.
MEASUREMENTS
..••.•...•..••....•••• • ••
TESTING OF HEARING AIDS .••... • . • . . . . . . . . . . . . .••...••
AUTOMATIC BH CURVE TRACER . . . . . . . . • . . . . . . . . . . . . • . • .
ELECTRONIC MEASUREMENT OF ROUGHNESS •....••.•••.•.•
VIBRATION
MEASUREMENTS
472
473
475
475
475
476
478
484
(FLOW
RESISTANCE) •..•••.••.•...•....•.•...••..•••..••••••••
MEASUREMENT OF REVERBERATION TIME •.....•••.....•.•
AUDIOMETRY
466
467
467
467
468
468
468
468
469
469
469
470
470
470
471
471
472
472
•..•.........•.•............
485
486
487
488
492
493
494
494
495
496
497
498
ARCHITECTURAL ACOUSTICS AND THE COLLECTION AND
DISPERSION OF SOUND
11.1
11.2
499
500
A. Sound Absorption and Reverberation ..... . ......... 500
B. Mechanism of Sound Absorption by Acoustical Mate­
rials ............................................ 503
INTRODUCTION
••.•...•••.•..•....•••.........•....•.•••
DISPERSION OF SOUND • .. • •••••..••...••....••••.•....••
CONTENTS
XVI
CHAPTER
PAGE
Functional Sound Absorbers ....................... 506
Resonator Sound Absorber......................... 509
Electronic Sound Absorber ......................... 511
Articulation and Reverberation Time ................ 511
Sound Motion Picture Reproducing System .......... 511
Sound Re-enforcing System ........................ 516
Theater Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 521
J. Reverberation Time of a Theater for the Reproduction
of Sound ......................................... 523
K. Power Requirements for Reproducing Systems.. . . . .. 524
L. Noise at Different Locations .. . . . . . . . . . . . . . . . . . . . .. 524
M. Public Address Systems ........................... 525
N. Sound Motion Picture "Drive In" Theater .......... 529
O. Orchestra and Stage Shell.................... . . . .. 530
P. General Announce and Paging Systems .............. 531
Q. Intercommunicating Systems ....................... 532
R. Radio Receiver Operating in a Living Room. . . . . . . .. 533
S. Radio Receiver Operating in an Automobile ........ 534
T. Absorption of Sound in Passing Through Air. . . . . . .. 535
U. Sound Transmission Through Partitions.... . . . . . . . .. 536
COLLECTION OF SOUND .................................. 538
A. Sound Collecting System ......................... ,. 538
B. Broadcasting Studios.............................. 542
C. Sound Pickup Arrangement for a Radio Broadcast. . .. 546
D. Scoring and Recording Studios.. . . . . . . . . . . . . . . . . . .. 546
E. Sound Pickup Arrangements for Orchestra. . . . . . . . .. 548
F. Vocal Studios .................................... 551
G. Reverberation Time of Broadcasting, Recording and
Scoring Studios............ . . . . . . . . . . . . . . . . . . . . . .. 552
H. Sound Stages for Motion Pictures and Television.... 553
I. Sound Pickup Arrangements for Sound Motion Pic­
tures and Television ............................... 554
C.
D.
E.
F.
G.
H.
I.
11.3
XII. SPEECH, MUSIC AND HEARING
12.1 INTRODUCTION .........................................
12.2 HEARING MECHANISM..................................
12.3 VOICE MECHANISM ....................................
12.4 ARTIFICIAL VOICE MECHANISMS.. . . . . . . . . . . . . . . . . . . . . . ..
A. Artificial Larynx .................................
B. Voder ............................................
C. V ocoder ........................................ "
D. Speech Synthesizers...............................
12.5 VISIBLE SPEECH .......................................
12.6 RESPONSE FREQUENCY CHARACTERISTICS OF EARS. . . . . . . . ..
12.7 LOUDNESS .............................................
12.8 PITCH ................................................
12.9 MASKING .............................................
12.10 NONLINEARITY OF THE EAR .............................
12.11 EFFECT OF PHASE RELATIONS AMONG THE HARMONICS .....
558
558
560
564
564
565
566
567
568
569
570
571
572
573
573
xvii
CONTENTS
CHAPTER
12.12
12.13
12.14
12.15
12.16
12.17
12.18
12.19
12.20
PAGE
MODULATION
MINIMUM PERCEPTIBLE DIFFERENCES • . . . . . . . . . . . . . . . . . . . .
TIMBRE (TONE QUALITy) . • . . . . . . . . . . . • . . • . . . . . . . . . . . . . .
DURATION . • • . . . • • . • . . . . • . . . . . . . . • • . . . . . . . . . . . . • . . . . • . •
GROWTH AND DECAy • . . . . . . . • . • • • . . . . . • • . . . • . . • . . . . • . • •
AUDITORY LOCALIZATION •..•••......•••.•....•••..•••••.
DELAY EFFECT • . . . . . . . . • . . . • . . • . . . • . . . . . . . • . . . . . . . . • . . .
HEARING ACUITY IN THE UNITED STATES POPULATION . . . . .
THE
FREQUENCY
MUSIC
12.21
(VIBRATO) • • . • . • . . . • . . . . . . • . • . . • • . . • . . • . . .
AND
VOLUME
RANGES
OF
SPEECH
574
574
576
576
576
576
577
577
AND
.••...•...•...•..............••.......•.....•.•.
579
THE EFFECT OF FREQUENCY DISCRIMINATION, AMPLITUDE,
FREQUENCY SHIFT, REVERBERATION, NONLINEAR DISTORTION
AND NOISE UPON THE ARTICULATION OF REPRODUCED SPEECH
12.22
THE
12.23
ABSOLUTE AMPLITUDES AND SPECTRA OF SPEECH, MUSICAL
EFFECT
OF
FREQUENCY
DISCRIMINATION
UPON
QUALITY OF REPRODUCED MUSIC . . . . . • . . . . . . . . • . . . . . . • . . .
INSTRUMENTS AND ORCHESTRAS . . . . . . . . . .. . . . . . . . . . . . . . ..
12.24
12.25
12.26
12.27
NOISE IN REPRODUCING SySTEMS . . . . . . . . . . . • . . . . . . . . • • . .
ROOM NOISE AND THE REPRODUCTION OF SOUND . . . . . . . • . . •
COMBINATION TONES AND NONLINEAR TRANSDUCERS . . . . . .
FREQUENCY RANGES OF S:>UND REPRODUCIN"G SYSTEMS . . . . .
FREQUENCY
RANGE
PREFERENCE
FOR
REPRODUCED
FREQUENCY
RANGE
PREFERENCE
FOR
LIVE
SPEECH
COMPARISON
OF
LIVE
AND
REPRODUCED
SYMPHONY
601
603
OR­
CHESTRA . . . • . . . . . . . . . . . . . . . • . • . . . . • . . • . . . . . • . • • • . • . . • •.
XIII.
600
FREQUENCY RANGE PREFERENCE FOR STEREOPHONIC REPRO­
DUCED SPEECH AND MUSIC............... . . . . . . . . . . . . . . .
12.32
595
598
AND
MUSIC • • • • . . . . • . • . . . • . • . • • . • • . . . . . . . . • . . . . • . • • . . . . . • • •
12.31
587
591
592
594
SPEECH
AND MUSIC . . . . • . . . . . . • . . . • • . • . . . . • . . . . . . . . • . . . • • . . . . . .
12.30
587
EFFECT OF NONLINEAR DISTORTION UPON THE QUALITY OF
REPRODUCED SPEECH AND MUSIC . . . . • . . . . . . • • • . . . . • . . . . . .
12.28
12.29
580
THE
606
12.33
FUNDAMENTAL
SICAL INSTRUMENTS •.. • . • • . • • . . . . . . • . . . . . • . • • . . . . . . . • ••
610
12.34
12.35
12.36
12.37
MUSICAL SCALE . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . • . . . . .
611
612
613
619
FREQUENCY
RANGES
OF
VOICES
AND
Mu-
ELECTRICAL MUSICAL INSTRUMENTS . • . . . . . . . . . . . . . . . . . . . •
ELECTRONIC MUSIC SyNTHESIZER . . . . . . . •. . . • . . . . . . . . . . . .
PHONETIC TyPEWRITER . . . . . . . . . . . . . . . . . • . . . . • • . . • . . . . . .
COMPLETE SOUND REPRODUCING SYSTEMS
13.1
13.2
13.3
INTRODUCTION
.....•........•.....•............••...••.
DISTORTION AND NOISE CONSIDERATIONS . . . • . . . . . . . . . . . . . .
SOUND SYSTEMS
.....................................•.
A. Monaural System .................................
B. Binaural System ..................................
C. Auditory Perspective System.......................
13.4
13.5
TELEPHONE . . • • . • • . • . . . . • . . • . • . . . • • . . • . . . . • . . • . • . • . . • . .
MAGNETIC TAPE SOUND REPRODUCING SYSTEM . . . . . . . . . . . .
A. Monaural Magnetic Tape Sound Reproducing System
625
626
628
629
631
633
633
634
634
XVlll
CONTENTS
CHAPTER
13.6
13.7
13.8
13.9
13.10
13.11
13.12
PAGE
B. Stereophonic Magnetic Tape Sound Reproducing Sys­
tern ..... . .................. . ....... . .... . ........ 636
C. Binaural Magnetic Tape Sound Reproducing System.. 637
DISK PHONOGRAPH REPRODUCING SYSTEM • . . . . . . . . • • . . . • 638
A. Monaural Disk Phonograph Reproducing System .... 638
B. Stereophonic Disk Phonograph Reproducing System .. 642
SOUND MOTION PICTURE REPRODUCING SYSTEM . . . . . . . . . . . 643
A. Single-Channel Sound Motion Picture Reproducing
System ...................................... . ... 643
B. Multiple-Channel Sound Motion Picture Reproducing
System (Stereophonic System) . ................... 648
RADIO SOUND REPRODUCING SYSTEM . . . . . . . . • • . . . . . . . . . . . 650
TELEVISION SOUND REPRODUCING SYSTEM . • . . . . . . . . . • . . • . 653
DICTATING MACHINES •. . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . 653
HEARING AIDS . • . . • . . • . . .. . . . • . . . . . . . . . . . . . . . . . . . • • . . . 653
SOUND REPRODUCING SYSTEM COMPONENTS . . . . . . • . • . . . . . 656
XIV. MEANS FOR THE COMMUNICATION OF INFORMATION
14.1
14.2
INTRODUCTION
VOICE
.•.....•...•..............•..• . ....•...••
•• • . • . . . • . • . . . . • . . . . . . . • • . . . . . . • . . . . . . . . . . • . . . • .
UNDEVELOPED SYSTEMS FOR THE TRANSMISSION
MATION
14.22
14.23
14.24
14.25
14.26
14.27
14.28
14.29
657
658
658
MANUAL SIGNALS . . • . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . • . . 658
FEEDBACK
. • . . . . . . . . . . . . . . . . • . . . . . • • . . . • . . • . • . • . . . . • . • 658
SOUND GENERATOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . • 658
SEMAPHORE . . . . . . . • . . . . . . . • . . • . . . . . . . . . . • . . . . . . . . • . . . . 660
ORTHOGRAPHY . . . . . . . . . . . . . • . . • . . . . . . . . . • . . . . . . . . . . . . . . . 660
TYPOGRAPHY • • . . . • . . . • • . . . • •• . • • . . . . . . . . . • . . . . . . . . . . . • 660
PHOTOGRAPHY • . . . . . • • • . . . . . . . . . . . • . . . . . • . . . . . . . . . . . . . . 660
TELEGRAPH
. . . . . . . . . . . . . • . . . . . . . . • . . . . . . . . . . . . . . . . . • . . 660
TELETYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. •... 660
MOTION PICTURE . . • . . . • . . . • . . . • • . . . • . . . . . . . . . . . . . . . . . . 661
TELEPHONE • . . • • • • . . . . . . . • . . . . . • . • • • . . . . . . . . . . . . . . . ... 661
PHONOGRAPH
. • • . . . . • . . . . . . . . . . . . . • . . . . . . . . . . . . . • . . . . . 661
RADIO . . . . • . • . . . . . . . . • . . . . . . . . . . . • . . . . . . . . .. . . . . . . . . . . 662
SOUND SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . •... • . 662
FACSIMILE ..• • • • . . • . • . • . . . • . . . . . . ... • . . . . . . • . . . . . • . . . . 662
SOUND MOTION PICTURE . • . . . . . . . . . . . . . . . . . . . . . . • . . . . . . 663
TELEVISION • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . • 663
INFORMATION
14.3
14.4
14.5
14.6
14.7
14.8
14.9
14.10
14.11
14.12
14.13
14.14
14.15
14.16
14.17
14.18
14.19
14.20
14.21
.....••....•.•.................••.•.....
EXISTING MEANS AND SYSTEMS FOR THE TRANSMISSION OF
OF INFOR­
. . . . . . . . . . • . . . . •••.••.•. . •....••....••.•••••••
VISIBLE SPEECH
..................................•.•..
SPEECH SYNTHESIZER . . . . . . . . . . . . . . . . . . . • . . . • . . . . . . . . • .
PHONETIC TYPEWRITER
PRINT READER
.................•........•.....
.••........•.............•..•...........
LANGUAGE TRANSLATOR
. . . . . . . . . . . . . . . . . . . . . . . . . . • . • . ..
MUSIC SYNTHESIZER • . . . . . . . . . . . . . . . • . . . • .. . • . . . . . . . • • •
CONTROL OF MACHINES BY SPEECH . . . . . . . . . . . . . . . . • . . . . .
MACHINE AND OBJECT SENSOR •. • . . . . . . . . • •...•.•.•••...
663
664
664
664
665
665
665
666
666
CONTENTS
XIX
CHAPTER
PAGE
14.30
LIST OF UNDEVELOPED SYSTEMS FOR THE TRANSMISSION OF
14.31
CONCLUSION
666
668
INFORMATION
xv.
UNDERWATER SOUND
15.1
15.2
15.3
15.4
15.5
INTRODUCTION
DIRECT RADIATOR DYNAMIC PROJECTOR • . . . . . • . . . . . . . . • . .
CONDENSER HYDROPHONE . . . . . . • . • . . . . . • . • • . . • ..••.•.•.•
XVI.
669
669
672
673
HIGH-FREQUENCY DIRECT RADIATOR PROJECTOR AND HYDRO­
PHONE
15.6
15.7
15.8
15.9
15.10
15.11
15.12
15.13
15.14
15.15
15.16
15.17
15.18
•••••...•.•.•• • . . . . . . . . . . . . • • . . . ...••.••.
SOUND WAVES IN WATER ••.•.. • .•.....••.•....•••• . •.•
. . . . . . . . • • . . . . . . . . . . . . . . . . . . . . . . • . . • . . .. ..••• • .
MAGNETIC PROJECTOR . . . . . . . . . . . . . . . . . . . . . . . . . . • . . • . . • .
MAGNETOSTRICTION PROJECTOR . . . . . • . . . . . • . . . . . • • . . • • • • •
MAGNETOSTRICTION
HYDROPHONE
....................•..
QUARTZ CRYSTAL PROJECTOR . . . . . . . . . . . . . . . .. • . • . . . . . . . •
QUARTZ CRYSTAL HYDROPHONE .•....•••..•••....•• • .•.••
QUARTZ CRYSTAL SANDWICH PROJECTOR AND HYDROPHONE
ROCHELLE SALT CRYSTAL PROJECTOR AND HYDROPHONE ...•
BARIUM TITANATE HYDROPHONE . . . • . . . . . . . . . . . . . . . .••••
PASSIVE SONAR . . . . • • . . . . . . . . • • . . . . . . . . . . . . . . . . . . . . . . • .
ECHO DEPTH SOUNDING SONAR . . . . . • • . . . . . . . • . . . . • . . • . .
ECHO DIRECTION AND RANGING SONAR . . . . . . . . . • . . • • • •• .
SCANNING ECHO DIRECTION AND RANGING SONAR . . . . . . • . .
COMMUNICATION
SONAR
. . . . . . . . . . . . . . . . . . . • . .. ... • ••.•
674
675
677
680
682
684
685
686
686
687
687
688
689
691
ULTRASONICS
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
16.9
16.10
16.11
16.12
16.13
16.14
16.15
16.16
INDEX
INTRODUCTION
...•... . ..•........•... . ........•...•••..
ULTRASONIC GENERATORS . • • . . . . . . . • . . . . . . . • . • ...•• • ••••
CAVITATION
• . . . • • . . . . . . • . . . • . . . . • . . . . . • • . ..•.. • . . . . . . .
DISPERSION DUE TO ULTRASONICS . . . . . . • . . . . . . . . • . • . . • . .
EMULSIFICATION DUE TO ULTRASONICS .•....• • .•••. • •..••
COAGULATION DUE TO ULTRASONICS . . . . . . . • • . . • .• • .•..•..
CHEMICAL EFFECTS OF ULTRASONICS •..•• . . . . . • . . . . . . . . •
BroLOGICALEFFECTS OF ULTRASONICS • . • . . . . • . . • . . • ...•.•
MEDICAL ApPLICATIONS OF ULTRASONICS . . . . . . . . . • . • • ••••
THERMAL EFFECTS OF ULTRASONICS . . • • . . . . . . . . . . . . . . . • • •
ULTRASONICS AS A DETERGENT •.•.••' . . . . . • . . . . . . . . . . • • • .
ULTRASONIC CLEANING AND DEGREASING . . . . . . . . . . . . . . . •
ULTRASONIC DRILLING
ULTRASONIC SOLDERING
....•.•...•.•.••• • . . . . • . • . . • . • •••
.•••• . •...•....•••••...•.••..••.
TESTING OF MATERIALS BY MEANS OF ULTRASONICS
•. •• .•
ULTRASONIC DELAY LINES AND FILTERS .. . ..•.. .• . . . . . . . •
692
692
697
697
698
698
699
699
700
700
701
701
701
702
702
703
705
r'"
I
I
1
SOUND WAVES
1.1. Introduction.-The term acoustics in its broadest sense is a term
used to designate an art and a science involving sound in all its manifold
fonns and manifestations. Specifically, acoustics includes the generation,
transmission, reception, absorption, conversion, detection, reproduction,
and control of sound.
An important division of acoustics is the reproduction of sound which
is the process of picking up sound at one point, and reproducing it at the
same point, or at some other point either at the same time, or at some
subsequent time. The most common sound reproducing systems are the
telephone, phonograph, radio, sound motion picture, and television.
The radio, phonograph, sound motion picture, and television have made
it possible for all the people of the world to hear famous statesmen, artists,
actors, and musical aggregations where only a relatively small number had
been able to hear them first hand. It is evident that the reproduction of
sound has produced in a relatively short time a great change in the education
and entertainment of this and other countries. The impact of the telephone,
phonograph, radio broadcasting, sound motion pictures, and television
upon the dissemination of information, art, and culture has been tremendous.
The reproduction of sound in these fields has been as important to the
advancement of knowledge as the printing press and the printed page.
The ultimate useful destination of all informative sound, direct or repro­
duced, is the human ear. In this connection, great strides have been made
in obtaining knowledge on the characteristics and action of the human hearing
mechanism.
Measurements play an essential part in the advancement of any scientific
field. Instruments have been developed and standards have been established
for the measurement of the fundamental quantities in acoustics.
The applications of acoustics in the field of music have led to a better
understanding of the stuff of which music is made. This knowledge has been
applied to the development of new musical instruments employing the latest
electronic and acoustic principles.
Accelerated by the requirements in World War II, tremendous advances
were made in underwater sound. The developments in underwater sound
have resulted in systems for detection and accurate location of underwater
1
2
ACOUSTICAL ENGINEERING
craft and obstacles over great distances, in depth sounders, and in other
acoustic applications in undersea communication.
The industrial applications of ultrasonics have unfolded a new field in
acoustics. Some of the important ultrasonic developments include the
cleaning of machined parts, drilling, and flaw detection.
The science of architectural acoustics has advanced to the point where
auditoriums, studios, and rooms can be designed to obtain excellent acoustics
under severe artistic conditions.
With ever-increasing industrial expansion comes an increase in noise.
Work is now under way actively to control noise by the use of a variety of
acoustic countermeasures.
The preceding brief introduction to the present status of acoustics shows
that it plays a very important part in our modern civilization. Further­
more, the fundamentals and applications of the science of acoustics are so
well formulated and substantiated that a large area of the field of acoustics
has attained an engineering status.
In this book the author has attempted to outline the essentials of modern
acoustics from the standpoint of the engineer or applied scientist. It has
been the aim and purpose to make the book as complete as possible by
covering all the major aspects of modern acoustics as outlined in the
preceding text of the introduction. In order to cover a wide range of
readers, the book has been written and illustrated so that the derivations
may be taken for granted. The concepts of mechanical and acoustical
impedance have been introduced and applied so that anyone who is familiar
with electrical circuits will be able to analyze the action of vibrating systems.
1.2. Sound Waves.-Sound is an alteration in pressure, particle dis­
placement or particle velocity propagated in an elastic material or the
superposition of such propagated alterations.
Sound is also the sensation produced through the ear by the alterations
described above.
Sound is produced when air is set into vibration by any means whatso­
ever, but sound is usually produced by some vibrating object which is in
contact with the air. If a string, such as one used in a banjo or similar in­
strument, is stretched between two solid supports and plucked, sound is
produced which dies down in a fairly short time. When the string is
plucked it tends to spring back into its rest position, but due to its weight
(mass) and speed (velocity) it goes beyond its normal position of rest.
Then, in returning it again goes beyond its normal position of rest. The
excursions become smaller and smaller and finally the string comes to rest.
As the string moves forward it pushes air before it and compresses it, while
air rushes in to fill the space left behind the moving string. In this way
air is set in motion. Since air is an elastic medium, the disturbed portion
transmits its motion to the surrounding air so that the disturbance is propa­
gated in all directions from the source of disturbance.
If the string is connected in some way to a diaphragm such as a stretched
drumhead of a banjo, the motion is transmitted to the drum. The drum,
r
i
~
SOUND WAVES
3
___________________________________________________________
having a large area exposed to the air, sets a greater volume of air in motion
and a much louder sound is produced.
If a light piston several inches in diameter, surrounded by a suitable
baffle board several feet across, is set in rapid oscillating motion (vibration)
by some external means, sound is produced (Fig. 1.1). The air in front of
the piston is compressed when it is driven forward, and the surrounding air
expands to fill up the space left by the retreating piston when it is drawn
back. Thus we have a series of compressions and rarefactions (ex­
pansions) of the air as the piston is driven back and forth. Due to the
elasticity of air these areas of compression and rarefaction do not remain
stationary but move outward in all directions. If a pressure gage were set
PISTON
BAFFLE BOARD
FIG. 1.1.
Production of sound waves by a vibrating piston.
up at a fixed point and the variation in pressure noted, it would be found
that the pressure varies in regular intervals and in equal amounts above
and below the average atmospheric pressure. Of course, the actual varia­
tions could not be seen because of the high rate at which they occur. Now,
suppose that the instantaneous pressure, along a line in the direction of
sound propagation, is measured and plotted with the ordinates representing
the pressure; the result would be a wavy line as shown in Fig. 1.1. The
points above the straight line represent positive pressures (compressions,
condensations); the points below represent negative pressures (expansions,
rarefactions) with respect to the normal atmospheric pressure represented
by the straight line.
From the above examples a few of the properties of sound waves and
vibrations in general may be defined.
Periodic Quantity.-A periodic quantity is an oscillating quantity the
values of which recur for equal increments of the independent variable.
Cycle.- One complete set of recurrent values of a periodic quantity
comprises a cycle; or, in other words, anyone set of variations starting
at one condition and returning once to the same condition is a cycle.
Period.-The period is the time required for one cycle of a periodic
quantity.
Frequency.-The number of cycles occurring per unit of time, or which
would occur per unit of time if all subsequent cycles were identical with
4
ACOUSTICAL ENGINEERING
the cycle under consideration, is the frequency. The unit is the cycle per
second.
Fundamental Frequency.-A fundamental frequency is the lowest com­
ponent frequency of a periodic wave or quantity.
Harmonic.-A harmonic is a component of a periodic wave or quantity
having a frequency which is an integral multiple of the fundamental fre­
quency. For example, a component, the frequency of which is twice the
fundamental frequency, is called the second harmonic.
Subharmonic.-A subharmonic is a component of a complex wave having
a frequency which is an integral submultiple of the basic frequency.
Wavelength.-The wavelength of a periodic wave in an isotropic medium
is the perpendicular distance between two wave fronts in which the dis­
placements have a phase difference of one complete cycle.
Octave.-An octave is the interval between two frequencies having a
ratio of two to one.
Transducer.-A transducer is a device by means of which energy may
flow from one or more transmission systems to one or more other trans­
mission systems. The energy transmitted by these systems may be of
any form (for example, it may be electrical, mechanical, or acoustical) and
it may be the same form or different forms in the various input and out­
put systems.
The example of Fig. 1.1 has shown graphically some of the properties
of wave motion. It is the purpose of the next section to derive the funda­
mental wave equation. It is not necessary that the reader digest all
the assumptions and processes involved in order to obtain valuable infor­
mation concerning the properties of a sound wave.
1.3. Acoustical Wave Equation.-The general case of sound propaga­
tion involves three dimensions. The general relation for sound propaga­
tion of small amplitudes in three dimensions will be derived and then these
relations will be applied to special problems.
A. Equation of Continuity.-The fundamental equation of hydrokinetics
is the equation of continuity. This equation is merely a mathematical
statement of an otherwise obvious fact that matter is neither created nor
destroyed in the interior of the medium. That is, the amount of matter
which enters the boundaries of a.small volume equals the increase of matter
inside. Consider the influx and efflux through each pair of faces of the
cube of dimensions ~x, ~y, and ~z, the difference between the latter and
the former for the whole cube is
_ [O( pI 1'{)
ax
+ o(p'v) + O(p'W)] ~x ~y ~z
oy
oz
where x, y, z = coordinates of a particle in the medium,
u, v, w = component velocities of a particle in the medium, and
p' = density of the medium.
1.1
5
SOUND WAVES
o:e' Dox Doy Doz in
The rate of growth of mass
the expression 1.1.
the cube must be equal to
This may be written as
op'
ot
+ o(p'u) + o(p'v) + o(p'w) =
ox
oy
0
1.2
oz
where t = time.
This is the equation of continuity which signifies the conservation of
matter and the three dimensionality of space.
B. Equation of Motion.-Referring again to the space Dox Doy Doz the
acceleration of momentum parallel to x is p' Dox Doy Doz
~~.
The mean pres­
sures on the faces perpendicular to x are
where Po'
=
DoX) Doy Doz and
( po' - opo'
ox 2
pressure in the medium.
The difference is a force
( Po'
+ opo'
DoX)DoY Doz
ox 2
°fx° Dox Doy Doz in the direction of increasing x.
Equating this to the acceleration of momentum, the result is the equation
of motion,
, ou
opo', OV
opo', ow
opo'
p at = - ox 'p ot = - oy' p dt = - Tz
1.3
The equation of motion may be written
dV
( Iuvw
t
+p
1 Grad P'
0 =
0
1.4
C. Compressibility of a Gas.-The next property of a gas which is used
to derive the wave equation depends upon the thermodynamic properties
of gases. The expansions and contractions in a sound wave are too rapid
for the temperature of the gas to remain constant. The changes in pres­
sure and density are so rapid that practically no heat energy has time to
flow away from the compressed part of the gas before this part is no longer
compressed. When the gas temperature changes, but its heat energy
does not, the compression is termed adiabatic.
In the case of an adiabatic process,
;;' =
where
Po =
p
Po'
=
=
p' =
y =
(~r
1.5
static pressure. The static pressure is the pressure that
would exist in the medium with no sound waves present.
The unit is the dyne per square centimeter.
static or original density,
total pressure (static
excess),
instantaneous density (static
change), and
ratio of specific heat at constant pressure to that' at con­
stant volume and has a value of 1.4 for air.
+
+
6
ACOUSTICAL ENGINEERING
D. Condensation.-A new term will now be introduced. Condensation
is defined as the ratio of the increment of density change to the original
density,
p' - p
s=---
1.6
p
Combining equations 1.5 and 1.6
(i)Y
or
Po' =
= (1 + s)y =
Po
p
Po' = Po + POys
1
+ ys
1.7
1.8
The excess pressure, or instantaneous sound pressure p, is Po'
- Po.
1.9
P = POys
The instantaneous sound pressure at a point is the total instantaneous
pressure at that point minus the static pressure. The unit is the dyne per
square centimeter. This is often called excess pressure.
The effective sound pressure at a point is the root-mean-square value of
the instantaneous sound pressure over a complete cycle, at that point.
The unit is the dyne per square centimeter. The term "effective sound
pressure" is frequently shortened to "sound pressure."
The maximum sound pressure for any given cycle is the maximum
absolute value of the instantaneous sound pressure during that cycle.
The unit is the dyne per square centimeter. In the case of a sinusoidal
sound wave this maximum sound pressure is also called the pressure am­
plitude.
The peak sound pressure for any specified time interval is the maxi­
mum absolute value of the instantaneous sound pressure in that interval.
The unit is the dyne per square centimeter.
A dyne per square centimeter is the unit of sound pressure.
E. D'Alembertian Wave Equation.-The three equations 1.2, 1.4, and
1.5 characterize disturbances of any amplitude. The first two are non­
linear save for small amplitudes. In general, acoustic waves are of in­
finitesimal amplitudes, the alternating pressure is small compared with
the atmospheric pressure and the wavelength is so long that u, v, W, and s
change very little with x, y, and z. Substituting equation 1.6 in 1.2 and
neglecting high order terms,
~ + : + :; + ~~ =
0
1.10
The type of motion to be considered is irrotational, that is Curl V uvw =
That is a necessary and sufficient condition for the existence of a
scalar velocity potential 7> which is defined as
o.
07>
u = -,
ox
07>
v = -,
oy
W
or
V uvw
=
Grad 7>
= -07>
oz
1.11
7
SOUND WAVES
Substitute equations 1.11 in 1.3 and multiply by dx, dy, and dz
:tde/>
=
1.12
;,dPo"
-
or integrating
8e/>
=
_
8t
JdPo'
p'
Since the density changes very little, the mean density, p, may be used.
The fdpo' is the excess pressure; then
1.13
where P = excess pressure.
From equations 1.9, 1.10, 1.11, and 1.13
f) 2e/>
YPo(f)2e/>
82e/>
8t2 - P 8x2 + 8y2
+
82e/»_
1.14
8z2 - 0
or this may be written
f) 2e/>
8t 2
=
c2 V2e/>
which is the standard D'Alembertian wave equation for e/>.
of propagation is
yPO
p
=
The velocity
1.15
c2
For the velocity of sound in various mediums see Table 1.1.
TABLE 1.1. YOUNG'S MODULUS Q, IN DYNES PER SQUARE CENTIMETER, POISSON'S RATIO (T,
DENSITY p, IN GRAMS PER CUBIC CENTIMETER, VELOCITY OF SOUND C, IN METERS PER
SECOND, AND THE SPECIFIC ACOUSTICAL RESISTANCE pC, IN GRAMS PER SECOND PER
SQUARE CENTIMETER
METALS
Substance
Aluminum
Antimony
Beryllium
Bismuth
Cadmium
Cobalt
Copper
Gold
Iridium
Iron Cast
Iron Wrought
Lead.
Magnesium.
Mercury
Nickel
Q
a
7.3 X 1011
7.8 x 1011
26.0 X 1011
3.19xl0ll
5.3 X 1011
19.0 X 1011
11.0 X lOll
8.0 X lOll
52.0 X 1011
9.0 X lOll
20.0 X 1011
1. 7 X 1011
4.0 X lOll
.33
.33
.33
.35
.30
.30
.35
.35
.33
.29
.28
.43
.33
21.0
.31
...
X
lOll
...
0
2.7
6.6
1.8
9.7
8.6
8.7
8.9
19.3
22.4
7.8
7.9
11.3
1.7
13.5
8.8
C
pC
5200
3400
12000
1800
2500
4700
3500
2000
4700
3400
5100
1200
4800
1400
4900
140 X 10 4
220 x 104
216 X 104
170 X 10 4
215 X 10 4
410 X 104
310 X 104
390 X 104
1050 X 104
270 X 104
400 X 10 4
130 X 10 4
82 X 104
190 X 10 4
430 X 10 4
8
ACOUSTICAL ENGINEERING
METALS
Substance
I
Q
Palladium
Platinum
Rhodium
Silver
Tantalum
Tin
Tungsten
Zinc
12.0
17.0
30.0
7.8
19.0
4.5
35.0
8.2
X
X
X
X
X
X
X
X
(continued)
1011
1011
1011
1011
1011
1011
1011
1011
u
.39
.33
.34
.37
.31
.33
.17
.17
p
c
pC
12.0
21.4
12.4
10.5
16.6
7.3
19.0
7.1
3200
2800
4900
2700
3400
2500
4300
3400
380
600
610
280
560
180
830
240
X
X
X
X
X
X
X
X
104
10 4
104
10 4
10 4
10 4
104
10 4
7.0
8.2
8.4
8.8
2.8
8.1
8.8
7.7
7.7
4900
3900
3400
3700
5000
3800
4500
5000
5100
340
320
290
330
140
310
400
390
390
X
X
X
X
X
X
X
X
X
104
10 4
104
104
104
104
10 4
104
104
1.8
2.2
2.6
2.4
2.4
2.7
2.4
2.6
2.6
2.4
2.7
2.7
2.7
2.9
.92
3700
3400
3100
6000
5000
6000
4600
3300
3800
4200
4400
6200
5400
4500
3200
67
75
81
144
120
162
110
86
99
102
118
168
146
131
29
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
10 4
104
104
104
104
10 4
104
104
104
104
104
104
104
104
104
4500
3900
500
4300
4700
4000
4300
4100
3600
4600
4300
4600
29
25
1.2
23
24
27
29
29
16
21
23
26
X
X
X
X
X
X
X
X
X
X
X
X
104
104
104
10 4
104
104
104
104
104
104
104
104
ALLOYS
Alnico
Beryllium Copper.
Brass.
Bronze Phosphor .
Duraluminum
German Silver
Monel
Steel C.08
Steel C.38
17.0
12.5
9.5
12.0
7.0
11.6
18.0
19.0
20.0
X
X
X
X
X
X
X
X
X
1011
1011
1011
1011
1011
1011
1011
1011
1011
.32
.33
.33
.35
.33
.37
.32
.27
.29
CERAMICS, ROCKS
Brick.
Clay Rock
Concrete
Glass, Hard
Glass, Soft
Granite
Isolantite
Limestone
Marble
Porcelain
Quartz, Fused
Quartz, 11 Optic
Quartz, 1 Optic
Slate
Ice
2.5
2.5
2.5
8.7
6.0
9.8
5.0
2.9
3.8
4.2
5.2
10.3
7.95
5.8
.94
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
1011
. ..
...
...
...
...
...
...
...
...
...
...
...
...
...
...
WOODS (WITH THE GRAIN)
Ash
Beech
Cork
Elm
Fir
Mahogany
Maple
Oak, White
Pine, White
Poplar
Sycamore
Walnut
Across the grain,
1.3 X 1011
1.0 X 1011
.0062 X 1011
1.0xl0l l
1.1 X 1011
1.1 X 1011
1.3 X 1011
1.2 X 1011
.6 X 1011
1.0 X 1011
1.0 X 1011
1.2 X 1011
...
...
...
...
...
...
...
...
...
...
...
...
t to t of the above values for c.
.64
.65
.25
.54
.51
.67
.68
.72
.45
.46
.54
.56
9
SOUND WAVES
PLASTICS
Substance
u
p
c
...
.
. ..
...
...
. ..
...
...
...
...
...
...
...
...
...
...
...
...
...
1.3
1000
13
1.3
1300
17 x 104
1.2
3700
44
X
10 4
1.5
1.1
1.8
3700
1400
2200
55
15
40
X
X
X
10 4
10 4
104
1.2
1700
20
X
104
1.2
1.0
.9
1500
2200
1300
18
22
12
X
X
X
104
10 4
104
1.35
2500
34 x 10 4
1.3
2300
30
X
104
1.35
2500
34
X
104
1.8
1.1
.95
.9
1.7
1.1
2400
1400
70
470
1500
1700
43
15
.67
4.2
26
19
X
X
X
X
X
X
10 4
104
104
10 4
10 4
104
.81
.90
1.5
.74
.68
.87
1.0
1.03
1240
1170
983
1020
1390
1330
1441
1540
10.0 X
10.5 X
14.7 X
7.6 X
9.4 X
11.6 X
14.4 X
15.5 x
.00129
.00120
.00125
.00198
.00317
.00009
.00072
.00125
.00143
.00058
331
344
337
258
205
1270
432
336
317
405
Q
Cellulose Acetate,
Sheet
Cellulose Acetate,
Molded
Cellulose Acetate,
Butyrate.
Cellulose Acetate, Pyroxylin .
Ethyl Cellulose
Ivory.
.
.
Methyl Metha-Crylate
Resin, Cast
Methyl Metha-Crylate
Resin, Molded .
Paper, Parchment
Paraffin, 16 ° C.
Phenol-Formaldehyde
Wood Filler
Phenol-Formaldehyde
Paper Base
Phenol-Formaldehyde
Fabric Base
Phenol-Formaldehyde
Mineral Filler
Rubber, Hard
Rubber, Soft
Sheepskin
Shellac Compound
Styrene Resin
.
1.4
X
1010
2.1
X
1010
17.0
X
1010
21.0
2.1
9.0
X
X
X
1010
1010
1010
3.5
X
1010
2.8
4.8
1.5
X
X
X
1010
1010
1010
8.4
X
1010
7.0
X
1010
8.4
X
1010
10.5
2.3
.5
2.0
3.8
3.1
X
X
X
X
X
X
1010
1010
108
109
1010
1010
"
pC
X
104
LIQUIDS
Alcohol, Methyl
Benzine
Chloroform .
Ether
Gasoline
Turpentine .
Water, 13° C.
Water, Salt
......... .
..........
..........
......... .
..........
......... .
..........
......... .
...
. ..
. ..
...
. ..
...
. ..
...
GASES
Air, 0° C.
Air, 20° C.
Carbon Monoxide
Carbon Dioxide
Chlorine
Hydrogen
Methane
Nitrogen
Oxygen
Steam.
..........
0
•••••••••
..........
......... .
..........
......... .
..........
......... .
..........
......... .
. ..
. ..
. ..
...
. ..
...
. ..
...
. ..
...
42.7
41.4
42.0
51.2
65.0
11.4
31.0
42.0
45.5
23.5
104
104
104
10 4
104
104
104
104
10
ACOUSTICAL ENGINEERING
1.4. Plane Sound Waves.-Assume that a progressive wave proceeds
along the axis of x. Then cp is a function of x and t only and the wave
equation 1.14 reduces to
8 2cp
8t 2
=
8 2cp
c2 8x 2
1.16
A solution of this equation for a simple harmonic wave traveling in the
positive x direction is
cp = A cos k(ct - x)
1.17
where A = amplitude of cp,
k = 27T/A,
A ==; wavelength, in centimeters,
c = fA = velocity of sound, in centimeters per second, and
f = frequency, in cycles per second.
A. Particle Velocity in a Plane Sound Wave.-The particle velocity, u,
employing equations 1.11 and 1.17 is
u
=
~~ =
kA sin k(ct - x)
1.18
The particle velocity in a sound wave is the instantaneous velocity of a
given infinitesimal part of the medium, with reference to the medium as a
whole, due to the passage of the sound wave.
B. Pressure in a Plane Sound Wave.-From equations 1.9, 1.13, and
1.15 the following relation may be obtained
~; =
-
c2s
1.19
The condensation in a plane wave from equations 1.19 and 1.17 is
given by
s
=
Ak sin k(ct - x)
c
1.20
From equations 1.9 and 1.15 the following relation may be obtained
p = c2 ps
1.21
Then, from equations 1.20 and 1.21 the pressure in a plane wave is
p
kcpA sin k(ct - x)
=
1.22
Note: the particle velocity, equation 1.18, and the pressure, equation
1.22, are in phase in a plane wave.
C. Particle Amplitude in a Plane Sound Wave.-The particle amplitude
of a sound wave is the maximum distance that the vibrating particles of the
medium are displaced from the position of equilibrium.
From equation 1.18 the particle velocity is
~
=
u
=
kA sin k(ct - x)
1.23
11
SOUND WAVES
e
where = amplitude of the particle from its equilibrium position, m
centimeters.
The particle amplitude, in centimeters, is
A
cos k(ct - x)
1.24
e+ - -c
From equations 1.20 and 1.24 the condensation is
s
=
8e
1.25
8x
-
1.5. Spherical Sound Waves.-Many acoustical problems are concerned
with spherical diverging waves. In spherical coordinates x = r sin 8 cos !f,
y = r sin 8 sin!f and z = r cos 8 where r is the distance from the center,
8 is the angle between r and the oz axis and !f is the angle between the
projection of r on the xy plane and ox. Then \1 28 becomes
\1 2if>
=
~:~ + ~ ~ + r2 s~n 8 :8
(sin 8)
~~ + r2 s:n2 8 :~
1.26
For spherical symmetry about the origin
82
\1 2if> = r8r 2 (rif»
1.27
The general wave equation then becomes,
82
82
8t 2 (rif» = c2 8r 2 (rif»
1.28
The wave equation for symmetrical spherical waves can be derived in
another way. Consider the flux across the inner and outer surfaces of
the spl:erical shell having radii of r - !1rJ2 and r
!1rJ2, the difference is
+
- 4-rr 8r
~ (plr2?!)
8t !1r
1.29
The velocity is
1.30
where if> = velocity potential.
The expression 1.29 employing equation 1.30 becomes
_ 47T ~(plr2 8if» !1r
8r
8r
The rate of growth of mass in the shell is
8
I
1.31
4-rrr2 _l!.. !1r
1.32
8t
The difference in flux must be equal to the rate of growth of mass, expres­
sions 1.31 and 1.32,
2 8 p' + ~(
28if» = 0
1.33
r 8t
8r p r 8r
I
12
ACOUSTICAL ENGINEERING
Using equations 1.6, 1.9, and 1.13, equation 1.33 may be written,
r2 82r/> _ c2 ~ (r2 Or/»
ot 2
or
or
0
=
1 34
.
Equation 1.34 may be written
o2(rr/» _ 2 o2(rr/» _ 0
ot2
C Or 2 ­
1.35
which is the same as equation 1.28. The solution of equation 1.35 for
diverging waves is
r/> = A- €1k(ct-r)
1.36
r
From equations 1.19 and 1.36 the condensation is given by
s
--
90
80
_.!. or/> = _ jkA €1k (ct-r)
=
c2
ot
"-or--......
70
<J)
!oJ
t.J
cr 60
"o
t.J
z 50
'"
f\
\
\
\
z
«
t.J
I
\
t.J
i3 40
30
<J)
«
I
Q.
"­
20
"
I'-... ......
10
o
.01
1.37
cr
.02
.04
.08.1
.8
.2
I
2
4
r-­
8 10
20
40
80100
kr
FIG. 1.2. Phase angle between the pressure and particle velocity in a spherical sound
wave in t erms of kr. where k = 21T/11, 11 = wavelength and r = distance from the source.
A. Pressure m a Spherical Sound Wave .- The pressure from equation
1.21 is
P=
c2 ps
1.38
The pressure then from equations 1.37 and 1.38 is
P=
_jkcAp
r
€1k(ct-r)
1.39
-13
SOUND WAVES
Retaining the real part of equation 1.39 the pressure is
p
p kcA sin k(ct - r)
=
1.40
r
B. Particle Velocity in a Spherical Sound Wave.-The particle velocity
from equations 1.11 and 1.36 is
u
=
G+ jk)~
-
1.41
€lk(ct-r)
Retaining the real part of equation 1.41 the particle velocity is
Ak [~ cos k(ct - r) - sin k(ct - r)]
1.42
r kr
C. Phase Angle between the Pressure and the Particle Velocity in a Spheri­
cal Sound Wave.-The particle velocity given by equation 1.42 may be
written
u
=
-
= ~J~
+ k2 sin
r r
u
1.43
[k(ct - r) - 8]
where tan 8 = l/kr.
o
100
90
80 '\
70 '\
\.
60
50
40
38
36
\
34
1\
32
\\
30
3o
\
20
U
Q.J
x
0
28
26
\ 1\
24
22
2o
Q 19
~
IX
'\
8
I8
\.
7
\.
6
I
5
\
4
3
I4
I2
\
Io
1\
2
6
'\
5
4
i"...
2
1••001
8
.02
.04
.08.1
.2
.4
kr
.81
2
r--­
2
4
8
0
10
FIG. 1.3. Ratio of the absolute magnitude of the particle velocity to the
pressure in a spherical sound wave in terms of kr, where k = 2,,/)..,
).. = wavelength and r = distance from the source.
14
ACOUSTICAL ENGINEERING
Comparing equation 1.43 with equation 1.40 for the pressure it will be
seen that the phase angle between the pressure and velocity in a spherical
wave is given by
8 = tan-1 ~
kr
1.44
For very large values of kr, that is, plane waves, the pressure and particle
velocity are in phase. The phase angle as a function of kr is depicted in
Fig. 1.2.
D. Ratio of the Absolute Magnitudes of the Particle Velocity and the Pres­
sure in a Spherical Sound Wave.-From equations 1.40 and 1.43 the ratio
of the absolute value of the particle velocity to the absolute value of the
pressure is given by
·
R a t 10
vI
+ k 2r2
1.45
= --;.,-­
pckr
The ratio in equation 1.45, as a function of kr, is depicted in Fig. 1.3.
1.6. Stationary Sound Waves.-Stationary waves are the wave system
resulting from the interference of waves of the same frequencies and are
characterized by the existence of nodes or partial nodes.
Consider two plane waves of equal amplitude traveling in opposite direc­
tions; the velocity potential may be expressed as
g, =
A [cos k(ct - x)
+ cos k(ct + x)]
1.46
The pressure in this wave system from equations 1.19 and 1.21 is
p = - p
a:; = kcpA [sin k(ct -
x)
+ sin k(ct + x)]
P = 2kcpA [sin kct cos kx]
1.47
1.48
The particle velocity in this wave system from equations 1.11 and
1.46 is
u
= ~: =
u
=
kA [sin k(ct - x) - sin k(ct
-2kA [cos kct sin kx]
u = 2kA [sin (kct -
~) cos (kX - ~)]
+ x)]
1.49
1.50
1.51
Equations 1.48 and 1.51 show that the maxima of the particle velocity
and pressure are separated by a quarter wavelength. The maxima of p
and u differ by 90° in time phase.
A stationary wave system is produced by the reflection of a plane wave
by an infinite wall normal to the direction of propagation. This is the
simplest type of standing wave system. Complex stationary wave sys­
tems are produced when a sound source operates in a room due to the re­
flections from the walls, ceiling, and floor.
SOUND WAVES
15
1.7. Sound Energy Density.-Sound energy density is the sound energy
per unit volume. The unit is the erg per cubic centimeter.
The sound energy density in a plane wave is
E
=
p2
pc 2
1.52
where p = sound pressure, in dynes per square centimeter,
p = density, in grams per cubic centimeter, and
c = velocity of sound, in centimeters per second.
The positive radiation pressure in dynes per square centimeter exerted
by sound waves upon an infinite wall is
p = (y + I)E
where E
=
y =
1.53
energy density of the incident wave train in ergs per cubic
centimeter, and
ratio of specific heats, 1.4 for air.
Instruments for measuring the radiation pressure have been built, con­
sisting of a light piston mounted in a large wall with means for measuring
the force on the piston. Since the radiation pressure is very small these
instruments must be quite delicate.
1.8. Sound Intensity.-The sound intensity of a sound field in a speci­
fied direction at a point is the sound energy transmitted per unit of time
in the specified direction through a unit area normal to this direction at
the point. The unit is the erg per second per square centimeter. It may
also be expressed in watts per square centimeter.
The intensity, in ergs per second per square centimeter, of a plane wave is
p2
I = -
pc
where
p=
=
u
c=
p=
=
pu
= pcu 2
1.54
pressure, in dynes per square centimeter,
particle velocity, in centimeters per second,
velocity of propagation, in centimeters per second, and
density of the medium, in grams per cubic centimeter.
The product pc is termed the specific acoustical resistance of the medium.
The specific acoustical resistance of various mediums is shown in Table 1.1.
1.9. Decibels (Bels).-In acoustics the ranges of intensities, pressures,
etc., are so large that it is convenient to use a scale of smaller numbers
termed decibels. The abbreviation db is used for the term decibel. The
bel is the fundamental division of a logarithmic scale for expressing the
ratio of two amounts of power, the number of bels denoting such a ratio
being the logarithm to the base ten of this ratio. The decibel is one tenth
16
ACOUSTICAL ENGINEERING
of a bel. For example, with PI and P 2 designating two amounts of power
and n the number of decibels denoting their ratio:
n
=
10 loglO ;~, decibels
1.55
When the conditions are such that ratios of currents or ratios of voltages
(or the analogous quantities such as pressures, volume currents, forces, and
particle velocities) are the square roots of the corresponding power ratios,
the number of decibels by which the corresponding powers differ is ex­
pressed by the following formulas:
n
=
n
=
~, decibels
1.56
20 log 10 ~, decibels
1.57
20 loglO
t2
e2
where il/i2 and el/e2 are the given current and voltage ratios, respectively.
For relation between decibels and power and current or voltage ratios,
see Table 1.2.
TABLE
1.2.
THE RELATION BETWEEN DECIBELS AND POWER AND CURRENT OR VOLTAGE
RATIOS
Power Ratio
Decibels
Current or
Voltage Ratio
Decibels
1
0
1
0
2
3.0
2
6.0
3
4.S
3
9.5
4
6.0
4
12.0
5
7.0
5
14.0
6
7.8
6
15.6
7
8.5
7
16 .9
8
9.0
8
lS.1
9
9.5
9
19.1
10
10
10
20
100
20
100
40
1000
30
1000
60
10,000
40
10,000
SO
100,000
SO
100,000
100
1,000,000
60
1,000,000
120
17
SOUND WAVES
1.10. Doppler Effect.1-The change in pitch of a sound due to the rela­
tive motion of the source and observer is termed the Doppler Effect.
When the source and observer are approaching each other the pitch ob­
served by the listener is higher than the actual frequency of the sound
source. If the source and observer are receding from each other the pitch
is lower.
The frequency at the observation point is
fo
where v
=
=
v - vo
--fs
Vs
1.58
v -
velocity of sound in the medium,
Vo = velocity of the observer,
Vs
fs
=
=
velocity of the source, and
frequency of the source.
All the velocities must be in the same units.
No account is taken of the effect of wind velocity or motion of the me­
dium in equation 1.58. In order to bring in the effect of the wind, the
velocity v in the medium must be replaced by v
w where w is the wind
velocity in the direction in which the sound is traveling. Making this
substitution in 1.58 the result is
+
_
V
JO -
v
f
+w +w _
vOl
Vs
8
1.59
Equation 1.59 shows that the wind does not produce any change in pitch
unless there is some relative motion of the sound source and the observer.
1.11. Refraction and Diffraction.-The change in direction of propa­
gation of sound, produced by a change in the nature of the medium which
affects the velocity, is termed refraction. Sound is refracted when the
density varies over the wave front (see equation 1.15). A sound wave
may be bent either downward or upward depending upon the relative
temperatures (densities) of the air,2 Fig. 1.4. The distance over which
sound may be heard is greater when the wave is bent downward than
when it is bent upward. The first condition usually obtains during the
early morning hours while. the latter condition prevails during the day.
Structures3 may be built which refract sound waves. Acoustic lenses
and prisms employing these structures may be used for various acoustical
applications, as for example, loudspeakers and microphones. See Secs.
1 Perrine, J. 0., Amer. Jour. Phys., Vol. 12, No. 1, p. 23,1944.
This paper describes
sixteen versions of the Doppler and Doppler Echo Effects. In addition to systems
given in the text above are systems involving moving and fixed reflectors.
2 For other phenomena of atmospheric acoustics such as the effects of wind and
temperature upon the propagation of sound waves and the applications to sound
ranging and signaling in air, see Stewart and Lindsay, .. Acoustics," D. Van Nostrand
Company, Princeton, N.J., 1930.
3 Koch and Harvey, Jour. Acous. Soc. Amer., Vol. 21, No.5, p. 471, 1949.
18
ACOUSTICAL ENGINEERING
SURf"ACE
Of" THE
EARTH
FIG. 1.4.
SURfACE
Of" THE
EARTH
The refraction of a sound wave in air.
oo
1--------~
0]0
SPHERES
PARALLEL PLATES
0 0
~
OJC 0
DISKS
C] (J
~
1--'~WAVE
--.,.
'­
~
~lo----l
SERPENTINE PLATES
~~
llJb II 1_~
~
STRIPS
SLANT PLATES
1.5. Obstacle and path length structures for
refracting sound waves.
FIG.
6.15 and 8.6B. Practical systems have been developed based upon obstacle
arrays and path length devices.
Obstacle arrays increase the effective density of the medium and thus
produce a reduced propagation velocity of sound waves passing through
the array. Three different obstacle arrays are shown in Fig. 1.5.
SOUND WAVES
19
The index of refraction n, of a spherical obstacle array as shown in
Fig. 1.5, is given by
1.60
n2 = 1
i7Ta3N
+
where a = radius of the sphere, and
N = number of spheres per unit volume.
The index of refraction n, of a disk obstacle array as shown in Fig. 1.5,
is given by
1.61
n 2 = 1 le 3N
+
where c = radius of the disk, and
N = number of disks per unit volume.
The index of refraction n, of a series of strips as shown in Fig. 1.5, is
given by
1.62
where b = half breadth of the strip normal to the direction of propagation
of the wave and
N = number of strips per unit area viewed endwise.
Path length devices increase the time of travel of the waves through the
path over that in free space. Three different path length devices are shown
in Fig. 1.5.
The index of refraction n, of parallel plates as shown in Fig. 1.5, is unity.
The index of refraction n, of the serpentine plates as shown in Fig. 1.5,
is given by
1
n=1.63
10
where 1 = path length through the plates, and
10= path in the absence of the plates.
The index of refraction n, of slant plates as shown in Fig. 1.5, is given by
1
1
cos ()
n=-=--
10
where ()
=
1.64
angle between the direction of propagation of the wave and
the plane of the plates.
In the examples in this book the serpentine system will be used.
An acoustic lens which converges the impinging sound wave is shown
in Fig. 1.6A. The sheet metal is arranged so that the path length through
the lens is the greatest at the center of the lens. The action of the lens is
depicted by the ray and wavefront diagram of Fig. 1.6A. The path lengths
of all the pencils of the incident sound wave are all the same at focus. An
acoustic lens which diverges the incident sound wave is shown in Fig. 1.6B.
The action of the lens is depicted by the ray diagram and wavefront diagram
of Fig. 1.6B.
20
ACOUSTICAL ENGINEERING
INCIDENT SOUND
~
~
SHEET _ _
METAL
---....--
_
i
~
~
~
:f-------~----------7FOCUS
~
~
----A
INCIDENT SOUND
~
SHEET~
METAL
.­
./"
FIG. 1.6.
ing lens.
Acoustic lenses.
A. Converging lens.
B. Diverg­
An acoustic prism is shown in Fig. 1.7. The acoustic prism changes the
direction of the impinging sound wave. The action of the prism is depicted
by the ray and wavefront diagrams of Fig. 1.7.
Diffraction is the change in direction of propagation of sound due to
the passage of sound around an obstacle. It is well known that sound
will travel around an obstacle. The larger the ratio of the wavelength
to the dimensions of the obstacle the greater the diffraction. The dif­
fraction around the head is important in both speaking and listening.
The diffraction of sound by microphones and loudspeakers is important
in the performance of these instruments. The diffraction4 of sound by a
sphere, a cube, and a cylinder as a function of the dimensions is shown in
Fig. 1.8. These data may be used to predict the diffraction of sound by
objects of these general shapes. As, for example, the sphere may be used
to predict the diffraction of sound by the human head.
4
Muller, Black, and Dunn, Jour. Acous. Soc. Amer., Vol. 10, No. 1, p. 6, 1938.
21
SOUND WAVES
~~~~r~
0~~
~
~
~
FIG. 1.7.
-
Acoustic prism.
I"
10~~~
\1} \1 \
~::or 3d'
CYLINDER
5
""'"""I"'" I
90.
60·
o
-51--+--+--'-....L...J-L..L.J .
-lOr-I---
rn
...-<
~
,
DIRECTION or
PLANE WAVE
NORMAL
-15
~
'T'
.Jl..0·
'\..
180"
/f\r--,­
H-t-tt--''''15~0•.!-""",*~-!.I'4It-1:\H
D/>­
10
,=0·
30
60
SPHERE
5
l's"d
o
-5
()
-10 f-- I--­
-IS
.02
.04
'JJ
.06
/::J.ill
120
DIRECTION or
PLANE WAVE
I"-"
'\
NORMAL
0.1
0.2
0.4
1500
I
0.6
1.0
2.0
4.0
6.0
D/>­
FIG. 1.8. The diffraction of a sound wave by a cylinder, cube,
and sphere. (After Muller, Black and Dunn.)
22
ACOUSTICAL ENGINEERING
There are other shapes 5 besides the cylinder, cube, and sphere that are
used for microphone and loudspeaker enclosures. In order to provide
additional information, the diffraction of sound by the shapes shown in
Fig. 1.9 were obtained experimentally. The dimensions of the ten different
SPHERE
HEMISPHERE
CYLINDER
CYLINDER
DOUBLE CONE
CONE
CUBE
1.9.
studies.
FIG.
TRUNCATED PYRAMID
ON PARALLELOPIPED
Structures used in sound diffraction
enclosures are shown in Fig. 1.9. The experimentally-determined diffrac­
tion of a sound wave by these different enclosures was obtained by com­
paring the response of a small loudspeaker in free space with the response
of the loudspeaker mounted in the enclosures in the position shown in
Fig. 1.9. The diameter of the diaphragm of the cone used in the loud­
speaker was ! inch. Since the upper frequency limit of the response was
made 4000 cycles, the diameter of the cone is less than one-quarter wave­
length. In other words the source is for all practical purposes nondirec­
tional. The diffraction characteristics for the ten shapes are shown in
Fig. 1.10. The response frequency characteristics shown in Fig. 1.10 are for
the dimensions shown in Fig. 1.9. The response frequency characteristics
5
Olson, H. F., Audio Eng., Vol. 35, No. 11, p. 34. 1951.
23
SOUND WAVES
_ERE
A
.
5
~'
0
o
0
B
•
......­ ......
5
600
0
10vu
FREQUENCY
2000
[J
CYLINDER
c
4
HEMISPHERE
0
_iHffiff8 ffi
FREQUENCY
CYLINOfR
D
i~glrut IPJ+I~I±
100
E
200 300
800
1000
0
FREQUENCY
DOUBLE CONE
2000
4000
100
F
ZOO 300
800
1000
FREQUENCY
2000
4000
CONE ( )
I¥JOOFt·~ j:jHWhffi
100
G
200 300
600 1000
p~~~':,~~ FR~
2000
4000
loa
H
2OO!OO
600 1000
2000
4000
PYRAMI:R(;jY
III10"]HHH.
iEHHJEt[
'I -,.
~;".~UENCY
CUBE
ON PARALLELDPIPED
0
J
liH1tt1HTI 1141 fim
100
2OO:3QO
600
1000
FREQUENCY
2000
4000
100
200 300
600
1000
fREQUENCY
2000
4000
FIG. 1.10. Response frequency characteristics depicting the
diffraction of sound by ten objects of different shapes. The
dimensions of the objects are given in Fig. 1.9.
for enclosures of other dimensions can be obtained by multiplying the
ratio of the linear dimensions of the enclosure of Fig. 1.9 to the linear
dimension of the enclosure under consideration by the frequency of Fig. 1.10.
24
ACOUSTICAL ENGINEERING
See Sec. 1.13. For example, if the linear dimensions of the new enclosure
are two times that of Fig. 1.9, the frequency scale of Fig. 1.10 should be
multiplied by one-half.
Another example of diffraction of sound is illustrated by the zone plate
shown in Fig. 1.11. The path lengths of the sound from the source to the
focus vary by an integral wavelength. As a consequence, all the pencils
of sound are in phase at the focus with the result that the sound pressure
is considerably greater at this point than any other position behind the
zone plate.
F
~
m
121
CROSS - SECTIONAL
FRONT VIEW
FIG. 1.11.
plate.
Zone plate.
VIEW
The source 5 and the focus F are equidistant from the zone
1.12. Acoustical Reciprocity Theorem. 6 ,7,8,9-The acoustical reciproc­
ity theorem, as developed by Helmholtz, states: If in a space filled with
air which is partly bounded by finitely extended bodies and is partly un­
bounded, sound waves may be excited at a point A, the resulting velocity
potential at a second point B is the same in magnitude and phase as it
would have been at A had B been the source of sound. It is the purpose of
this section to derive the acoustical reciprocity theorem.
Consider two independent sets of pressures P' and P" and particle veloci­
ties v' and v". Multiply equation 1.4 by the P and v of the other set.
" dv'
, dv"
v dt - v (l[
+ 1"
pv grad P0, - 1,
pv grad P"
0 =
0
1.65
If P and v vary as a harmonic of the time, equation 1.65 becomes
1"
1,
dp"
-vgradp'
o--vgra
0= 0
p
p
1.66
There is the following relation:
v grad p = div vp 6
7
8
9
P div v
1.67
Rayleigh, "Theory of Sound," Macmillan and Company, London, 1926.
Ballentine, S., Proc., I.R.E., Vol. 17, No.6, p. 929, 1929.
Olson, H. F., RCA Review, Vol. 6, No. 1, p. 36, 1941.
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J., 1943.
25
SOUND WAVES
From equations 1.9 and 1.10
J...
ap + div v =
yPO at
0
1.68
From equations 1.66, 1.67, and 1.68,
div (v"
P' -
PH)
0
1.69
The relation of equation 1.69 is for a point.
1.69 over a region of space gives
Integration of equation
ff
v'
(v"P' - v'P")ds
=
=
0
1.70
If, in an acoustical system comprising a medium of uniform density
and propagating irrotational vibrations of small amplitude, a pressure
P' produces a particle velocity v' and a pressure P" produces a particle
velocity v", then
II
(v"P' - V'P")n ds
=
1.71
0
where the surface integral is taken over the boundaries of the volume.
In the simple case in which there are only two pressures, as illustrated
in the free field acoustical system of Fig. 1.12, equation 1.71 becomes
p'v"
=
P"v'
1.72
where p', P" and v', v" are the pressures and particle velocities depicted
in the free field acoustical system of Fig. 1.12.
1??2??
uun anna???? ?222222UU22222222222222?222Zd
ZA. p"X'
P')('ZAI
!a a
p'v'
p'v"
11111 I?? ?? 111112?2?1?2?2?Z21
FIELD
FIG. 1.12.
LUM PED
CONSTANTS
Reciprocity in field and lumped constant acoustical systems.
The above theorem is applicable to all acoustical problems. However,
it can be restricted to lumped constantslO as follows: In an acoustical sys­
tem composed of inertance, acoustical capacitance, and acoustical resist­
ance, let a set of pressures Pt', h', P3' ... Pn', all harmonic of the same
frequency acting in n points in the system, produce a volume current dis­
tribution Xl' X 2 ', X3' ... Xn', and let a second set of pressures Pt", h",
P3
Pn", of the same frequency as the first, produce a second volume
current distribution Xl", X 2 ", X3" ... X n". Then
N
•••
n
2. Pi'K" =
j=l
10
n
2.p/,X/
1.73
j=l
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.].,
1943.
26
ACOUSTICAL ENGINEERING
This theorem is valid provided the acoustical system is invariable,
contains no internal source of energy or unilateral device, linearity in
the relations between pressures and volume currents, and complete re­
versibility in the elements and provided the applied pressures PI, h,
P3 ... Pn are all of the same frequency.
In the simple case in which there are only two pressures, as illustrated
in the acoustical system of lumped constants in Fig. 1.7, equation 1.73
becomes
plC" = p"lC
1.74
where p', pIt and lC, X" are the pressures and volume currents depicted
in the acoustical system of lumped constants in Fig. 1.12.
1.13. Acoustical Principle of Similarity.ll-The principle of similarity
in acoustics states: For any acoustical system involving diffraction phe­
nomena it is possible to construct a new system on a different scale, which
will exhibit similar performance, providing the wavelength of the sound is
altered in the same ratio as the linear dimensions of the new system.
The principle of similarity is useful in predicting the performance of
similar acoustical systems from a single model. A small model can be
built and tested at very high frequencies to predict the performance of
similar large systems at lower frequencies. For example: in the diffraction
of sound by objects, if the ratio of the linear dimensions of the two objects
is X : 1, the corresponding configuration of the frequency characteristics
will be displaced 1 : X in frequency. This is illustrated in Figs_ 1.8 and
1.10. Other examples are the directional characteristics of various sound
sources Figs. 2.3 to 2.23 inclusive, the air load upon a diaphragm, Fig. 5.2,
etc.
1.14. Lon~itudinal Waves in a Rod.-The preceding considerations have
been concerned with sound waves in gases and fluids. In the case of solids,
longitudinal waves in rods are of practical interest in many applications.
It is the purpose of this section to derive the equations for longitudinal
sound waves in a rod of homogeneous material and constant cross section.
The longitudinal axis of the bar will be assumed to coincide with the x
axis. Consider an element of the bar ox, determined by two planes per­
ox from x = O. Assume
pendicular to x and initially at distances x and x
that the planes are displaced by distances g and g
The distance
between the planes is now
+
ox + og = ox +
+ Dr
:! ox
:! ox .
1.75
The increase in distance between the planes is
The increase in length of the bar per unit length at this point is
11
Olson, H. F., RCA Review, Vol. 6, No. 1, p. 36, 1941.
~.
SOUND WAVES
27
Young's modulus is defined as the ratio of the longitudinal stress to the
corresponding extension. At the first face of the element Young's modulus
1S
F
S
Q = og
1.76
ox
where Q = Young's modulus, in dynes per square centimeter,
F = force, in dynes,
S = cross-sectional area of the rod, in square centimeters, and
og
ax =
.
extenSlOn.
The force acting on the element across the first face is
F=QS~
oX
1.77
The force acting across the second face of the element is
F
+ SF =
=
~+~
Sg) Sx
ox
ax (QS ox
QS ?.ff. + QS o2g Sx
ox
ox2
QS
1.78
1.79
The resultant force on the element is
SF =
QS
o2g
ox 2 Sx
1.80
The acceleration of momentum of the element is
SpSx
o2g
ot 2
1.81
where p = density, in grams per cubic centimeter.
Equating the resultant force on the element to the acceleration of mo­
mentum, the result is
o2g Q o2g
ot 2 = Pox 2
This is the wave equation for g. Equation
1.82
1.82 is analogous to equation
1.16 for plane waves in a gas and the solution of the differential equation
is similar. The velocity of propagation, in centimeters per second, of
longitudinal waves is in a rod
c=J~
1.83
where Q = Young's modulus. in dynes per square centimeter (see Table
1.1), and
p = density, in grams per cubic centimeter (see Table 1.1).
28
ACOUSTICAL ENGINEERING
The velocity of sound, Young's modulus and the density for vanous
solids are given in Table 1.1.
1.15. Torsional Waves in a Rod.-A rod may be twisted about an
axis of the rod in such a manner that each transverse section remains in its
own plane. If the section is not circular there will be motion parallel to
the axis of the bar. For a circular cross section and a homogeneous bar the
equations of motion are analogous to those of longitudinal waves in the rod.
The velocity of propagation, in centimeters per second, of torsional waves
in a rod, is
c-J
-
Q+ 1)
2p (0'
1.84
where Q = Young's modulus, in dynes per square centimeter (see Table
1.1),
p = density, in grams per cubic centimeter (see Table 1.1), and
a = Poisson's ratio (see Table 1.1).
1.16. Cylindrical Sound Waves,12-From a practical standpoint, the
important waves in acoustics are plane and spherical waves. However,
it may be interesting as an addition in the chapter on sound waves to
indicate some of the characteristics of cylindrical sound waves.
The consideration will be the sound pressure and particle velocity pro­
duced by a long cylinder expanding and contracting radially with a velocity
Uo, in centimeters per second, given by
1.85
where U 0
= maximum velocity of the vibration in centimeters per second,
w = 2TTj,
j = frequency, in cycles per second, and
t = time, in seconds.
When the radius of the cylinder is small compared to the wavelength,
the sound pressure p, in dynes per square centimeter, at a distance large
compared to the radius of the cylinder may be expressed as
p = TTpaU 0
J¥
€jk(r-ct)-jw/4
1.86
where p = density of the medium, in grams per cubic centimeter,
c = velocity of sound in the medium, in centimeters per second,
k = 2TTj"A
"A = wavelength, in centimeters,
a = radius of the cylinder, in centimeters, and
r = distance from the axis of the cylinder.
12 Morse, "Vibration and Sound," McGraw-Hill Book Company, New York, N.Y.,
1948.
SOUND WAVES
29
The particle velocity u, in centimeters per second, under the same con­
ditions is given by
U =
7TaU o
JZ
cr
ff.i k (r-ct)-i7T/4
1.87
It will be seen that the pressure and particle velocity decrease inversely
as the square root of the distance from the cylinder.
The product of the pressure and the particle velocity gives the flow of
energy per square centimeter as follows,
1.88
It will be seen that the intensity falls off inversely as the distance.
2
ACOUSTICAL RADIATING SYSTEMS
2.1. Introduction.-There are almost an infinite number of different
types of sound sources. The most common of these are the human voice,
musical instruments, machinery noises, and loudspeakers. The most
important factors which characterize a sound source are the directional
pattern, the radiation efficiency, and the output as a function of the fre­
quency. In the case of some sound sources as, for example, musical instru­
ments, it is almost impossible to analyze the action. However, in the case
of most sound reproducers the action may be predicted with amazing
accuracy. It is the purpose of this chapter to consider some of the simple
sound sources that are applicable to the problems of sound reproduction.
2.2. Simple Point Source.-A point source is a small source which
alternately injects fluid into a medium and withdraws it.
A. Point Source Radiating into an Infinite Medium. Solid Angle of
4rr Steradians.-Consider a point source having a maximum rate of fluid
emission of 4rrA cubic centimeters per second. The momentary rate at a
time t is 4rr A cos wt. The maximum rate of fluid emission may be written
- 4rrA
where S
=
go =
=
sgo
2.1
area of the surface of the source, in square centimeters, and
maximum velocity, in centimeters per second over the sur­
face S.
The velocity potential of a point source from equation 1.36 is
cpr
= -A
r
"jk(ct-r)
2.2
The particle velocity at a distance r from equation 1.42 is
u
=
-
A: [lr cos k (ct - r) - sin k(ct - r]
2.3
The pressure at a distance r from equation 1.40 is
p = pkcA sin k(ct - r)
r
30
2.4
ACOUSTICAL RADIATING SYSTEMS
31
The intensity or average power, in ergs per second, transmitted through
a unit area at a distance r, in centimeters, is the product of p and u and is
given by
pck 2A2
P = ----zT2
2.5
The total average power in ergs per second emitted by the source is
PT = 217pck2A2
2.6
where p = density of the medium, in grams per cubic centimeter,
c = velocity of sound, in centimeters per second,
k = 217/A,
A= wavelength, in centimeters, and
A is defined by equation 2.1.
B. Point Source Radiating into a Semi-Infinite Medium. Solid Angle
of 21T Steradians.-The above example considered a point source operating
in an infinite medium. The next problem of interest is that of a point
source operating in a semi-infinite medium, for example, a point source
near an infinite wall.
In this case we can employ the principle of images as shown in Fig. 2.1.
The pressure, assuming the same distance from the source, is two times
that of the infinite medium. The particle velocity is also two times that
of the infinite medium. The average power transmitted through a unit
area is four times that of the infinite medium. The average power out­
put of the source, however, is two times that of a simple source operating
in an infinite medium.
C. Point Source Radiating into a Solid Angle of 17 Steradians.-Em­
ploying the method of images Fig. 2.1 the pressure is four times, the par­
ticle velocity is four times, and the average power transmitted through a
unit area is sixteen times that of an infinite medium for the same distance.
The average power output of the source is four times that of a simple
source operating in an infinite medium.
D. Point Source Radiating into a Solid Angle of 17/2 Steradians.-Em­
ploying the method of images, Fig. 2.1, the pressure is eight times, the
particle velocity eight times, and the average power transmitted through a
unit area is sixty-four times that of the same source operating in an infinite
medium at the same distance. The average power output is eight times
that of the same simple source operating in an infinite medium.
E. Application of the Simple Source.-The above data may be applied
to acoustic radiators in which the dimensions are small compared to the
wavelength and located close to the boundaries indicated above. For
example, A would correspond to a loudspeaker, which acts as a simple
source, suspended in space at a large distance from any walls or boundaries.
B would correspond to a loudspeaker placed on the floor in the center of
the room. C would correspond to a loudspeaker placed on the floor along
a wall, and D would correspond to a loudspeaker placed in the comer of
32
ACOUSTICAL ENGINEERING
the room. Of course, as pointed out above, these examples hold only when
the dimensions of the radiator and the distance from the wall are small
compared to the wavelength.
SOLID ANGLE
OF SOUND
EMISSION
•
PRESSURE
AT A
DISTANCE r
41T
dF-
ENERGY
DENSITY
DISTANCEr
P
W
2Tf
2p
2W
41
1T
4p
4W
161
1!
8p
8W
641
SOURCE
,.~+~,
POWER
OUTPUT
IMAGES
SOURCE
2
IMAGES
2.1. The sound pressure, total power output, and energy
density delivered by a point source operating in solid angles of
47T, 27T, 7T, and 7T/2 steradians.
FIG.
2.3. Double Source (Doublet Source).l,2,3.4-A double source consists
of two point sources equal in strength ±4n-A', but opposite in phase sepa­
rated by a vanishingly small distance Sr. The strength of the doublet is
47TA'Sr. Let A'ar = A. In these considerations A' corresponds to A of
equation 2.1, that is 47TA' = 5 go.
At a distance r in a direction inclined at an angle a to the axis of the
doublet the velocity potential is
cfo
=
(!+ jk)A
r
r
€jk(ct-r)
cos a
2.7
Lamb, "Dynamical Theory of Sound," E. Arnold, London, 1931.
Davis, "Modern Acoustics," The Macmillan Co., New York, N.Y., 1934.
3 Wood, .. A Textbook of Sound," Bell and Sons, London, 1930.
4 Crandall, "Theory of Vibrating Systems and Sound," D. Van Nostrand Company,
Princeton, N.J., 1926.
1
2
33
ACOUSTICAL RADIATING SYSTEMS
The pressure from equation 2.7 is
P=
p 8e/>
_
8t
= _
j pckA
r
(!r + jk) €jk(ct-r) cos a
2.8
Retaining the real parts of equation 2.8
p = pc:A
Gsin k(ct -
r)
+ k cos k(ct - r)] cos a
2.9
At a very large distance
A
pIX. -kC
OSa
r
2.10
pIX. 2kA
r
2.11
2
At a very small distance
cos a
The particle velocity has two components, the radial
verse
!r 88e/>·a
~t
and the trans­
The radial component of the particle velocity from equation
2.7 is,
~=
u=
[(~ +~~)
-
+jk (h +~)] A €1k(ct-r) cos a
2.12
Retaining the real parts of equation 2.12
u
=
-
A
[(~ - ~) cos k(ct -
r) -
!:
sin k(ct - r)] cos a
2.13
At a very large distance
Ak2
cos a
r
2.14
raA cos a
2.15
U IX. -
At a very small distance
U IX.
The transverse component of the particle velocity is
U
18
j
r 8a
= -
=
-
(_;1+_ _J.k) A€1k(ct-r) sin a
r2
2.16
Retaining the real parts of equation 2.16
U
=
-
A
[Ja cos k(ct -
r) -
~ sin k(ct -
r)] sin a
2.17
At a very large distance
Ak .
U IX.
Y2 sma
2.18
34
ACOUSTICAL ENGINEERING
At a very small distance
2.19
Fig. 2.2 shows the velocity components and the pressure for various
points around a doublet source. A common example of a doublet source
is a direct radiator loudspeaker mounted in a small baffle. (Dimensions
of the baffle are small compared to the
wavelength.) If the response of such a
/
loudspeaker
is measured with a pressure
~,
,
I
microphone for various angles at a con­
distance, the result will be a cosine
~\lIi"/PARTICLE stant
If the response is measured
characteristic.
' /
• ./""'\ VELOCITY
with
a
velocity
microphone keeping the
I .,/// '
axis pointed toward the loudspeaker,
,
the result will be a cosine directional
characteristic. If the same is repeated
- -!\,',,'
keeping the axis of the velocity microphone
normal to the line joining the microphone
.
"'--,"-- '
and the loudspeaker, the result will be a
~ PRESSURE sine
directional characteristic.
The total power, in ergs, emitted by a
doublet source is
~---.I
"
,
""
\
'//'
~'---'1I' '''--,
\
I \
i\.. '
r-X
2.20
FIG. 2.2. The sound pressure and
particle velocity at a constant dis­
tance from a doublet source. The
magnitude of the pressure is in­
dicated by the circle. The particle
velocity has two components, a
'radial and a transverse component.
The direction and magnitude of
these two components are indicated
by vectors.
where p
p
=
==
c=
dS =
pressure, in dynes per square
centimeter,
density, in grams per cubic
centimeter,
velocity of sound, in centi­
meters per second, and
area, in square centimeters,
over which the pressure is p.
Taking the value of p from .equation 2.9 (for r very large), the total
average power in ergs per second emitted by a doublet source is
i'"
pCk4A2
PT
=
27T1'2
PT
=
i1Tpck4A2
o
- - - cos 2 e;; sin e;; de;;
2r2
where p = density, in grams per cubic centimeter,
k = 217/1..,
I.. = wavelength, in centimeters,
c = velocity of sound, in centimeters per second, and
A is defined in the first paragraph of this section.
2.21
2.22
35
ACOUSTICAL RADIATING SYSTEMS
The power output from a simple source (equation 2.6) is proportional
to the square of the frequency, while the power output from a doublet
source (equation 2.22) is proportional to the fourth power of the frequency.
For this reason the power output of a direct radiator loudspeaker falls off
rapidly with frequency when the dimensions of the baffle are small compared
to the wavelength (see Sec. 6.8).
DISTANCE =7).
DISTANCE =
1.0
1.0
DISTANCE =~"
t"
01 STANCE = 2"
DISTANCE=>­
1.0
DISTANCE =
~"
1.0
1.0
FIG. 2.3. Directional characteristics of two separated equal small sources vibrating in
phase as a function of the distance between the sources and the wavelength. The polar
graph depicts the sound pressure, at a fixed distance, as a function of the angle. The
sound pressure for the angle 0° is arbitrarily chosen as unity. The direction correspond­
ing to the angle 0° is perpendicular to the line joining the two sources. The directional
characteristics in three dimensions are surfaces of revolution about the line joining the
two sources as an axis.
:'U. Series of Point Sources.-The directional characteristicS ,6,7 of a
source made up of any number of equal point sources, vibrating in phase,
located on a straight line and separated by equal distances is given by
. (nwd. )
. (wd.
nsm
TsmlX )
sm TsmlX
Ra
where Ra
=
n=
d=
A=
=
2.23
ratio of the pressure for an angle IX to the pressure for an
angle IX = O. The direction IX = 0 is normal to the line,
number of sources,
distances between the sources, in centimeters, and
wavelength, in centimeters.
The directional characteristics of a two-point source are shown in Fig.
2.3. It will be noted that the secondary lobes are equal to the main lobe.
~Wolff,
I., and Malter, L., Jour. Acous. Soc. Amer., Vol. 2, No.2., p. 201, 1930.
Stenzel, H., Elek. Nach. Tech., Vol. 4, No.6, p. 239, 1927.
7 Stenzel, H., Elek. Nach. Tech., Vol. 6, No.5, p. 165, 1929.
8
36
ACOUSTICAL ENGINEERING
2.5. Straight-Line Source.-A straight-line source may be made up
of a large number of points of equal strength and phase on a line separated
by equal and very small distances. If the number of sources n approach
infinity and d, the distance between the sources, approaches zero in such a
way that
nd = 1
the limiting case
becomes
IS
the line source.
If this is carried out, equation 2.23
. (TTl.
sm
Xsma )
Ra
= ----;.-----=­
TTl .
1" sma
2.24
The directional characteristics of a continuous line source are shown in
Fig. 2.4. The directional characteristics are symmetrical about the line
iII 1i • 3.'.0
~0
z
z .•
a
60
.
60
oLENGTH.8A30
90
eoLENGTI;l=4\0
we
~
60
4
•
eo
80
90
t09
LENGTH- 8~
90
~O
~~: '0
~
•
.e
30
~
~
eo
,0LENGTH-lf~
. z•
:o
30
eo
60
90
eo
eo
to
60
90
90
LENGTH.
3
L1.I
2>­
30
00
90
.LENGT~4>­
3
30
•
60
80
~A
•
.
90
LENGT'~_'tA
60
z',:'O
~Q
60
LENGTH-3A
30
30
zo
LENGTH-~
•
.6
00
30
Z ,6
to
30
60
90
vee
90
LENGT~_'~A
LENGTH-A
LENG1tt.fA30
Z
eo
60
00
'm'O'0 .'.00
90
1,0
LENGTH .2~"
60
90
60
90
90
LENGTH
eo
90
90
-6>­
60
90
FIG. 2.4.
Directional characteristics of a line source as a function of the length and the
wavelength. The polar graph depicts the sound pressure, at a large fixed distance, as a
function of the angle. The sound pressure for the angle 0° is arbitrarily chosen as unity.
The direction corresponding to the angle 0 0 is perpendicular to the line. The directional
characteristics in three dimensions are surfaces of revolution about the line as an axis.
as an axis. Referring to Fig. 2.4, it will be seen that there is practically no
directivity when the length of the line is small compared to the wave­
length. On the other hand, the directional characteristics are sharp when
the length of the line is several wavelengths.
2.6. Beam Tilting by Phase Shifting.-The direction and shape of the
wavefront produced by a series of sound sources may be altered by the
introduction of a delay pattern in the excitation of the sources. An example
of a series of point sources of sound equally spaced along a straight line in
ACOUSTICAL RADIATING SYSTEMS
combination with a delay system is shown in Fig. 2.5.
in Fig. 2.5 the distance x, in centimeters, is given by
37
In the system shown
x = ct
2.25
where c = velocity of sound, in centimeters, and
t = time delay, in seconds.
The angle 8, the angle by which the wavefront is shifted by the delay
system, is given by
8
sin-1
=
j
2.26
where d = distance between the units, in centimeters.
Phase shifting can be used in many other ways besides beam tilting. For
example, practically any wavefront shape can be obtained by introducing
the appropriate phase shift in the sound sources.
SOUNO
SOURCES
DELAY
UNITS
INPUT
FIG. 2.5. A delay system for tilting the direc­
tional characteristic of a line of sound sources.
2.7. Tapered Straight-Line Source.-The directional characteristicS of a
line source, all parts vibrating in phase, in which the strength varies linearly
from its value at the center to zero at either end, is given by
sin 2
Ra
where Ra
=
1=
,\ =
=
(~
(l
sin
;;>.. sin a
a)
)2
2.27
ratio of the pressure for an angle a to the pressure for an
angle a = O. The direction a = 0 is normal to the line,
total length of the line in centimeters, and
wavelength, in centimeters.
The directional characteristics of a tapered line source are shown in
Fig. 2.6. Comparing the directional characteristics of Fig. 2.6 with those
of the uniform line of Fig. 2.4, it will be seen that the main lobe is broader
and the secondary lobes are reduced in amplitude.
8
Menges, Karl, Akus Zeit., Vol. 6, No. 2, p . 90,1941.
38
ACOUSTICAL ENGINEERING
.~ • "'~~'i'
LENG H: >0
",o,,~.,
••
to.oL]~~~~~lIJ_,C;~~~:;t:l>-.O
8
''rENGTH:2X'°
I~
If
90LENGTH:4X'"
3~
30_
.0
.~
to,ll:1~~~~~:IJ
'LENGTH:6>­
'0
.0
901
'0
10
90
FIG. 2.6. Directional characteristics of a tapered line source as a function of the length
and the wavelength. The volume current output along the line varies linearly from a
maximum at the center to zero at the two ends. The polar graph depicts the sound
pressure, at a fixed distance, as a function of the angle. The sound pressure for the
angle 0° is arbitrarily chosen as unity. The direction corresponding to the angle 0° is
perpendicular to the line. The directional characteristics in three dimensions are sur­
faces of revolution about the line as an axis.
2.8. Nonuniform Straight-Line Source.-The directional characteristic
of a line, all parts vibrating in phase, in which the strength varies as a
function of the distance x along a line is given by
2.28
where
x = distance from the center of the line, in centimeters,
d = total length of the line, in centimeters,
f(x) = strength distribution function and the other quantities are
the same as those in equation 2.27.
2.9. End Fired Line Source.-An end fired line source is one in which
there is progressive phase delay between the elements of the line. In the
case in which the time delay of excitation between the elements corresponds
to the time of wave propagation in space for this distance the maximum
directivity occurs in direction corresponding to the line joining the elements.
The directional characteristics of an end fired line of this type and of uniform
strength is given by
sin
X(I -
I cos a)
Ra=------7T
X (I - I cos)
a
2.29
39
ACOUSTICAL RADIATING SYSTEMS
where Ra
ratio of the pressure for an angle a to the pressure for the
angle a = O. The direction a = 0 is along the line,
I = length of the line, and
A = wavelength.
=
The directional characteristics 9 of a uniform end fired line with progressive
time delay between elements corresponding to the time of wave propagation
over this distance in free space are shown in Fig. 2.7. The maximum direc­
tivity occurs along the direction corresponding to the line. The directional
characteristics are symmetrical about the line as an axis.
'80
.80
LENGTH'
t
LENGTH.
i
LENGTH .).
'.0
.10
LENGTH '2).
LENGTH' 4).
FIG. 2.7. Directional characteristics of an end fired line.source as a
function of the length and wavelength. The polar graph depicts the
sound pressure, at a large fixed distance, as a function of the angle.
The sound pressure for the angle 0° is arbitrarily chosen as unity. The
direction corresponding to angle 0° coincides with the line. The
directional characteristics in three dimensions are surfaces of revolution
about the line as an axis.
2.10. Super Directivity Source.-From the preceding examples of
directional systems it will be seen that in order to obtain some order of
directivity the dimensions of the radiator must be greater than a wavelength.
It is possible to obtain a high order of directivity from a source which is
smaller in dimension than wavelength. These systems have been termed
super directional sources.1 0 A super directional system may be considered
to be the difference between two patterns each of which is due to a conven­
tionallinear array employing in phase excitation. A super directional array
H. F .• Jour. [nst. Rad. Eng.• Vol. 27. No.7. p. 438. 1939.
Pritchard. R. L.. Jour. Acous. Soc. Amer., Vol. 25, No.5, p. 879, 1953.
9 Olson.
10
40
ACOUSTICAL ENGINEERING
is shown in Fig. 2.8. It will be seen that alternate elements are oppositely
phased. Comparing the directional pattern of the super directivity source
with the directivity pattern of the simple line source of Fig. 2.4, it will be
seen that approximately the same
30
directivity is obtained with a line of
one-third the length of the simple line
20
source. This added directivity is
o
obtained at the expense of some
...o~ 10
other factors. The reversed phase
::>
::; 0
excitation results in a loss in
0.
:<
efficiency. Close tolerances must be
"-10
maintained upon the strength of the
elements and the operating wave­
length, otherwise the directivity pat­
~2~ELATIVE AMPLITUDES
tern will not be maintained. Thus
FIG. 2.8. Directional characteristics of a
a
super directional system is sensitive
super directivity source consisting of five
to frequency changes and is, there­
sources spaced 1 wavelength apart. The
relative amplitudes and the phases of the
fore, not suitable for broad band
sources are shown in the diagrams above.
operation. Therefore, the applica­
The polar graph depicts the sound pressure
at a large fixed distance from the array.
tions for a super directional source
The sound pressure for the angle 0° is
are
where a narrow frequency band
arbitrarily chosen as unity. The direc­
width and low efficiency can be
tion corresponding to the angle 0° is
perpendicular to the line joining the
tolerated in exchange for smaller
sources. The directional characteristics
space
requirements.
in three dimensions are surfaces of revolu­
tion about the line as an axis.
2.11. Curved-Line Source (Arc
of a Circle).-A curved-line source
may be made up of a large number of point sources vibrating in phase on the
arc of a circle separated by very small distances. The directional charac­
teristics of such a)ine in the plane of the arc are,
II)
..
Ra
= 2m 1+
lr'lmm cos [2~R cos
(a
+ kB)]
+ j kklmm sin [2~R
where
Ra
a
,\.
R
2m + 1
(j
k
cos (a
+ kB)] I
2.30
ratio of the pressure for an angle a to the pressure for
an angle a = 0,
= angle between the radius drawn through tlie central
point and the line joining the source and the distant
observation point,
= wavelength, in centimeters,
= radius of the arc, in centimeters,
= number of points,
= angle subtended by any two points at the center of the
arc, and
= variable.
=
41
ACOUSTICAL RADIATING SYSTEMS
Another methodl l is to break up the arc into a large number of equal
chords. The strength is assumed to be uniform over each chord. Also the
phase of all the chords is the same. In this case the result takes the form,
R.
=
1
2m + 1
k=m
2: mcos
{27TR
T
k= -
. k=m
2:
+J
k= -
where
Rr. =
A=
k =
R
=
2m + 1 =
8=
d=
.
SIll
{27TR
T
}Sin[~sin(a+k8)]
cos (a + k8) --=di------...;:...
~ sin (a + k8)
A
cos (a + k8)
m
} sin [7T;Sin(a +
k8)]
d
~ sin (a + k8)
2.31
A
ratio of the pressure for an angle a to the pressure for
an angle a = 0,
wavelength, in centimeters,
variable,
radius of the arc, in centimeters,
number of chords,
angle subtended by any of the chords at the center of
circumscribing circle, and
length of one of the chords, in centimeters.
RADIUS- ).
RADIUS. ;
1.0
to
Ll.LL.L:-::::£.~!f:3:::::::LL...l.JUoo 90LLL::£~~I:::::L--L-.J90
RADIUS" 2),
RADIUS- 4),
RADIUS - 8),
RADIUS -18).
FIG. 2.9. Directional characteristics of a 60° arc as a function of the radius and the
wavelength. The polar graph depicts the sound pressure, at a large fixed distance, as a
function of the angle in the plane of the arc. The sound pressure for the angle 0° is
arbitrarily chosen as unity.
The directional characteristics for an arc of 60°, 90°, and 120° are shown
in Figs. 2.9, 2.10, and 2.11. The interesting feature of the directional
11
Wolff, I., and Malter, L., Jour. Acous. Soc. Amer., Vol. 2, No.2, p. 201, 1930.
42
ACOUSTICAL ENGINEERING
characteristics of an arc is that the directional characteristics are very
broad for wavelengths large compared to the dimensions, and are narrow for
wavelengths comparable to the dimensions and are broad again for wave­
lengths small compared to the dimensions of the arc. The arc must be
several wavelengths in length in order to yield a "wedge-shaped" direc­
tional characteristic.
RADIUS
1.0
·i
RADIUS-i
RADIUS->"
•.0
1.0
FIG. 2.10.
Directional characteristics of a 90° arc as a function of the radius and the
wavelength. The polar graph depicts the sound pressure at a large fixed distance, as a
function of the angle in the plane of the arc. The sound pressure for the angle 0° is
arbitrarily chosen as unity.
RADIUS=~
'0
'II
:"
RADIUS=~
o
60
60
RADIUS-'>'
•
60
•
go
RADIUS 4)..
RADIUS 8>.
FIG. 2.11. Directional characteristics of a 120° arc as a function of the radius and the
wavelength. The polar graph depicts the sound pressure, at a large fixed distance, as a
function of the angle in the plane of the arc. The sound pressure for the angle 0° is
arbitrarily chosen as unity.
ACOUSTICAL RADIATING SYSTEMS
43
2.12. Circular-Rin~ Source.-The directional characteristics l2 ,13 of a
circular-ring source of uniform strength and the same phase at all points on
the ring is
2.32
where Ra
=
J0 =
R
=
a =
ratio of the pressure for an angle a to the pressure for an
angle a = 0,
Bessel function of zero order,
radius of the circle, in centimeters, and
angle between the axis of the circle and the line joining the
point of observation and the center of the circle.
The directional characteristics of a circular-ring source as a function of
the diameter and the wavelength are shown in Fig. 2.12. The shapes are
DIAMET.~R
-t
DIAME1~R -ll>.
•
•
30
z .•
60
.,
90 DIAMETt,R
30
90
90
..
a
3
•
w.s
A
-1>. 90
00
:
Z
0.­
..,
90
,~":~:
00
'10
•
"oIAMET~R_l~). '.~IAMET5R_~
0
30
gO
,"
,~~.~
30
.0
gO
30
..
..
eo
•
DIAMET~R_12~
•
DIAMET~R_l
5
JlDIAMET
R -3>.
3
30 3
60
90
10
0
eo
'1090
eo
90
2.12. Directional characteristics of a circular-line or ring source as a function of the
diameter and wavelength. The polar graph depicts the sound pressure, at a large fixed
distance, as a function of the angle. The sound pressure for the angle 0 0 is arbitrarily
chosen as unity. The direction corresponding to the angle 0° is the axis. The axis is
the center line perpendicular to the plane of the circle. The directional characteristics
in three dimensions are surfaces of revolution about the axis.
FIG.
quite similar to those of a straight line. The characteristic is somewhat
sharper than that of a uniform line of length equal to the diameter of the
circle, but has almost the same form. The amplitudes of the secondary
lobes are greater than those of the uniform line.
2.13. Plane Circular -Piston Source.-The directional characteris­
tics14 , 15 of a circular-piston source mounted in an infinite baffle with all
Stenzel, H., Elek. Nach. Tech., Vol. 4, No.6, p. 1. 1927.
Wolff, I., and Malter, L., Jour. Acous. Soc. Amer., Vol. 2, No.2, p. 201. 1930.
14 Stenzel, H., Elek. Nach. Tech., Vol. 4, No.6, p. 1. 1927.
15 Wolff, I., and Malter, L., Jour. Acous. Soc. Amer., Vol. 2, No.2, p. 201. 1930.
12
13
ACOUSTICAL ENGINEERING
parts of the surface of the piston vibrating with the same strength and phase
are
2lI (2;R sin IX)
Ra
where Ra
=
II =
R
IX
=
=
A=
=
2.33
271'R.
-A- SIll IX
ratio of the pressure for an angle IX to the pressure for an
angle IX = 0,
Bessel function of the first order,
radius of the circular piston, in centimeters,
angle between the axis of the circle and the line joining the
point of observation and the centre of the circle, and
wavelength, in centimeters.
DIAME~~R. f >.
DIAMETER· >.
DIAMEUR"~>'
DIAMETER·3>.
3$
00 .:~.o
LJ~~~~;;r:Jw
90
30
80
•
00 00
~4
eo
go
ZA
00
-~>.
.'.~ ]I
981AMETER
&0
en.4
80
•
90
liO
90
·00
ao
60
B\iY
; 30 31i'.030
00
00
90
'li1.0
60
90
60
IKt
00
.:W:
·~IAMETER.6~
60
80
80
90
FIG. 2.13. Directional characteristics of a circular-piston source mounted in an infinite
baffle as a function of the diameter and wavelength. The polar graph depicts the sound
pressure, at a large fixed distance, as a function of the angle. The sound pressure for
the angle 0° is arbitrarily chosen as unity. The direction corresponding to the angle 0°
is the axis. The axis is the center line perpendicular to the plane of the piston. The
directional characteristics in three dimensions are surfaces of revolution about the axis.
The directional characteristics of a plane circular-piston source mounted
in an infinite baffle as a function of the diameter and wavelength are shown
in Fig. 2.13. The characteristic is somewhat broader than that of the
uniform line of length equal to the diameter of the circle, but has approxi­
mately the same form. The amplitudes of the secondary lobes are smaller
than those of the uniform line.
2.14. Nonuniform Plane Circular-Surface Source.1 6-The integration
of the expression for a plane circular-surface source in which the strength
varies as a function of the distance from the center cannot be obtained in
16 Jones, R. Clark. Jour. Acous. Soc. Amer.• Vol. 16. No.3. p. 147. 1945.
This is a
comprehensive paper on the study of directional patterns of plane surface sources with
specified normal velocities. A number of directional patterns and tables are given.
45
ACOUSTICAL RADIATING SYSTEMS
simple terms. An approximate method may be employed in which the
plane circular surface with nonuniform strength is divided into a number of
rings with the proper strength assigned to each ring. An alternative method
may be employed in which the strength distribution is obtained by super­
posing a number of plane circular-surface sources of different radii with
the proper strength assigned to each surface.
2.15. Plane Circular-Piston Source Set in the End of an Infinite
Pipe,17,18_The directional characteristics of a plane circular-piston set in
the end of an infinite pipe with all parts of the piston vibrating with the
same amplitude and phase as a function of the diameter and wavelength
are shown in Fig. 2.14.
1.0
180·
DIAMETER· -}
1.0
180"
DIAMETER. -}
1.0
I O·
DIAMETER.
>.
FIG. 2.14.
Directional characteristics of a circular-piston source located in the
end of an infinite pipe as a function of the diameter and wavelength. The
polar graph depicts the sound pressure, at a large fixed distance, as a function
of the angle. The sound pressure for the angle 0° is arbitrarily chosen as
unity. The direction corresponding to the angle 0° is the axis. The axis is
the center line perpendicular to the plane of the piston. The directional
characteristics in three dimensions are surfaces of revolution about the axis.
An example of a vibrating piston set in a tube is that of a loudspeaker
mechanism set in a completely enclosed cabinet having a face area not
appreciably larger than the loudspeaker mechanism.
2.16. Plane Circular-Piston Source in Free Space,19-The directional
characteristics of a plane circular piston in free space with all parts of the
piston vibrating with the same amplitude and phase as a function of the
diameter and wavelength are shown in Fig. 2.15. In the low-frequency
range the directional pattern is the same as that of a doublet source because
it is doublet in this frequency range.
An example of a vibrating piston in free space is a loudspeaker mechanism
operating in free space without a baffle, cabinet, etc.
2.17. Plane Square-Surface Source.-The directional characteristics
of a plane square-surface source, with all parts of the surface vibrating with
the same intensity and phase, in a normal plane parallel to one side, is the
Levine and Schwinger, Phys. Rev., Vol. 73, No.4, p. 383, 1948.
Beranek, " Acoustics," McGraw-Hill Book Company, New York, N.Y.. 1954.
19 Wiener, F. M., Jour. Acous. Soc. Amer., Vol. 23, No.6, p. 697,1951.
17
18
46
ACOUSTICAL ENGINEERING
same as that of a uniform line source having a length equal to one side of
the square (equation 2.24).
The directional characteristics of a plane square-surface source, with all
parts of the surface vibrating with the same strength and phase, in a normal
plane containing the diagonal is the same as that of the tapered line source
having a length equal to the diagonal (equation 2.28).
180·
DIAMETER'
t
180"
DIAMETER·
A
"2
180·
DIAMETER'
A
H+8Jl8+f--+
0'
180·
DIAMETER
180·
aliA
DIAMETER •
180'
2A
DIAMETER:
4A
2.15. Directional characteristics of a circular piston located in free space
as a function of the diameter and wavelength. The polar graph depicts the
sound pressure, at a large fixed distance, as a function of the angle. The
sound pressure for the angle 0° is arbitrarily chosen as unity. The direction
corresponding to the angle 0° is the axis. The axis is the center line perpendi­
cular to the plane of the piston. The directional characteristics in three
dimensions are surfaces of revolution about the axis.
FIG.
2.18. Plane Rectan~ular -Surface Source.-The directional characteris­
tics of a rectangular-surface source with all parts of the surface vibrating
with the same strength and phase are
. (7Tl
. a)
T a S111
SIll
Ra = ---:---..:..
7Tla .
T
2.34
SIlla
where la = length of the rectangle,
lb = width of the rectangle,
a = angle between the normal to the surface source and the pro­
jection of the line joining the middle of the surface and the
observation point on the plane normal to the surface and
parallel to la, and
ACOUSTICAL RADIATING SYSTEMS
47
f3 angle between the normal to the surface source and the pro­
jection of the line joining the middle of the surface and the
observation point on the plane normal to the surface and
parallel to lb.
The directional characteristic of a plane rectangular-surface source with
uniform strength and phase is the same as the product of the characteristic
of two line sources at right angles to each other and on each of which the
strength and phase are uniform.
2.19. Horn Source.-The directional characteristics of a horn depend
upon the shape, mouth opening, and the frequency. It is the purpose of
this section to examine and consider some of the factors which influence the
directional characteristics of a horn.
The phase and particle velocity of the various incremental areas which
may be considered to constitute the mouth determines the directional
characteristics of the horn. The particular complexion of the velocities
and phases of these areas is governed by the flare and dimensions and shape
of the mouth. In these considerations the mouth will be of circular cross
section and mounted in a large flat baffle. The mouth of the horn plays
a major role in determining the directional characteristics in the range
where the wavelength is greater than the mouth diameter. The flare is
the major factor in determining the directional characteristics in the range
where the wavelength is less than the mouth diameter.
A. Exponential Horns.-The effect of the diameter of the mouth for a
constant flare upon the directional characteristics 20 ,21 of an exponential
horn is depicted in Fig. 2.16. At the side of each polar diagram is the
diameter of a vibrating piston which will yield approximately the same
directional characteristic. It will be seen that up to the frequency at which
the wavelength becomes comparable to the mouth diameter, the directional
characteristics are practically the same as those of a piston of the size of
the mouth. Above this frequency the directional characteristics are
practically independent of the mouth size and appear to be governed
primarily by the flare.
To further illustrate the relative effects of the mouth and flare, Fig. 2.17
shows the effect of different rates of flare, for a constant mouth diameter,
upon the directional characteristics of an exponential horn. These results
also show that, for the wavelengths larger than the mouth diameter, the
directional characteristics are approximately the same as those of a vibrating
piston of the same size as the mouth. Above this frequency the directional
Olson, H. F., RCA Review. Vol. 1. No.4. p. 68. 1937.
Goldman. S .. Jour. Acous. Soc. Amer.• Vol. 5. p. 181. 1934. reports the results of an
investigation upon the directional characteristics of exponential horns at 15.000 and
25.000 cycles. A comparison can be made with the results shown in Figs. 2.16 and 2.17
by increasing the dimensions of the horns used by him to conform with those shown
here and decreasing the frequency by the factor of increase in dimensions. Such a
comparison shows remarkable agreement between the two sets of data.
20
21
48
ACOUSTICAL ENGINEERING
characteristics are broader than those obtained from a piston the size of the
mouth. From another point of view, the diameter of the piston which will
yield the same directional characteristic is smaller than the mouth. These
results also show that the directional characteristics vary very slowly with
frequency at these smaller wavelengths. Referring to Fig. 2.17 it will be
seen that for any particular high frequency, 4000, 7000, or 10,000 cycles per
second, the directional characteristics become progressively sharper as the
rate of flare decreases .
'. ",0
,.
30 . .
:0.IJ"
.
6
•
oe
en -4
Ii'.
45
6 45
°8 I.
. ,
0
.
I.
•
z
If.
545
.4
°
.5
4545
45
5.2
5.2
IS
..
.e
4.2
°
2000""
4000",
7000",
10,000",
FIG. 2.16. The directional characteristics of a group of exponential horns, with a con­
stant flare and throat diameter of t inch as a function of the mouth diameter. The
number at the right of each polar diagram indicates the diameter of a circular piston
which will yield the same directional characteristic. The polar graph depicts the sound
pressure, at a fixed distance. as a function of the angle. The sound pressure for the
angle 0° is arbitrarily chosen as unity. The direction corresponding to 0° is the axis
of the horn. The directional characteristics in three dimensions are surfaces of revolu­
tion about the horn axis.
B. Conical Horns.-In the case of the circular conical horn the direc­
tional pattern should be the same as that of a circular, spherical surface
source. The radius of the spherical surface is the distance along the side
of the horn from the apex to the mouth. The directional characteristics of
two conical horns are shown in Fig. 2.18. At the lower frequencies the
directional pattern is approximately the same as that of a piston of the same
size as the mouth. The directional pattern becomes sharper with an increase
of the frequency. However, at the higher frequencies where the diameter
of the mouth is several wavelengths, the pattern becomes broader as would
be expected from a spherical surface source. The directional characteristics
of a conical horn as depicted in Fig. 2.18 are practically the same as those of
a spherical surface source.
C. Parabolic Horns.-In the parabolic horn the sectional area is pro­
portional to the distance from the apex. This horn may be constructed
49
ACOUSTICAL RADIATING SYSTEMS
FIG. 2.17. The directional characteristics of a group of exponential horns. with a
mouth diameter of 12 inches and a throat diameter of %inch, as a function of the
flare. The number at the right of each polar diagram indicates the diameter of a
circular piston which will yield the same directional characteristic. The polar
graph depicts the sound pressure, at a fixed distance, as a function of the angle.
The sound pressure for the angle 0° is arbitrarily chosen as unity. The direction
corresponding to 0° is the axis of the horn. The directional characteristics in
three dimensions are surfaces of revolution about the horn axis.
~?1"
:.=.=
"~• •
~3~\.5r~~a~t}\~~JO
Sll" _
8
8
,,"
1000'V
. . . .
:~ ~ ~ ~
2000<\.0
4000<\.0
7000<\.0
100001'\.0
FIG. 2.18. The directional characteristics of two conical horns with mouth
diameters of 12 inches and throat diameters of %inch and lengths of 12 inches and
24 inches. The polar graph depicts the sound pressure, at a fixed distance, as a
function of the angle. The sound pressure for the angle 0° is arbitrarily chosen as
unity. The direction corresponding to 0° is the axis of the horn. The directional
characteristics in three dimensions are surfaces of revolution about the horn axis.
as shown in Fig. 2.19 in which two opposite horn walls are parallel and the
other two are inclined at an angle with respect to each other. The direc­
tional characteristics of a 90° parabolic horn are shown in Fig. 2.19. The
50
ACOUSTICAL ENGINEERING
source at the mouth is essentially a curved-line source described in Sec.
2.11. Therefore, the directional characteristics in a plane parallel to the
two parallel sides of the horn should be essentially the same as that of a
90° arc. Comparing Fig. 2.19 with the 90° arc source of Fig. 2.10 it will
be seen that the two directional patterns are quite similar.
From the directional patterns of horn-type radiators described in the
preceding sections, it is evident that a wide range of directional patterns
is possible in simple horns by variations in the shape of the horn and the
mouth opening.
The results of Figs. 2.16, 2.17, 2.18, and 2.19 are applicable to other geo­
metrically similar horns by changing the wavelength (or the reciprocal
of the frequency) in the same ratio as the linear dimensions in accordance
with the principle of similarity of Sec. 1.13.
A commercial application 22 of the principles of the parabolic horn is
shown in Fig. 2.20. The horn is of the exponential rate of flare with straight
sides on two boundaries and curved sides on the other two boundaries.
The directivity patterns in the plane normal to the straight sides are the
same as those of the parabolic horn of Fig. 2.19. The coverage in the
vertical plane can be obtained by using the proper number of horn units as
shown in Fig. 7.18.
2.20. Curved-Surface Source.-A sphere vibrating radially radiates
sound uniformly outward in all directions. A portion of a spherical surface,
large compared to the wavelength and vibrating radially, emits uniform
sound radiation over a solid angle sub tended by the surface at the center
of curvature. To obtain uniform sound distribution over a certain solid
angle, the radial air motion must have the same phase and amplitude over
the spherical surface intercepted by the angle having its center of curvature
at the vertex and the dimensions of the surface must be large compared
to the wavelength. When these conditions are satisfied for all frequencies,
the response characteristic will be independent of the position within the
solid angle.
A loudspeaker 23 ,24,25 consisting of a large number of small horns with
the axis passing through a common point will satisfy, for all practical
purposes, the requirement of uniform phase and amplitude over the spherical
surface formed by the mouths of the horns. A cellular or multihorn of this
type is shown in Fig. 2.21A. This particular horn system consists of fifteen
horns arranged in five vertical rows and three horizontal rows. The mouth
opening of each horn is 8 X 8 inches. The horizontal and vertical angle
between the axis of the individual horn is 17°.
The directional characteristics of a multihorn loudspeaker may be
predicted theoretically 25 from the directional characteristics of an in­
dividual horn and the geometrical configuration of the assembly of horns.
Volkmann, J. E., Unpublished Report.
Wente, E. C., and Thuras, A. L., Jour. A. I. E. E., Vol. 53, No. !, p . 17, 1934.
24 Hilliard, J. K., Tech. Bull. Acad. Res. Council, March, 1936.
25 Olson, H. F., RCA Review, Vol. 1, No.4, p. 68, 1937.
22
23
51
ACOUSTICAL RADIATING SYSTEMS
FRONT
VICW
0
/ ~ ..
~: ~
B~ foilPLA.N
SIDE
VIEW
VIEW
.....
:0:"
a~
° ••
Di§~~3::~90
eo
4400"\1
R;4)"
.
gO
8800'\.1
R ;8>..
,ot::.~~~~j
132001'\J
17600ru
R= 18>..
R :.12>.
FIG. 2.19. The directional characteristics of a parabolic horn of the shape and the
dimensions shown in the sketches on the left. The patterns were obtained in the plane
midway between and parallel to the two parallel sides. The polar graph depicts the
sound pressure, at a fixed distance, as a function of the angle. The sound pressure for
the angle 0° is arbitrarily chosen as unity. The direction corresponding to 0° is spaced
midway between the two nonparallel sides of the horn. R = 12 inches. The ratio of
RIA is also given for comparison with Fig. 2.10.
FRONT VIEW
SIDE VIEW
TOP VIEW
FIG. 2.20. A horn of exponential flare with two
straight sides.
y
A
A,
~----x
z
e,
B
FIG. 2.21. A. A spherical radiating surface consisting of 15 individual
exponential horns. B. Geometry for predicting the directional characteristics
of a cluster of small horns.
52
ACOUSTICAL ENGINEERING
Assume that the point of observation is located on the OY axis, Fig. 2.21B,
at a distance several times the length of the horn. The amplitude of the
vector contributed by an individual horn for the angle cp can be deter­
mined from its individual directional characteristic. In this illustration,
the plane XOZ is chosen as reference plane for the phase of the vector.
The phase angle of the vector associated with an individual horn is
8=
where d
=
,.\ =
~ 360°
2.35
the distance between the center of the mouth of the horn and
the reference plane X'O'Z', in centimeters, and
wavelength, in centimeters.
The vectors, having amplitudes AI, A 2 , A3, A4, etc., determined from the
directional characteristics and having phase angles 81 , 82 , 83 , 84 , etc., de­
termined from equation 2.35, are added vectorially as shown in Fig. 2.21B.
This method of predicting the directional characteristics assumes that
there is no interaction between individual horns which changes the com­
plexion of the velocities at the mouth from that which obtains when
operating an individual horn. Obviously, this condition is not absolutely
satisfied. Apparently, the discrepancy has no practical significance because
it has been found that this method of analysis agrees quite well with experi­
mental results.
The directional characteristics of the cellular horn of Fig. 2.21A are
shown in Figs. 2.22 and 2.23. Above 2000 cycles the dimensions of the
total mouth surface are several wavelengths and the directional character­
istics are fairly uniform and defined by the total angular spread. Where
the dimensions are comparable to the wavelength the directional charac­
teristics become very sharp, as shown by the polar curves for 500 and 1000
cycles. Then, as the dimensions of the surface become smaller than the
wavelength, 250 cycles, the angular spread broadens, as is illustrated by
the larger spread for the smaller vertical dimension when compared to the
smaller spread for the larger horizontal dimension.
The directional characteristics of a cellular horn show a striking resem­
blance to those of an arc of the same angular spread. For example, the
angular spread of the horn of Fig. 2.21 in the plane containing the line
AA' and the axis is 87!0. This may be compared to the arc of Fig. 2.10.
In this case ,.\/4, "12, "\, 2"\, 4"\, and 8"\ will correspond to 145, 290, 580,
1160, 2320, and 4640 cycles. The angular spread in the plane containing
the line BB' and the axis is 52!0. This may be compared to the 60° arc
of Fig. 2.9 with the same relation between the wavelengths and frequencies,
as noted above. It will be seen that there is a marked resemblance between
corresponding frequencies. Of course, there is some variation due to the
fact that the frequencies do not correspond exactly. Further, there is some
difference in the angular spread. For most spherical surfaces of this type
the directional characteristics in various planes correspond very closely to
the directional characteristics of the corresponding arc.
ACOUSTICAL RADIATING SYSTEMS
"B
"...
•
10
1000N
500N
250N
53
"O
30
~4
tt~
eo
60
60
~
•
"
~
8000N
40001V
20001V
00
e
00
.
-
~
eo
•0
~
~
FIG. 2.22. Directional characteristics of the IS-cell cellular horn
shown in Fig. 2.21A in a plane containing the line B-B' and the
axis of the center horn. The polar graph depicts the sound
pressure, at a fixed distance, as a function of the angle. The
sound pressure for the angle 0° is arbitrarily chosen.
E
2SON
o
eo
1.0
~.
4
90
'"
'111.0
z.
30
2000'"
O
~.
~4
w
60
90
'ml.
loo01V
SOON
~
60
90
90
4000'"
'"
60
60
90
90
8000IV
FIG. 2.23. Directional characteristics of the IS-cell cellular horn
shown in Fig. 2.21A in a plane containing the line A-A' and the
axis of the center horn. The polar graph depicts the sound
pressure, at a fixed distance, as a function of the angle. The
sound pressure for the angle 0° is arbitrarily chosen.
2.21. Cone-Surface Source. 26-The directional characteristics 27 of a
paper or felted paper cone used in the direct radiator-type loudspeaker
may be predicted theoretically from the dimensions and shape of the cone
and the velocity of sound propagation in the material. For this type of
analysis the cone is divided into a number of ring-type radiators as shown
in Fig. 2.24. The dimension of the ring along the cone should be a small
Carlisle, R. W., Jour. Acous. Soc. Amer., Vol. 15, No.1, p. 44,1943.
The analysis in this section assumes that there is no reflected wave at the outer
boundary. In order to obtain a uniform response frequency characteristic the reflected
wave must be small. If the reflected wave is small, the effect upon the directional
pattern may be neglected.
26
27
54
ACOUSTICAL ENGINEERING
fraction of the wavelength of sound in the paper. The output of the cone
at any angle is the vector sum of the vectors A o, AI, A2 ... An where the
A's are the amplitudes of the individual rings.
The phase angle of the amplitude of the first ring is
80 = 0
2.36
The phase angle of the amplitude of the second ring is
fh
27T(dl - DI) cos IX
AA
Ap
The phase angle of the amplitude of the third ring is
82
27T(dl
=
2.37
=
+ d2 _
AA
DI
+
D2) cos IX
Ap
2.38
Ao
SECTIONAL
FIG. 2.24.
VIEW
VECTOR
DIAGRAM
Geometry for obtaining the directional pattern of a cone-type radiator.
The phase angle of the amplitude of the nth ring is
2 ... dn DI + D2 ... Dn)
8n -- 27T (dl + dAA
AP
cos IX
where dl, d2 , •••
Dl, D2, ...
2 39
•
axial distances shown in Fig. 2.24 in centimeters, and
distances along the cone shown in Fig. 2.24 in centi­
meters,
AA = wavelength of sound in air, in centimeters,
Ap = wavelength of the sound in the paper cone, in centi­
meters, and
IX = angle between the axis of the cone and the line joining
the observation point and the center of the first ring.
The relative amplitude of the vector A n is given by
=
=
An
where
T"
Dn
AA
IX
=
=
=
=
J0 =
= 27T1'nDnJo
e:
n
sin IX)
2.40
radius of the nth ring, in centimeters,
width of the nth ring along the cone, in centimeters,
wavelength of sound in air, in centimeters,
angle between the axis of the cone and the line joining the
observation point and the center of the cone, and
Bessel function of zero order.
ACOUSTICAL RADIATING SYSTEMS
55
The directional characteristic of the cone is
K=n
R _
2: A
K=n
K
cos (} K
-
j
K=O
2: A
K-O
K-n
a -
2: A
K
sin (} K
2.41
K
K=O
where Ra
=
ratio of the pressure for an angle a to the pressure for an angle
a =
o.
A consideration of equation 2.41 shows that the directional pattern is
a function of the frequency and becomes sharper as the frequency increases.
For a particular frequency, cone angle, and material the directional patterns
are practically similar for the same ratio of cone diameter to wavelength.
For a particular frequency and the same cone material the directional
pattern becomes broader as the cone angle is made larger. For a particu­
lar frequency and cone angle the directional pattern becomes broader as
the velocity of propagation in the material decreases (see Sec. 6.2).
3
MECHANICAL VIB RATING SYSTElVlS
3.1. Introduction.-The preceding chapters have been confined to the
considerations of simple systems, point sources, homogeneous mediums,
and simple harmonic motion. Sources of sound such as strings, bars,
membranes, and plates are particularly liable to vibrate in more than one
mode. In addition, there may be higher frequencies which mayor may not
be harmonics. The vibrations in solid bodies are usually termed as longi­
tudinal, transverse, or torsional. In most cases it is possible to confine
the motion to one of these types of vibrations. For example, the vibrations
of a stretched string are usually considered as transverse. It is also possible
to excite longitudinal vibrations which will be higher in frequency. If the
string is of a fairly large diameter torsional vibrations may be excited.
The vibrations of a body are also affected by the medium in which it is
immersed. Usually, in the consideration of a particular example it is
necessary to make certain assumptions which will simplify the problem.
The mathematical analysis of vibrating bodies is extremely complex and
it is beyond the scope of this book to give a detailed analysis of the various
systems. For complete theoretical considerations, the reader is referred
to the treatises which have been written on this subject. It is the purpose
of this chapter to describe the most common vibrators in use today, to
illustrate the form of the vibrations, and to indicate the resonant frequencies.
3.2. Strin~s.-In all string instruments the transverse and not the
longitudinal vibrations are used. In the transverse vibrations all parts
of the string vibrate in a plane perpendicular to the line of the string. For
the case to be described it is assumed that the mass per unit length is a
constant, that it is perfectly flexible (the stiffness being negligible), and
that it is connected to massive nonyielding supports, Fig. 3.1. Since the
string is fixed at the ends, nodes will occur at these points. The funda­
mental frequency of the string is given by
/= !..J!
2l m
where T = tension, in dynes,
m = mass per unit length, in grams,
l = length of the string, in centimeters.
S6
3.1
57
MECHANICAL VIBRATING SYSTEMS
The shape of the vibration of a string is sinusoidal. In addition to the
fundamental, other modes of vibration may occur, the frequencies being
2, 3, 4, 5, etc., times the fundamental. The first few modes of vibration
of a string are shown in Fig. 3.1. The points which are at rest are termed
nodes and are marked N. The points between the nodes where the ampli­
tude is a maximum are termed antinodes or loops and are marked L.
-.............. N
L
FI RST
fUNDAMENTAL
HARMON IC
•
L
·
·
FIRST
SECOND
THIRD
-....::::::::::---­
SECOND HARMONIC
OVERTONE
..............:---~OVERTONE
THIRD
~
OVERTONE
L
L
HARMONIC
::""""'N
FOURTH
L
•
N
~
HARMONIC
L
L
N~~~~N
•
FOURTH
OVERTONE
fiFTH
HARMONIC
•
N~NL~LN~.NL
~~~
FI frH
OVERTONE
SIXTH
N
~
HARMONIC
FIG. 3.1. Modes of vibration of a stretched string.
and loops are indicated by Nand L.
The nodes
The above example is the simplest form of vibration of a string. A few
of the problems which have been considered by different investigators 1,2,3,4,5
are as follows: nonuniform strings, loaded strings, stiff strings, nonrigid
supports, the effect of damping, and the effect of different types of excitation.
These factors of course alter the form of vibration and the overtones.
3.3. Transverse Vibration of Bars. 1 ,3,4,5-In the preceding section
the perfectly flexible string was considered where the restoring force due
to stiffness is negligible compared to that due to tension. The bar under
no tension is the other limiting case, the restoring force being entirely due
to stiffness. For the cases to be considered it is assumed that the bars
are straight, the cross section is uniform and symmetrical about a central
Rayleigh, " Theory of Sound," Macmillan and Company, London, 1926.
Crandall, " Theory of Vibrating Systems and Sound," D. Van Nostrand Company,
Princeton, N.J., 1926.
3 Wood, " A Text Book of Sound," Bell and Sons, London, 1930.
4 Morse, "Vibration and Sound," McGraw-Hill Book Company, New York, N.Y.,
1936.
5 Lamb, "Dynamical Theory of Sound," E. Arnold, London, 1931.
1
2
S8
ACOUSTICAL ENGINEERING
plane and, as in the case of the string, only the transverse vibrations will
be considered.
A. Bar Clamped at One End.-Consider a bar clamped in a rigid support
at one end with the other end free (Fig. 3.2A). The fundamental frequency
is given by
/I
.S596JQK 2
l2
p
=
3.2
where 1 = length of the bar, in centimeters,
p = density, in grams per cubic centimeter, see Table 1.1,
Q = Young's modulus, in dynes per square centimeter, see Table 1.1:
and
K = radius of gyration.
A
-"""'"
riRST
SECOND
~
D
~
SECOND
~
' .............................,
OVERTONE
SECOND
OVERTONE
OVERTONE
~
,.........
OVERTONE
­
FIRST
OVERTONE
~~
-----------­
~..........
SECOND
SECOND OVERTONE
THIRD
----------­
~
fUNDAMENTAL
~
r--­
~
OVERTONE
F
fUNDAMENTAL
fiRST
.-­
OVERTONE
~,
THIRD
}""'="'"-~~
THIRD
l"'" -
OVERTONE
~
fUNDAMENTAL
fiRST
SECOND
E
L
OVERTONE
~~
OVERTONE
----'="...--........-1
THIRD
THI RD OVERTONE
fiRST
FIRST OVERTONE
~~<=-i:
OVERTONE
.,.........,'"='""'...........,
.........
fUNDAMENTAL
OVERTONE
~ ---==----~/
c
B
I
~
---- .......
THIRD
~
OVERTONE
~
OVERTONE
,.........
~
OVERTONE
FIG. 3.2. Modes of transverse vibrations of bars.
A. A bar clamped at one end and
free at the other. B. A bar clamped at one end and supported at the other. C. A bar
supported at one end and free at the other. D. A bar free at both ends. E. A bar sup­
ported at both ends. F. A bar clamped at both ends.
For a rectangular cross section the radius of gyration is
K=~
v'll
where a = thickness of the bar, in centimeters, in the direction of vibration.
For a circular cross section,
where a
=
radius of the ba.r, in centimeters.
59
MECHANICAL VIERATING SYSTEMS
For a hollow circular cross section,
+ a1 2
2
outside radius of the pipe, in centimeters, and
inside radius of the pipe, in centimeters.
K = Va 2
where a
=
al =
The modes of vibration of a bar clamped at one end are shown in Fig.
3.2A. The table below gives the position of the nodes and the frequencies
of the overtones.
No. of Tone
No. of Nodes
1
0
Distances of Nodes from Free
End in Terms of the Length of
the Bar
Frequencies
as a Ratio of the
Fundamental
11
,
2
1
.2165
3
2
.1321, .4999
17.5511
4
3
.0944, .3558 .. 6439
34.3911
6.26711
It will be seen that the overtones are not harmonics.
The first overtone
of a bar or reed has a higher frequency than the sixth harmonic of a string.
The tuning fork is the most common example of a bar clamped at one end,
because it can be considered to be two vibrating bars clamped at the lower
ends. The overtone or the high-frequency sound of a tuning fork is quickly
damped out leaving almost a pure sound.
B. Bar Free at Both Ends.-Consider a perfectly free bar (Fig. 3.2D).
The fundamental frequency is given by
/1 =
l.1i231T JQ~2
where 1 = length of the bar, in centimeters.
the same as in equation 3.2.
3.3
All the other quantities are
The modes of vibration of a perfectly free bar are shown in Fig. 3.2D.
The table which follows gives the position of the nodes· and the frequencies
of the overtones.
No. of Tone
I No. of Nodes
I
Distances of Nodes from One
End in Terms of the Length of
the Bar
Frequencies as
a Ratio of the
Fundamental
11
1
2
.2242, .7758
2
3
.1321. .50, .8679
2.75611
3
4
.0944, .3558 •. 6442.. 9056
5.40411
4
5
.0734.. 277•. 05, .723, .9266
9.93311
60
ACOUSTICAL ENGINEERING
C. Bar Clamped at Both Ends.-Consider a bar rigidly clamped at both
ends (Fig. 3.2F). The same tones are obtained as in the case of the per­
fectly free bar.
D. Bar Supported at Both Ends.-Consider a bar supported on knife
edges at the two edges at the two ends (Fig. 3.2E). The fundamental
frequency is given by
7T jQK2
/1 = 212
3.4
P
where 1 = length of the bar, in centimeters. All the other quantities are
the same as in equation 3.2.
The overtones are
/2 = 4/1
fa = 9/1
/4 = 16/1 etc.
The nodes are equidistant as in case of the string.
E. Bar Clamped at One End and Supported at the Other.-Consider a
bar clamped at one end and supported at the other end (Fig. 3.2B). The
fundamental frequency is given by
/1
=
2.45jQK2
12
P
3.5
The overtones are
/2
=
fa =
/4 =
3.25/1
6.75/1
11.5/1,
and
Is = 17.7/1
The modes of vibration are shown in Fig. 3.2B.
F. Bar Supported at One End and Free at the Other.-Consider a bar
supported at one end and free at the other (Fig. 3.2C) . The fundamental
frequency is zero. The first overtone is given by
/2
=
2.45jQK2
12
P
3.6
The overtones are
/1
fa =
/4 =
Is =
0
3.25/2
6.75/2
11.5/2,
/6 =
17.7/2
=
and
The modes of vibration are shown in Fig. 3.2C.
G. Tapered Cantilever Bars.-In the preceding, considerations have
been concerned with bars of uniform cross section. It is the purpose of
MECHANICAL VIBRATING SYSTEMS
61
this section to give the formulas for the resonant frequencies of tapered
cantilever bars.
The resonant frequency of a wedge-shaped bar vibrating normal to the
two parallel sides of the wedge, Fig. 3.3A, is
f=
3.7
li!4ji~;
where b = thickness of the bar in the direction of vibration, in centimeters.
SIDE
VIEWS
END VIEWS
FIG. 3.3. Tapered cantilever bars,
that is, bars clamped at one end and
free at the other. A. A wedge-shaped
bar vibrating in a direction normal to
the two parallel sides. B. A wedge­
shaped bar vibrating in a direction
parallel to the two parallel sides.
C. A conical bar.
The resonant frequency of a wedge-shaped bar vibrating parallel to the
two parallel sides of the wedge, Fig. 3.3B, is
f=
'~25ji~;
3.8
The resonant frequency of a conical bar, Fig. 3.3C, is
f= 1.39jQa 2
l2
4p
3.9
where a = radius of the cone at the base, in centimeters.
3.4 Stretched Membranes. 6 ,7,8,9,lO-The ideal membrane is assumed to
be flexible and very thin in cross section, and stretched in all directions by a
Lamb, " Dynamical Theory of Sound," E. Arnold, London, 1931.
Rayleigh, " Theory of Sound," Macmillan and Company, London, 1926.
8 Morse, "Vibration and Sound," McGraw-Hill Book Company, New York, N.Y.,
6
7
1936.
Wood, " A Text Book of Sound," Bell and Sons, London, 1930.
Crandall, " Theory of Vibrating Systems and Sound," D. Van Nostrand Company,
Princeton, N.J., 1926.
9
10
62
ACOUSTICAL ENGINEERING
force which is not affected by the motion of the membrane. Complete
theoretical analyses have been made of circular, square, and rectangular
membranes. For cases of practical interest the membrane is assumed to be
rigidly clamped and stretched by a massive surround. It is the purpose of this
section to consider circular, square, and rectangular stretched membranes.
A. Circular M embrane.-The fundamental frequency of a circular stretched
membrane is given by
/01 =
where m
R
=
=
T =
.382J!
R m
3.10
mass, in grams per square centimeter of area,
radius of the membrane, in centimeters, and
tension, in dynes per centimeter.
FIG. 3.4. Modes of vibration of a stretched circular mem­
brane. Shaded segments are displaced in opposite phase to
unshaded.
The fundamental vibration is with the circumference as a node and a
maximum displacement at the center of the circle (Fig. 3.4A). The fre­
quencies of the next two overtones with nodal circles are
/02 = 2.30/01
/03 = 3.60/01
and are shown in Figs. 3.4B and 3.4C. The frequencies of the first, second
and third overtones with nodal diameters are
/11
=
/21
=
!al
=
1.59/01
2.14/01
2.65/01
MECHANICAL VIBRATING SYSTEMS
63
These nodes are shown in Figs. 3.4D, 3.4E, and 3.4F. Following these
simpler forms of vibration are combinations of nodal circles and nodal
diameters. The frequency of one nodal circle and one nodal diameter,
Fig. 3.4G, is
/12 = 2.92/01
The frequency of one nodal circle and two nodal diameters, Fig. 3.4H, is
/22
=
3.50/
The frequency of two nodal circles and one nodal diameter, Fig. 3.41, is
/13 =
4.22/01
The stretched circular membrane is used in the condenser microphone
(see Sec. 8.2B). The fundamental resonance frequency is placed at the
upper limit of the frequency range. A resistive load is coupled to the
diaphragm for damping the response in the neighborhood of the funda­
mental resonance frequency. This resistance is incorporated in the back
plate which serves as the stationary electrode.
A stretched circular membrane is also used in all types of drums. In
this case the air enclosure as well as the characteristics of the membrane
controls the modes of vibration.
B. Square Membrane.-The fundamental frequency of a square stretched
membrane is given by
/=.705J7:
a
where m
=
a
=
m
3.11
mass, in grams per square centimeter of area,
length of a side, in centimeters, and
T = tension, in dynes per centimeter.
C. Rectangular Membrane.-The fundamental frequency of a rectangu­
lar stretched membrane with the sides in the ratio of 1 to 2 is given by
-
.7~J!
/ - vab
where m
a
m
3.12
mass, in grams per square centimeter,
2b = length of the long side, in centimeters,
length of the short side in centimeters, and
tension, in dynes per centimeter.
3.5. Circular Plates. 1l ,12,13,14,15-The circular plates shown in Fig. 3.5
are assumed to be of uniform cross section and under no tension. I t is
=
=
b=
T =
Rayleigh, "Theory of Sound," Macmillan and Company, London, 1926.
Morse, "Vibration of Sound," McGraw-Hill Book Company, New York, N.Y.,
1936.
13 Wood, " A Text Book of Sound," Bell and Sons, London, 1930.
14 Crandall, " Theory of Vibrating Systems and Sound," D. Van Nostrand Company,
Princeton, N.J., 1926.
IS Lamb, " Dynamical Theory of Sound," E. Arnold, London, 1931.
11
12
64
ACOUSTICAL ENGINEERING
the purpose of this section to consider the vibration of circular plates for
the various support means of Fig. 3.5.
A. Circular Clamped Plate.-Consider a circular clamped plate as shown
in Fig. 3.5A. The fundamental frequency is given by
]01
=
.467tJ
R2
Q
p(1 _ a2)
3.13
where t
R
thickness of the plate, in centimeters,
radius of the plate up to the clamping boundary, in centimeters,
density, in grams per cubic centimeters (see Table 1.1),
Poisson's ratio (see Table 1.1), and
Young's modulus, in dynes per square centimeter (see Table
1.1).
The fundamental frequency is with the circumference as a node and a
maximum displacement at the center (Fig. 3.6A).
The frequency of the next two overtones with nodal circles, Fig. 3.6B
and 3.6C, are,
]02 = 3.91]01
]03 = 8.75]01
=
=
p=
a =
Q=
The frequencies of the first, second, and third overtones with nodal diame­
ters are
]11 = 2.09]01
121 = 3.43]01
!al = 4.95]01
These nodes are shown in Figs. 3.6D, 3.6E, and 3.6F.
Following these simpler forms of vibration are combinations of nodal
circles and nodal diameters. The frequency of one nodal circle and one
nodal diameter, Fig. 3.6G, is
!I2 = 5.98]01
The frequency of one nodal circle and two nodal diameters, Fig. 3.6H, is
122 =
8.74]01
The frequency of two nodal circles and one nodal diameter, Fig. 3.61, is
!I3 =
11.9]01
The clamped plate is used in electromagnetic telephone receivers in which
the steel diaphragm serves as the armature (see Sec. 9.2A). It is used in
carbon microphones (see Sec. 8.2A) . It is used in the subaqueous condenser
microphone (Sec. 15.4) and the magnetic subaqueous loud speaker (sec. 15.6).
Clamped plate diaphragms have been used in miniature condenser micro­
phones. The disadvantage of a plate is the difficulty of mounting a thin
plate to give a small mass per unit area for high sensitivity and still have
sufficient stiffness to yield a high fundamental frequency.
MECHANICAL VIBRATING SYSTEMS
65
!(Yo
~
8
I
CLAMPED EDGE.
C
0
I
SUPPORTED [DGE
0
D
I
SUPPORTED CENTER
§Sj
F"REE
FIG. 3.5. Circular plates. A. A circular plate clamped at the
edge. B. A circular plate supported at the edge. C. A cir­
cular plate supported at the center. D. A free circular plate.
FIG. 3.6. Modes of vibration of a clamped circular plate.
Shaded segments are displaced in opposite phase to unshaded.
In telephone receivers, microphones, and loudspeakers employing a
clamped diaphragm, the effective mass and effective area of the diaphragm,
in terms of the velocity at the center, are needed when the system is reduced
to a lumped element representation. The effective mass or effective area
for this condition is one third of the total mass or total area of the diaphragm.
66
ACOUSTICAL ENGINEERING
The air or water load on the diaphragm can be determined by assuming the
effective radius of the equivalent piston to be .55 times the radius of the
diaphragm (see Sec. 5.8).
B. Circular Free Plate.-Consider a circular plate under no tension,
uniform in cross section and perfectly free (Fig. 3.5D). For a vibration
with nodal circle, as depicted in Fig. 3.4B, the frequency is
f
where t
=
=
p =
a =
R
Q=
=
.412t)
R2
Q
p(1 _ a 2)
3.14
thickness of the plate, in centimeters,
radius of the plate, in centimeters,
density, in grams per cubic centimeter (see Table 1.1),
Poisson's ratio (see Table 1.1), and
Young's modulus, in dynes per square centimeter (see Table
1.1).
For a vibration with two nodal diameters, as depicted in Fig. 3.4E, the
frequency is
.193t)
Q
3.15
f = R2 (pl - a2)
C. Circular Plate Supported at the Center.-Consider a circular plate under
no tension, uniform in cross section, edges perfectly free and supported at
the center (Fig. 3.5C). The frequency, for the umbrella mode, is
f
=
.1ntJ
R2
Q
p(1 _ a2)
3.16
D. Circular Plate Supported at the Outside.-Consider a plate under no
tension, uniform in cross section, edges simply supported at the periphery
(Fig. 3.SB). The fundamental frequency is
f
.233tJ
=
R2
Q
p(1 _
a 2)
3.17
3.6. Longitudinal Vibration of Bars.1 6 ,17,18,19-Consider an entirely
free rod of homogeneous material and constant cross section (see Sec. 1.14).
The simplest mode of longitudinal vibration of a free rod is one in which
a loop occurs at each end and a node in the middle, that is, when the length
of the rod is one-half wavelength. The fundamental frequency of longi­
tudinal vibration of a free rod, Fig. 3.7, may be obtained from equation
1.78 as follows,
3.18
Rayleigh, "Theory of Sound," Macmillan and Company, London, 1926.
Morse, " Vibration and Sound," McGraw-Hill Book Company, New York, N.Y.,
1936.
18 Wood, " A Text Book of Sound," Bell and Sons, London, 1930.
19 Lamb, " Dynamical Theory of Sound," E. Arnold, London, 1931.
16
17
67
MECHANICAL VIERATING SYSTEMS
where l
=
p
=
Q=
length of the rod, in centimeters,
density of the material, in grams per cubic centimeter (see
Table 1.1),
Young's modulus, in dynes per square centimeter (see Table
1.1),
c = velocity of sound, in centimeters per second (see Table 1.1,
and equation 1.78), and
A = wavelength of the sound wave, in centimeters.
The overtones of the free rod are harmonics of the fundamental; that is
2h,fa = 3h,f4 = 4h, etc., Fig. 3.7.
fz =
L
N
------)
fUNDAMeNTAL
+
OVERTONE
FIRST
0=
---
~
::::
N
SECOND
=
HARMONIC
OVERTONE
~
--
SECOND
N
L
:::
L
N
L
N
L
(1 -L
F' I RST
-::
. . --
L
~
THIRD
:;
=~)
HARMONIC
N
L
-- - .- ­ :=: ~)
HARMONIC
FIG. 3.7. Modes of longitudinal vibrations of a free rod.
nodes and loopes are indicated by Nand L.
The
The fundamental resonance frequency occurs when the length of the rod
is one-half wavelength. This fact provides a means of computing the
velocity of sound when the density, Young's modulus, and the frequency
are known, or the frequency of sound when the velocity, density, and
Young's modulus are known.
Rods in which the longitudinal waves are excited by striking the ends
are used as standards of high-frequency sounds, 5000 cycles and above,
where a tuning fork is not very satisfactory.
Longitudinal waves in a rod may be set up by electromagnetic, electro­
static, or magnetostriction means. In the first case, if the rod is of mag­
netic material and is held near an electromagnet in which an alternating
current is flowing a longitudinal force will be set up in the rod. If the
frequency of the driving current is continuously variable, the rod will be
set into violent vibrations at the fundamental resonant frequency. If the
plane end of a rod is placed near a metallic disk, the two plane surfaces
may be used to serve as plates of a condenser. An alternating current sent
through the condenser will cause an alternating force to be exerted upon the
end of the rod. The rod will be sent into violent vibrations when the
frequency of the impressed alternating current corresponds to the funda­
mental frequency or one of the overtones. Magnetization of magnetic
materials produces small changes in the dimensions of these materials.
68
ACOUSTICAL ENGINEERING
A rod of magnetic material placed in a coil of wire will experience a change
in length corresponding to the alterations in the actuating current. If the
coil is part of the circuit of a vacuum tube oscillator the rod will vibrate
and the vacuum tube will oscillate at the fundamental frequency of the rod.
Such a system is termed a magnetostriction sonic, ultrasonic, or supersonic
generator 20 and may be used to produce sound waves in air or any other
medium (see Secs. 15.7 and 15.8).
3.7. Torsional Vibration of Bars. 21 ,22-A solid bar or tube may be
twisted about the axis of the rod in such a manner that each transverse
section remains in its own plane (see Sec. 1.15). If the section is not circular
L
N
(]'J~tt'tttt
...
F"IRST
F"UNDAMENTAL
L
N
(] j
t
t ++
F"IRST
L
•
L
()) t
t +•
SECOND
N
t
L
~)) )~t •• •
+ t
SECOND
OVERTONE
N
HARMONIC
L
N
L
~J~ ~
+ • +
t
N
L
! j ~ , , • + ~})
THIRD
OVERTONE
+, tti))
HARMONIC
HARMONIC
FIG. 3.8. Modes of torsional vibration of a free rod.
nodes and loops are indicated by Nand L.
The
there will be motion parallel to the axis of the bar. Consider an entirely
free rod of homogeneous material and circular cross section. The simplest
or fundamental mode of torsional vibration occurs when there is a node in
the middle and a loop at each end, that is, when the length of the rod is one­
half wavelength. The fundamental resonant frequency, Fig. 3.8, may be
obtained from equation 1.79, as follows
C
C
IJ
11 = X= 2t = 2i
Q
2p(a
+ 1)
3.19
where 1 = length of the rod, in centimeters,
p = density, in grams per cubic centimeter (see Table 1.1),
Q = Young's modulus, in dynes per square centimeter (see Table
1.1),
a = Poisson's ratio (see Table 1.1),
C = velocity of propagation of torsional waves, m centimeters per
second, see equation 1.79, and
A = wavelength of the torsional wave, in centimeters.
Pierce, G. W., Proc. Am. Acad. Arts and Sci., Vol. 63, p. 1, 1928.
Wood, " A Text Book of Sound," Bell and Sons, London, 1930.
22 Rayleigh, "Theory of Sound," Macmillan and Company, London, 1926.
20
21
69
MECHANICAL VIBRATING SYSTEMS
The overtones, as in the case of longitudinal vibrations, are harmonics of
the fundamental. That is, h = 2/1, fa = 3/1, /4 = 4/1, etc. The nodes
and antinodes for the various harmonics are formed as in the case of longi­
tudinal vibrations.
Torsional vibrations may be set up in bars by any means which applies
tangential forces to the free end. From a comparison of the longitudinal
and torsional vibrations in the same bar, Poisson's ratio may be determined.
3.8. Open and Closed Pipes.-The vibrations of a column of gas or
fluid in a cylindrical tube are analogous to the longitudinal vibrations in
a solid bar. For the open pipe there must be a loop of displacement at
the open ends.
The fundamental resonant frequency of a pipe, open at both ends,
Fig. 3.9, is
c
c
3.20
1=X=U
where 1 = length of the pipe, in centimeters,
c = velocity of sound, in centimeters per second (see Table 1.1), and
.:\ = wavelength, in centimeters.
L
FlRST HARMONIC
FUNDAMENTAL
L
N
L
SECOND OVERTONE
SECOND
N
L
FUNDAMENTAL
FIRST HARMONIC
N
L
N
F'I RST OVERTONE
L
L
N
HARMON Ie
L
THIRD
N
L
b
IS:::::
HARMONIC
L
><::
FIRST OVERTONE
N
L
THIRO HARMONIC
3.9. Modes of vibration of the air column in a pipe open at both ends and
in a pipe closed at one end and open at the other end. The velocity nodes and
loops are indicated by Nand L.
FIG.
The overtones of an open pipe are harmonics of the fundamental. That
is,12 = 2/1, fa = 3/1,14 = 4/1, etc.
The fundamental resonant frequency of a pipe closed at one end and
open at the other end, Fig. 3.9, is
c
c
1=>"=41
3.21
The overtones of the pipe closed at one end are the odd harmonics.
That is h = 311, fa = 5!I, etc.
In the above examples the end connection has been omitted. Rayleigh 23
23
Rayleigh, "Theory of Sound," Macmillan and Company, London, 1926.
70
ACOUSTICAL ENGINEERING
shows the added length at the open end to be .82R where R is the radius
of the pipe. If the pipe is terminated in a large flange the end connection
will be that given in Sec. 5.12.
Organ pipes and whistles have been built to cover the range from 16
cycles to 30,000 cycles. The frequency of open and closed pipes may be
computed from the above equations. The sound vibrations in the pipe
are set up by the stream of air which is controlled by the vibration in the
pipe. It is an oscillatory system fed by a direct current of air or gas.
r
4
DYNAMICAL ANALOGIES
4.1. Introduction.-Analogies are useful when it is desired to compare
an unfamiliar system with one that is better known. The relations and
actions are more easily visualized, the mathematics more readily applied,
and the analytical solutions more readily obtained in the familiar system.
Analogies make it possible to extend the line of reasoning into unexplored
fields.
A large part of engineering analysis is concerned with vibrating systems.
Although not generally so considered, the electrical circuit is the most
common example and the most widely exploited vibrating system. The
equations of electrical circuit theory may be based on Maxwell's dynamical
theory in which the currents play the role of velocities. Expressions for
the kinetic energy, potential energy, and dissipation show that network
equations are deducible from general dynamic equations. In other words,
an electrical circuit may be considered to be a vibrating system. This
immediately suggests analogies between electrical circuits and other
dynamical systems as, for example, mechanical and acoustical vibrating
systems.
The equations of motion of mechanical systems were developed a long
time before any attention was given to equations for electrical circuits.
For this reason, in the early days of electrical circuit theory, it was natural
to explain the action in terms of mechanical phenomena. However, at
the present time, electrical circuit theory has been developed to a much
higher state than the corresponding theory of mechanical systems. The
number of engineers and scientists versed in electrical circuit theory is
many times the number equally familiar with mechanical systems.
Almost any work involving mechanical or acoustical systems also includes
electrical systems and electrical circuit theory. The acoustical engineer
is interested in sound reproduction or the conversion of electrical or mechan­
ical energy into acoustical energy, the development of vibrating systems,
and the control of sound vibrations. This involves acoustical, electro­
acoustical, mechanoacoustical, or electromechanoacoustical systems. The
mechanical engineer is interested in the development of various mechanisms
or vibrating systems involving masses, springs, and friction.
71
72
ACOUSTICAL ENGINEERING
Electrical circuit theory is the branch of electromagnetic theory which
deals with electrical oscillations in linear electrical networks'! An electrical
network is a connected set of separate circuits termed branches or meshes.
A circuit may be defined as a physical entity in which varying magnitudes
may be specified in terms of time and a single dimension. 2 The branches
or meshes are composed of elements. Elements are the constituent parts
of a circuit. Electrical elements are resistance, inductance, and capacitance.
Vibrations in one dimension occur in mechanical systems made up of
mechanical elements, as, for example, various assemblies of masses, springs,
and brakes. Confined acoustical systems in which the dimensions are small
compared to the wavelength are vibrations in a single dimension.
The number of independent variables required to completely specify the
motion of every part of a vibrating system is a measure of the number of
degrees of freedom of the system. If only a single variable is needed the
system is said to have a single degree of freedom. In an electrical circuit
the number of degrees of freedom is equal to the number of independent
closed meshes or circuits.
The use of complex notation has been applied extensively to electrical
circuits. Of course, this operational method can be applied to any analyti­
cally similar system.
Mathematically the elements in an electrical network are the coefficients
in the differential equations describing the network. When the electric
circuit theory is based upon Maxwell's dynamics, the network forms a
dynamical system in which the currents play the role of velocities. In the
same way the coefficients in the differential equations of a mechanical or
acoustical system may be looked upon as mechanical or acoustical elements.
Kirchhoff's electromotive force law plays the same role in setting up the
electrical equations as D'Alembert's principle does in setting up the mechan­
ical and acoustical equations. That is to say, every electrical, mechanical,
or acoustical system may be considered as a combination of electrical,
mechanical, or acoustical elements. Therefore, any mechanical or acoustical
system may be reduced to an electrical network and the problem may be
solved by electrical circuit theory.
In view of the tremendous amount of study which has been directed
toward the solution of circuits, particularly electrical circuits, and the
engineer's familiarity with electrical circuits, it is logical to apply this
knowledge to the solution of vibration problems in other fields by the same
theory as that used in the solution of electrical circuits.
It is the purpose of this chapter to develop the analogies between elements 3
in electrical, mechanical, and acoustical systems.
1 The use of the terms" circuit" and" network" in the literature is not established.
The term" circuit" is often used to designate a network with several branches.
2 The term" single dimension" implies that the movement or variation occurs along
a path. In a field problem there is a variation in two or three dimensions.
3 For further considerations of analogies see Olson, " Dynamical Analogies," D . Van
Nostrand Company, Princeton, N.J., 1943.
DYNAMICAL ANALOGIES
73
4.2. Definitions.-A few of the terms used in dynamical analogies will
be defined in this section. Terms not listed below will be defined in sub­
sequent sections.
Abvolt-An abvolt is the unit of electromotive force.
Instantaneous Electromotive Force-The instantaneous electromotive
force between two points is the total instantaneous electromotive force.
The unit is the abvolt.
Effective Electromotive Force-The effective electromotive force is the
root mean square of the instantaneous electromotive force over a complete
cycle between two points. The unit is the abvolt.
Maximum Electromotive Force-The maximum electromotive force for
any given cycle is the maximum absolute value of the instantaneous electro­
motive force during that cycle. The unit is the abvolt.
Peak Electromotive Force-the peak electromotive force for any specified
time interval is the maximum absolute value of the instantaneous electro­
motive force during that interval. The unit is the abvolt.
Dyne-A dyne is the unit of force or mechanomotive force.
Instantaneous Force (Instantaneous Mechanomotive Force)-The in­
stantaneous force at a point is the total instantaneous force. The unit is
the dyne.
Effective Force (Effective Mechanomotive Force)-The effective force
is the root mean square of the instantaneous force over a complete cycle.
The unit is the dyne.
Maximum Force (Maximum Mechanomotive Force)-The maximum
force for any given cycle is the maximum absolute value of the instanta­
neous force during that cycle. The unit is the dyne.
Peak Force (Peak Mechanomotive Force)-The peak force for any
specified interval is the maximum absolute value of the instantaneous
force during that interval. The unit is the dyne.
Dyne Centimeter-A dyne centimeter is the unit of torque or rotato­
motive force.
Instantaneous Torque (Instantaneous Rotatomotive Force)-The in­
stantaneous torque at a point is the total instantaneous torque. The unit
is the dyne centimeter.
Effective Torque (Effective Rotatomotive Force)-The effective torque
is the root mean square of the instantaneous torque over a complete cycle.
The unit is the dyne centimeter.
Maximum Torque (Maximum Rotatomotive Force)-The maximum
torque for any given cycle is the maximum absolute value of the instanta­
neous torque during that cycle. The unit is the dyne centimeter.
Peak Torque (Peak Rotatomotive Force)-The peak torque for a speci­
fied interval is the maximum absolute value of the instantaneous torque
during that interval. The unit is the dyne centimeter.
Dyne per Square Centimeter-A dyne per square centimeter is the unit
of sound pressure.
Static Pressure-The static pressure is the pressure that would exist
74
ACOUSTICAL ENGINEERING
in a medium with no sound waves present. The unit is the dyne per square
centimeter.
Instantaneous Sound Pressure (Instantaneous Acoustomotive Force)­
The instantaneous sound pressure at a point is the total instantaneous
pressure at the point minus the static pressure. The unit is the dyne per
square centimeter.
Effective Sound Pressure (Effective Acoustomotive Force)-The effective
sound pressure at a point is the root mean square value of the instantaneous
sound pressure over a complete cycle at the point. The unit is the dyne
per square centimeter.
Maximum Sound Pressure (Maximum Acoustomotive Force)-The
maximum sound pressure for any given cycle is the maximum absolute value
of the instantaneous sound pressure during that cycle. The unit is the
dyne per square centimeter.
Peak Sound Pressure (Maximum Acoustomotive Force)-The peak sound
pressure for any specified time interval is the maximum absolute value of
the instantaneous sound pressure in that interval. The unit is the dyne per
square centimeter.
Abampere-An abampere is the unit of current.
Instantaneous Current-The instantaneous current at a point is the total
instantaneous current at that point. The unit is the abampere.
Effective Current-The effective current at a point is the root mean
square value of the instantaneous current over a complete cycle at that
point. The unit is the abampere.
Maximum Current-The maximum current for any given cycle is the
maximum absolute value of the instantaneous current during that cycle.
The unit is the abampere.
Peak Current-The peak current for any specified time interval is the
maximum absolute value of the instantaneous current in that interval.
The unit is the abampere.
Centimeter per Second-A centimeter per second is the unit of velocity.
Instantaneous Velocity-The instantaneous velocity at a point is the total
instantaneous velocity at that point. The unit is the centimeter per second.
Effective Velocity-The effective velocity at a point is the root mean
square value of the instantaneous velocity over a complete cycle at that
point. The unit is the centimeter per second.
Maximum Velocity-The maximum velocity for any given cycle is the
maximum absolute value of the instantaneous velocity during that cycle.
The unit is the centimeter per second.
Peak Velocity-The peak velocity for any specified time interval is the
maximum absolute value of the instantaneous velocity in that interval.
The unit is the centimeter per second.
Radian per Second-A radian per second is the unit of angular velocity.
Instantaneous Angular Velocity-The instantaneous angular velocity
at a point is the total instantaneous angular velocity at that point. The
unit is the radian per second.
DYNAMICAL ANALOGIES
75
Effective Angular Velocity-The effective angular velocity at a point
is the root mean square value of the instantaneous angular velocity over
a complete cycle at that point. The unit is the radian per second.
Maximum Angular Velocity-The maximum angular velocity for any
given cycle is the maximum absolute value of the instantaneous angular
velocity during that cycle. The unit is the radian per second.
Peak Angular Velocity-The peak angular velocity for any specified
time interval is the maximum absolute value of the instantaneous angular
velocity in that interval. The unit is the radian per second.
Cubic Centimeter per Second-A cubic centimeter per second is the unit
of volume current.
Instantaneous Volume Current-The instantaneous volume current
at a point is the total instantaneous volume current at that point. The
unit is the cubic centimeter per second.
Effective Volume Current-The effective volume current at a point is
the root mean square value of the instantaneous volume current over a
complete cycle at that point. The unit is the cubic centimeter per second.
Maximum Volume Current-The maximum volume current for any
given cycle is the maximum absolute value of the instantaneous volume
current during that cycle. The unit is the cubic centimeter per second.
Peak Volume Current-The peak volume current for any specified time
interval is the maximum absolute value of the instantaneous volume current
in that interval. The unit is the cubic centimeter per second.
Electrical Impedance-Electrical impedance is the complex quotient
of the alternating electromotive force applied to the system by the resulting
current. The unit is the abohm.
Electrical Resistance-Electrical resistance is the real part of the elec­
trical impedance. This is the part responsible for the dissipation of energy.
The unit is the abohm.
Electrical Reactance-Electrical reactance is the imaginary part of the
electrical impedance. The unit is the abohm.
Inductance-Inductance in an electrical system is that coefficient which,
when multiplied by 277 times the frequency, gives the positive imaginary
part of the electrical impedance. The unit is the abhenry.
Electrical Capacitance-Electrical capacitance in an electrical system
is that coefficient which, when multiplied by 277 times the frequency, is
the reciprocal of the negative imaginary part of the electrical impedance.
The unit is the abfarad.
Mechanical Rectilineal Impedance 4 (Mechanical Impedance)-Mechan­
ical rectilineal impedance is the complex quotient of the alternating force
4 The word" mechanical" is ordinarily used as a modifier to designate a mechanical
system with rectilineal displacements and the word .. rotational" is ordinarily used as
a modifier to designate a mechanical system with rotational displacements. To avoid
ambiguity in this book, where both systems are considered concurrently, the words
" mechanical rectilineal " are used as modifiers to designate a mechanical system with
rectilineal displacements and the words .. mechanical rotational " are used as modifiers
to designate a mechanical system with rotational displacements.
76
ACOUSTICAL ENGINEERING
applied to the system by the resulting linear velocity in the direction of the
force at its point of application. The unit is the mechanical ohm.
Mechanical Rectilineal Resistance (Mechanical Resistance)-Mechanical
rectilineal resistance is the real part of the mechanical rectilineal impedance.
This is the part responsible for the dissipation of energy. The unit is the
mechanical ohm.
Mechanical Rectilineal Reactance (Mechanical Reactance)-Mechanical
rectilineal reactance is the imaginary part of the mechanical rectilineal
impedance. The unit is the mechanical ohm.
Mass-Mass in a mechanical system is that coefficient which, when
multiplied by 21T times the frequency, gives the positive imaginary part of
the mechanical rectilineal impedance. The unit is the gram.
Compliance-Compliance in a mechanical system is that coefficient which,
when multiplied by 21T times the frequency, is the reciprocal of the negative
imaginary part of the mechanical rectilineal impedance. The unit is the
centimeter per dyne.
Mechanical Rotational Impedance (Rotational Impedance)-Mechanical
rotational impedance is the complex quotient of the alternating torque
applied to the system by the resulting angular velocity in the direction of
the torque at its point of application. The unit is the rotational ohm.
Mechanical Rotational Resistance (Rotational Resistance)-Mechanical
rotational resistance is the real part of the mechanical rotational impedance.
This is the part responsible for the dissipation of energy. The unit is the
rotational ohm.
Mechanical Rotational Reactance (Rotational Reactance)-Mechanical
rotational reactance is the imaginary part of the mechanical rotational
impedance. The unit is the rotational ohm.
Moment of Inertia-Moment of inertia in a mechanical rotational system
is that coefficient which, when multiplied by 21T times the frequency, gives
the positive imaginary part of the mechanical rotational impedance. The
unit is the gram centimeter to the second power.
Rotational Compliance-Rotational compliance in a mechanical rota­
tional system is that coefficient which, when multiplied by 21T times the
frequency, is the reciprocal of the negative imaginary part of the
mechanical rotational impedance. The unit is the radian per centimeter
per dyne.
Acoustical Impedance-Acoustical impedance is the complex quotient
of the alternating pressure applied to the system by the resulting volume
current. The unit is the acoustical ohm.
Acoustical Resistance-Acoustical resistance is the real part of the
acoustical impedance. This is the part responsible for the dissipation of
energy. The unit is the acoustical ohm.
Acoustical Reactance-Acoustical reactance is the imaginary part of the
acoustical impedance. The unit is the acoustical ohm.
Inertance-Inertance in an acoustical system is that coefficient which,
when multiplied by 21T times the frequency, gives the positive imaginary
DYNAMICAL ANALOGIES
77
part of the acoustical impedance. The unit is the gram per centimeter
to the fourth power.
Acoustical Capacitance-Acoustical capacitance in an acoustical system
is that coefficient which, when multiplied by 2rr times the frequency, is the
reciprocal negative imaginary part of the acoustical impedance. The unit
is the centimeter to the fifth power per dyne.
Element-An element or circuit parameter in an electrical system defines
a distinct activity in its part of the circuit. In the same way, an element
in a mechanical rectilineal, mechanical rotational, or acoustical system
defines a distinct activity in its part of the system. The elements in an
electrical circuit are electrical resistance, inductance, and electrical capaci­
tance. The elements in a mechanical rectilineal system are mechanical
rectilineal resistance, mass, and compliance. The elements in a mechanical
rotational system are mechanical rotational resistance, moment of inertia,
and rotational compliance. The elements in an acoustical system are
acoustical resistance, inertance, and acoustical capacitance.
Electrical System-An electrical system is a system adapted for the
transmission of electrical currents consisting of one or all of the electrical
elements: electrical resistance, inductance, and electrical capacitance.
Mechanical Rectilineal System-A mechanical rectilineal system is a
system adapted for the transmission of vibrations consisting of one or all
of the following mechanical rectilineal elements: mechanical rectilineal
resistance, mass, and compliance.
Mechanical Rotational System-A mechanical rotational system is a
system adapted for the transmission of rotational vibrations consisting of
one or all of the following mechanical rotational elements: mechanical
rotational resistance, moment of inertia, and rotational compliance.
Acoustical System-An acoustical system is a system adapted for the
transmission of sound consisting of one or all of the following acoustical
elements: acoustical resistance, inertance, and acoustical capacitance.
Electrical Abohm-An electrical resistance, reactance, or impedance is
said to have a magnitude of one abohm when an electromotive force of one
abvolt produces a current of one abampere.
Mechanical Ohm-A mechanical rectilineal resistance, reactance, or
impedance is said to have a magnitude of one mechanical ohm when a force
of one dyne produces a velocity of one centimeter per second.
Rotational Ohm-A mechanical rotational resistance, reactance, or
impedance is said to have a magnitude of one rotational ohm when a torque
of one dyne centimeter produces an angular velocity of one radian per second.
Acoustical Ohm-An acoustical resistance, reactance, or impedance is
said to have a magnitude of one acoustical ohm when a pressure of one
dyne per square centimeter produces a volume current of one cubic centi­
meter per second.
4.3. Elements. 5-An element or circuit parameter in an electrical system
5 The symbols used in this book conform with American Standards Association,
"Letter Symbols for Acoustics," Yl0.11-1953.
ACOUSTICAL ENGINEERING
78
defines a distinctive activity in its part of the circuit. In an electrical system
these elements are resistance, inductance, and capacitance. They are dis­
tinguished from the devices: resistor, inductor, and capacitor. A resistor,
inductor, or capacitor idealized to have only resistance, inductance, and
capacitance is a circuit element. As indicated in the preceding chapter,
the study of mechanical and acoustical systems is facilitated by the intro­
duction of elements analogous to the elements of an electric circuit. In
this procedure, the first step is to develop the elements in these vibrating
systems. It is the purpose of this chapter to define and describe electrical,
mechanical rectilineal, mechanical rotational, and acoustical elements.
4.4. Resistance.-A. Electrical Resistance.-Electrical energy is changed
into heat by the passage of an electrical current through a resistance.
Energy is lost by the system when a charge q of electricity is driven through
a resistance by a voltage e. Resistance is the circuit element which causes
dissipation.
Electrical resistance rE, in abohms, is defined as
4.1
where e
i
voltage across the electrical resistance, in abvolts, and
current through the electrical resistance, in abamperes.
=
=
Equation 4.1 states that the electromotive force across an electrical
resistance is proportional to the electrical resistance and the current.
B. Mechanical Rectilineal Resistance.-Mechanical rectilineal energy
is changed into heat by a rectilinear motion which is opposed by linear
resistance (friction). In a mechanical system dissipation is due to friction.
Energy is lost by the system when a mechanical rectilineal resistance is
displaced a distance x by a force 1M.
Mechanical rectilineal resistance (termed mechanical resistance) rM,
in mechanical ohms, is defined as
4.2
where 1M
u
=
=
applied mechanical force, in dynes, and
velocity at the point of application of the force, in centimeters
per second.
Equation 4.2 states that the driving force applied to a mechanical recti­
lineal resistance is proportional to the mechanical rectilineal resistance
and the linear velocity.
C. Mechanical Rotational Resistance.-Mechanical rotational energy
is changed into heat by a rotational motion which is opposed by a rotational
resistance (rotational friction) . Energy is lost by the system when a
mechanical rotational resistance is displaced by an angle cp by a torque 1R.
DYNAMICAL ANALOGIES
79
Mechanical rotational resistance (termed rotational resistance) rR, in
rotational ohms, is defined as
JR
rR=O
where JR
=
() =
4.3
applied torque, in dyne centimeters, and
angular velocity at the point of application about the axis,
in radians per second.
Equation 4.3 states that the driving torque applied to a mechanical
rotational resistance is proportional to the mechanical rotational resistance
and the angular velocity.
D. Acoustical Resistance.-In an acoustical system dissipation may be
due to the fluid resistance or radiation resistance. At this point the former
type of acoustical resistance will be considered. Acoustical energy is
changed into heat by the passage of a fluid through an acoustical resistance.
The resistance is due to viscosity. Energy is lost by the system when a
volume X of a fluid or gas is driven through an acoustical resistance by a
pressure p.
Acoustical resistance r A, in acoustical ohms, is defined as
rA =
1.u
4.4
where p = pressure, in dynes per square centimeter, and
U = volume current, in cubic centimeters per second.
Equation 4.4 states that the driving pressure applied to an acoustical
resistance is proportional to the acoustical resistance and the volume current.
4.5. Inductance, Mass, Moment of Inertia, Inertance.-A. Inductance.
-Electromagnetic energy is associated with inductance. Electromagnetic
energy increases as the current in the inductance increases. I t decreases
when the current decreases. It remains constant when the current in
the inductance is a constant. Inductance is the electrical circuit element
which opposes a change in current. Inductance L, in abhenries, is defined
as
4.5
where
e = electromotive or driving force, in abvolts, and
di/dt = rate of change of current, in abamperes per second.
Equation 4.5 states that the electromotive force across an inductance is
proportional to the inductance and the rate of change of current.
B. Mass.-Mechanical rectilineal inertial energy is associated with
mass in the mechanical rectilineal system. Mechanical rectilineal energy
increases as the linear velocity of a mass increases, that is, during linear
acceleration. It decreases when the velocity decreases. It remains
80
ACOUSTICAL ENGINEERING
constant when the velocity is constant. Mass is the mechanical element
which opposes a change of velocity. Mass m, in grams, is defined as
JM
where du /dt
du
= m dt
4.6
acceleration, in centimeters per second per second, and
=
JM = driving force, in dynes.
Equation 4.6 states that the driving force applied to the mass is pro­
portional to the mass and the rate of change of velocity.
C. Moment oj Inertia.-Mechanical rotational inertial energy is associated
with moment of inertia in the mechanical rotational system. Mechanical
rotational energy increases as the angular velocity of a moment of inertia
increases, that is, during angular acceleration. It decreases when the angular
velocity decreases. I t remains a constant when the angular velocity is a con­
stant. Moment of inertia is the rotational element which opposes a change
in angular velocity. Moment of inertia I, in gram (centimeter)4, is given by
!R = I~~
where dO/dt
4.7
angular acceleration, in radians per second per second, and
torque, in dyne centimeters.
Equation 4.7 states that the driving torque applied to the moment of
inertia is proportional to the moment of inertia and the rate of change of
angular velocity.
D. Inertance.-Acoustical inertial energy is associated with inertance
in the acoustical system. Acoustical energy increases as the volume current
of an inertance increases. It decreases when the volume current decreases.
It remains constant when the volume current of the inertance is a constant.
Inertance is the acoustical element that opposes a change in volume current.
Inertance M, in grams per (centimeter)4, is defined as
=
JR =
p = Mdd~
where
M
dU/dt
4.8
inertance, in grams per (centimeter)4,
rate of change of volume current, in cubic centimeters
per second per second, and
p = driving pressure, in dynes per square centimeter.
Equation 4.8 states that the driving pressure applied to an inertance is
proportional to the inertance and the rate of change of volume current.
Inertance 6 may be expressed as
m
M = 52
4.9
where m
S
6
=
=
=
=
mass, in grams,
cross-sectional area in square centimeters, over which the
driving pressure acts to drive the mass.
See Sec. 5.6.
DYNAMICAL ANALOGIES
81
The inertance of a circular tube is
M=L
7TR2
4.10
where R = radius of the tube, in centimeters,
1 = effective length of the tube, that is, length plus end correction,
in centimeters, and
p = density of the medium in the tube, in grams per cubic centi­
meter.
4.6. Electrical Capacitance, Rectilineal Compliance, Rotation Com­
pliance, Acoustical Capacitance.-A. Electrical Capacitance.-Electro­
static energy is associated with the separation of positive and negative
charges, as in the case of the charges on the two plates of an electrical
capacitance. Electrostatic energy increases as the charges of opposite
polarity are separated. It is constant and stored when the charges remain
unchanged. It decreases as the charges are brought together and the
electrostatic energy released. Electrical capacitance is the electrical cir­
cuit element which opposes a change in voltage. Electrical capacitance
CE, in abfarads, is defined as
.
1
=
C de
E dt
4.11
Equation 4.11 may be written
e
=
C1E
f'd t
1
=
q
CE
4.12
where q = charge on electrical capacitance, in abcoulombs, and
e = electromotive force, in abvolts.
Equation 4.12 states that the charge on an electrical capacitance is pro­
portional to the electrical capacitance and the applied electromotive force.
B. Rectilineal Compliance.-Mechanical rectilineal potential energy is
associated with the compression of a spring or compliant element. Mechan­
ical energy increases as the spring is compressed. It decreases as the
spring is allowed to expand. It is a constant, and is stored, when the spring
remains immovably compressed. Rectilineal compliance is the mechanical
element which opposes a change in the applied force. Rectilineal compliance
CM (termed compliance) is defined as
4.13
where x
=
1M =
displacement, in centimeters, and
applied force, in dynes.
Equation 4.13 states that the displacement of a compliance is propor­
tional to the compliance and the applied force.
Stiffness is the reciprocal of compliance.
82
ACOUSTICAL ENGINEERING
C. Rotational Compliance.-Mechanical rotational potential energy is
associated with the twisting of a spring or compliant element. Mechanical
energy increases as the spring is twisted. It decreases as the spring is
allowed to unwind. It is constant, and is stored when the spring remains
immovably twisted. Rotational compliance is the mechanical element
which opposes a change in the applied torque. Rotational compliance
CR, or moment of compliance, is defined as
/R=i...
C
4.14
R
where c/> = angular displacement, in radians, and
JR = applied torque, in dyne centimeters.
Equation 4.14 states that the rotational displacement of the rotational
compliance is proportional to the rotational compliance and the applied
torque.
D. Acoustical Capacitance.-Acoustical potential energy is associated
with the compression of a fluid or gas. Acoustical energy increases as the
gas is compressed. It decreases as the gas is allowed to expand. It is
constant, and is stored when the gas remains immovably compressed.
Acoustical capacitance is the acoustic element which opposes a change in
the applied pressure. The pressure, in dynes per square centimeter, in
terms of the condensation, is from equation 1.21
p=
c2 ps
4.15
where c = velocity, in centimeters per second,
p = density, in grams per cubic centimeter, and
s = condensation, defined in equation 4.16.
The condensation in a volume V due to a change in volume from V to
V'is
V-V'
s=
4.16
V
The change in volume V - V', in cubic centimeters,
volume displacement, in cubic centimeters.
V - V' = X
IS
equal to the
4.17
where X = volume displacement, in cubic centimeters.
From equations 4.15,4.16, and 4.17, the pressure is
2
P=~X
V
4.18
Acoustical capacitance C A is defined as
X
p=­
C
A
4.19
83
DYNAMICAL ANALOGIES
where p
X
sound pressure in dynes per square centimeter, and
volume displacement, in cubic centimeters.
=
=
Equation 4.19 states the volume displacement in an acoustical capaci­
tance is proportional to the pressure and the acoustical capacitance.
From equations 4.18 and 4.19 the acoustical capacitance of a volume is
V
4.20
CA=­
2
pc
where V
volume, in cubic centimeters.
=
rE
rA
-.NvVV­
~
nIffff!r
IZiZz::II
IZZZZZI
~
cA
cE
~
D
CN
:J
~f-
I
m
M
L
ELECTRICAL
TR
fN
IZZZZZI
ACOUSTICAL
=-=
•
CR
RECTILINfAL
ROTATIONAL
MECHANICAL
4.1. Graphical representation of the three basic elements in
electrical, mechanical rectilineal, mechanical rotational, and acousti­
cal systems.
FIG.
I'E
=
electrical resis­
tance
mechanical ro­
tational resis­
tance
m . . mass
Yli •.
CA = acoustical ca­
pacitance
I'A
= acoustical re­
sistance
L = inductance
I = moment of
CM
inertia
= compliance
I'M
=
M =
mechanical rec­
tilineal resis­
tance
inertance
C E = electrical capa­
citance
C R = rotational com­
pliance
4.7. Representation of Electrical, Mechanical Rectilineal, Mechan­
ical Rotational, and Acoustical Elements.-Electrical, mechanical
rectilineal, mechanical rotational, and acoustical elements have been defined
in the preceding sections. Fig. 4.1 illustrates schematically the four elements
in each of the four systems.
The electrical elements, electrical resistance, inductance, and electrical
capacitance are represented by the conventional symbols.
Mechanical rectilineal resistance is represented by sliding friction which
causes dissipation. Mechanical rotational resistance is represented by a
84
ACOUSTICAL ENGINEERING
wheel with a sliding friction brake which causes dissipation. Acoustical
resistance is represented by narrow slits which cause dissipation due to
viscosity when fluid is forced through the slits. These elements are
analogous to electrical resistance in the electrical system.
Inertia in the mechanical rectilineal system is represented by a mass.
Moment of inertia in the mechanical rotational system is represented by a
flywheel. Inertance in the acoustical system is represented as the fluid
contained in a tube in which all the particles move with the same phase
when actuated by a force due to pressure. These elements are analogous
to inductance in the electrical system.
Compliance in the mechanical rectilineal system is represented as a spring.
Rotational compliance in the mechanical rotational system is represented
as a spring. Acoustical capacitance in the acoustical system is represented
as a volume which acts as a stiffness or spring element. These elements are
analogous to electrical capacitance in the electrical system.
In the preceding discussion of electrical, mechanical rectilineal, mechanical
rotational, and acoustical systems it was observed that the four systems are
analogous. As pointed out in the introduction, using the dynamical concept
for flow of electrical currents in electrical circuits the fundamental laws are
of the same nature as those which govern the dynamics of a moving body.
In general, the three fundamental dimensions are mass, length, and time.
These quantities are directly connected to the mechanical rectilineal system.
Other quantities in the mechanical rectilineal system may be derived in
terms of these dimensions. In terms of analogies the dimensions in the
electrical circuit corresponding to length, mass, and time in the mechanical
rectilineal system are charge, self-inductance, and time. The corresponding
analogous dimensions in the rotational mechanical system are angular
displacement, moment of inertia, and time. The corresponding analogous
dimensions in the acoustical system are volume displacement, inertance, and
time. The above-mentioned fundamental dimensions in each of the four
systems are shown in tabular form in Table 4.1. Other quantities in each
of the four systems may be expressed in terms of the dimensions of Table 4.1.
A few of the most important quantities have been tabulated in Table 4.2.
Tables 4.1 and 4.2 depict analogous quantities in each of the four systems.
Further, they show that the four systems are dynamically analogous.
The dimensions given in Table 4.1 should not be confused with the classical
dimensions of electrical, mechanical, and acoustical systems given in Table
4.3. Table 4.3 uses mass M, length L, and time T. In the case of
electrical units dielectric and permeability constants are assumed to be
dimensionless.
For further considerations of dynamical analogies, as, for example,
electrical, mechanical rectilineal, mechanical rotational, and acoustical
systems of one, two, and three degrees of freedom, corrective networks,
wave filters, transients, driving systems, generating systems, theorems and
applications, the reader is referred to Olson, "Dynamical Analogies," D.
Van Nostrand Company, Princeton, N.]., 1943.
I
Time
Lt-I
L -lt2
CE
WE
PE
Electrical
Capacitance
Energy
Power
Compliance
Mechanical
Resistance
I Lq2t-a
Power
Lq 2t- 2 Energy
-----
rE
Force
Lqt-2
Electrical
Resistance
i
.
Linear
Velocity
e
__
I
I
t
x
CM
rM
JM
.:i: orv
Symbol
PM
I
-
Quantity
Rotational
Compliance
m-1t 2
Power
Energy
Rotational
Resistance
mt- I
mx 2t- 3
I
Sy
~I
PB
I
CR
rB
JB
~ or IJ
---
Symbol
1
1
D
Mechanical Rotation
Angular
Velocity
I
4.2
Time
Angular Displacement
mxt- 2 Torque
xt-1
Dimension
----mx 2t- 2
WM
I
Quantity
Mechanical Rotationa
Moment of Inertia
4.1
TABLE
I SY:bOI
Mechanical Rectilineal
Quantity
iTime
Linear Displacement
IMass
Quantity
Mechanical Rectilineal
qt- 1
sion
Electromotive
Force
i
I Dimen-
i
Symbol
Current
Quantity
Electrical
q
I
Electrical Charge
t
L
I
I Symbol
I
Self-Inductance
Quantity
Electrical
TABLE
86
ACOUSTICAL ENGINEERING
TABLE
4.3
Electrical
Quantity
Electromotive Force
Mechanical Rectilineal
Isym- Dimension
bol
Unit
Volts x 10 8
I
e
Ml/2L3/2
T-2
Quantity
Unit
Force
SymDimension
bol
--
Dynes
JM
MLT-2
-Charge or
Quantity
Mlj2Vj2 Linear Dis- Centimeters
placement
Coulombs x 10-1
q
10-1
i
Mlj2Vj2
T-l
ZE
LT- l
x
L
---Current
Electrical
Impedance
Amperes
Ohms
X
X
10 9
Linear
Velocity
Centimeters i or v
per Second
Mechanical Mechanical
Impedance
Ohms
ZM
LT-l
MT-l
-Electrical
Resistance
Ohms
X
10 9
1'E
LT-l
Mechanical Mechanical
Resistance
Ohms
1'M
MT-l
Mechanical Mechanical
Reactance
Ohms
XM
MT-l
Mass
m
M
- - -----
Electrical
Reactance
XE
LT-l
10 9
L
L
10- 9
CE
L-IT2
Ergs per Second
IPE
Ohms
Inductance
Henries
Electrical
Capacitance
Farads
Power
X
X
X
109
ML2T-3
Grams
Compliance Centimeters CM
per Dyne
Power
Ergs per
Second
PM
M-IT2
ML2T-3
87
DYNAMICAL ANALOGIES
TABLE
4.3-Continued.
Acoustical
Mechanical Rotational
Quantity
Unit
Quantity
Isym
bol - Dimension
Unit
- ­ - - ­- -
---­
Torque
Angular
Displacement
Dyne
Centimeter
Radians
Dynes per
Square
Centimeter
p
ML-IT-2
Volume Dis- Cubic Cen­
placement timeters
X
L3
ML2T-2
JR
1
'"
Symbol Dimension
Pressure
-Volume
Current
Cubic Centimeter per
Second
X
or
Angular
Velocity
Radians per
Second
~ or 8
T-l
Rotational
Impedance
Rotational
Ohms
ZR
ML2T-l
Acoustical Acoustical
Impedance
Ohms
ZA
ML-4T-l
ML2T-l
Acoustical Acoustical
Resistance
Ohms
rA
ML-4T-l
---
-----
XA
ML-4T-l
U
L3T-l
-Rotational
Resistance
Rotational
Ohms
rR
----Rotational
Reactance
Rotational
Ohms
XR
Acoustical Acoustical
Reactance
Ohms
ML2T-l
I
Moment of
Inertia
(Gram) (Centimeter) 2
I
ML2
-Inertance
Grams per
(Centimeter) 4
Power
ML-4
--
-Rotational
Compliance
M
Radians per
Dyne per
Centimeter
CR
M-IL-2T2
Acoustical
Capacitance
(Centime­
ter)5 per
Dyne
CA
M-IL4T2
Ergs per
Second
PR
I ML2T-3
Power
Ergs per
Second
PA
I ML2T-3
5
ACOUSTICAL ELEMENTS
5.1. Introduction.-The preceding chapter is devoted to analogies be­
tween electrical, mechanical, and acoustical systems. The purpose of draw­
ing these analogies is to facilitate the solution of problems in mechanical
and acoustical vibrating systems by converting these problems into the
corresponding electrical analogies and solving the resultant electrical
circuits by conventional electrical circuit theory. An electrical circuit is
composed of electrical elements. In the same way the acoustical system
is composed of acoustical elements. The type of element, that is, acousti­
cal resistance, inertance, or acoustical capacitance, will depend upon the
characteristic manner in which the medium behaves for different sources of
sound and in the different ways of confining the medium. It is the purpose
of this chapter to consider acoustical elements and combination of elements.
5.2. Acoustical Resistance.-Acoustical resistance may be obtained by
forcing air through a small hole. The resistance is due to viscosity which
may be considered as friction between adjacent layers of air. In the
ordinary transmission of sound in a large tube the motion of all the particles
in a plane normal to the axis is the same, therefore the frictional losses are
small. When sound travels in a small tube the particle velocity varies
from zero at the boundary to a maximum at the center. The same is true
when a steady stream of air is forced through a small hole or tube, the
velocity of adjacent layers varies from zero at the boundary to a maximum
at the center. The smaller the hole the higher will be the resistance because
of the greater effect of the sides.
A small tube also has inertance. Therefore, the reactive component
increases with frequency. The inertive reactance increases as the size of
the hole decreases as does the acoustical resistance, but at a slower rate.
Therefore, the inertive reactance may be made negligible compared to the
acoustical resistance if the hole is made sufficiently small.
Acoustical resistance employing viscosity may be made in various forms
as, for example, a large number of small holes or a large number of slits.
The acoustical impedance of fine holes and slits will be considered in the
next two sections.
5.3. Acoustical Impedance of a Tube of Small Diameter.- The trans­
mission of sound waves or direct currents of air in a small tube is influenced
88
ACOUSTICAL ELEMENTS
89
by acoustical resistance due to viscosity. The diameter is assumed to be
small compared to the length so that the end correction may be neglected.
The length is assumed to be small compared to the wavelength.
The acoustical impedance, in acoustical ohms, of a small-diameter
tube1 ,2,3 is given by
5.1
where R
=
fL =
w =
I=
p=
radius of the tube, in centimeters,
viscosity coefficient, 1.86 X 10-4 for air,
27Tj,J = frequency, in cycles per second,
length of the tube, in centimeters, and
density, in grams per cubic centimeter.
The effect of viscosity is to introduce acoustical resistance in the form of
dissipation as well as to add to the acoustical reactance.
The acoustical resistance of a single hole is ordinarily much too high.
The desired acoustical resistance may be obtained by using a sufficient
number of holes.
5.4. Acoustical Impedance of a Narrow Slit.-A narrow slit acts in a
manner quite similar to the narrow tube. The length is assumed to be
small compared to the wavelength. The thickness is assumed to be small
compared to the length.
The acoustical impedance, in acoustical ohms, of a narrow slit 4 ,5,6 is
given by
5.2
where fL
=
p
=
d=
viscosity coefficient, 1.86 X 10-4 for air,
density, in grams per cubic centimeter,
thickness of the slit normal to the direction of flow, in centi­
meters,
I = width of the slit normal to the direction of flow, in centimeters,
= length of the slit in the direction of the flow, in centimeters,
w
27Tj, and
j = frequency, in cycles per second.
w =
1
Crandall, " Vibrating Systems and Sound," D. Van Nostrand Company, Princeton,
N.J., 1926.
Lamb, " Dynamical Theory of Sound," E. Arnold, London, 1931.
Rayleigh, " Theory of Sound," Macmillan and Company, London, 1926.
4 Crandall, " Vibrating Systems and Sound," D. Van Nostrand Company, Princeton,
N.J., 1926.
5 Lamb, " Dynamical Theory of Sound," E. Arnold, London, 1931­
6 Rayleigh, " Theory of Sound," Macmillan and Company, London, 1926.
2
3
90
ACOUSTICAL ENGINEERING
In equation 5.2 the acoustical resistance varies inversely as the cube of d
and the inertance inversely as d. Therefore, practically any ratio of
inertance to acoustical resistance may be obtained. The magnitude may
be obtained by a suitable choice of wand t. A slit type of acoustical re­
sistance may be formed by using a pile of washers spaced by small shims.
Another form consists of a spiral of tape with adjacent turns very close
together.
5.5. Acoustical Resistance of Silk Cloth.-Silk cloth provides a simple
means of obtaining an acoustical resistance. The magnitude of the acousti­
cal resistance is governed by the size and nature of the holes in the material
so~-+--~--+-~--~--~--~-+--~~
«
'"«~45~-+--~--+-~--~--~--~-+--~~
i40~-1--~--+---~-+--~--+--1--~~
=>
~35r--+---r--+-~---r--~-.~-+---r--1
'"
Q.
",30~-+--~--+-~--~--~--~-+---r~
u
z
«25r--+---r--+-~--~--~--~-+--~~
lii
~20r--+---r--+-~---r--~--~-+---r--1
~
~ISr--+---r--+-~---r--~--~-+---r--1
u
~IOr--+---r--+-~---r--~--~-+---r--4
o=>
u
«
5r--±---r--+-~---r--~--r--+---r--i
oL-~--~2~~3--~4--~5--~6--~7~~8--~9~~IO·
LAYERS
OF
SILK CLOTH
FIG. 5.1. The acoustical resistance, per square
centimeter, of sheer silk cloth as a function of the
number of layers.
and the number of layers of the cloth. The acoustical resistance of sheer
silk cloth as a function of the number of layers of the material is shown in
Fig. 5.1. As in the case of the small tube and narrow slit, the ratio of
acoustical resistance to inertance is governed by the size of the holes (see
equations 5.1 and 5.2).
Silk cloth has been used as an acoustical resistance element in micro­
phones, telephone receivers, and loudspeakers for many years. The struc­
tural simplicity and the high ratio of acoustical resistance to inertance
make it particularly desirable as an acoustical resistance.
ACOUSTICAL ELEMENTS
91
5.6. Inertance.-Inertance is defined, in Sec. 4.5D, as
M
where S
=
=
mass
S2
5.3
area, in square centimeters, over which the driving pressure
acts to drive the mass, in grams.
The acoustical impedance of various types of systems will be considered
in Secs. 5.8, 5.9, 5.10, 5.11, 5.12, 5.13 and 5.14. The imaginary part of these
expressions is due to the inertance of the systems.
For closed systems the acoustical resistance term of Secs. 5.8, 5.9, 5.10,
5.11,5.12, 5.13 and 5.14 should be omitted because there is no radiation.
In this case the entire acoustical impedance is positive acoustical reactance.
The acoustical reactance term of equations 5.1 and 5.2 is due to inertance.
5.7. Acoustical Capacitance.-The most common type of acoustical
capacitance used in acoustical systems consists of a cavity or volume with
rigid boundaries. The linear dimensions of the enclosure are assumed to
be small compared to the wavelength.
For equation 1.21 the sound pressure is
p = pc 2s
5.4
where p = density of air, in grams per cubic centimeter,
c = velocity of sound, in centimeters per second, and
s = condensation.
The condensation, from Sec. 1.3D, is
dV
s=V
5.5
where dV is the change in the original volume V.
dV=Sx=X
5.6
displacement, in centimeters, over the area S, in square centi­
meters, and
X = volume displacement, in cubic centimeters.
From equations 5.4, 5.5, and 5.6.
where x
=
X
V
P
pc 2
5.7
The ratio Xjp is termed the acoustical capacitance by definition (see
Sec. 4.6D). Therefore the acoustical capacitance of a volume is
CA
V
=-2
pc
5.8
The next consideration will be an acoustical capacitance combined with
an acoustical resistance. The acoustical impedance of a cavity in which
92
ACOUSTICAL ENGINEERING
the boundaries or a portion of the boundary is terminated in an acoustical
resistance is
5.9
where r A = acoustical resistance of the boundary, in acoustical ohms,
CA = acoustical capacitance of the volume, in cubic centimeters per
second,
w = 2'TTJ, and
f = frequency, in cycles per second.
5.8. Mechanical and Acoustical Impedance Load upon a Vibrating
Piston. 7 ,8,9-The mechanical impedance, in mechanical ohms, of the air
load upon one side of a vibrating piston set in an infinite baffle is
ZM
where R
=
'TTR2 pc
[1 _h(2kR)]
+ J.'TTWPK
kR
2k 3 (2kR)
1
radius of piston, in centimeters,
p = density, in grams per cubic centimeter,
c = velocity of sound, in centimeters per second,
k = 2'TT/A,
A = wavelength, in centimeters,
W = 2'TTJ, and
f = frequency, in cycles per second.
hand Kl may be found in treatises10 ,l1 on Bessel functions.
are also defined by the series,
h(2kR)
k2R2
k4R4
k 6R6
1 - ---rR = 2
- 22.3 22.32.4'"
5.10
=
They
+
K (2kR)
=
1
3[(2kR)3 _ (2kR)5
'TT
3
32.5
+ 32.5
(2kR)7
]
2.7' . .
5.11
The acoustical impedance, in acoustical ohms, of the air load upon one
side of a vibrating piston in an infinite baffle is
ZA
=
~~
'TTR2
[1 - h(2kR)]
+ 2'TTR4k3
jwp K (2kR)
kR
1
5 12
.
The acoustical impedance per unit area of the piston is
Zl =
pc [ 1 -
h(2kR)]
jwp
-w- + 2R2k3K1(2kR)
5.13
Rayleigh, " Theory of Sound," Macmillan and Company, London, 1926.
Crandall, " Vibrating Systems and Sound," D. Van Nostrand Company, Princeton,
N.J., 1926.
9 Stewart and Lindsay, " Acoustics," D. Van Nostrand Company, Princeton, N.J.,
1930.
10 Watson, " Theory of Bessel Functions," Cambridge Press, London, 1922.
11 Jahnke and Emde, "Tables of Function," Teubner, Berlin, 1928.
7
8
93
ACOUSTICAL ELEMENTS
The resistive and reactive acoustical impedance components of the air
load per unit area on one side of a vibrating piston set in an infinite baffle
are shown in Fig. 5.2. These characteristics are useful in determining the
radiation resistance and reactive component of the air load on the cone in a
2
t­
I
.8
~
.4
.'
~
,,;V ..,
«
"'a:«
.."'
«
"
",
-'" j',
\"
'.
1/'~
.'
.0
w
<J
Z
........
,,~ .....' -' II/
.1
.08
.01
x.
"
" ....
XI
/
00­
/2 J,
XI
",y" .......
\fl
.­
.­
3
I
2
.04
a:
1//
..'
.2
I·
.'
11
:3
,<!,
: r,
a .008
..
w
"
:f
.­
.004
..J
«
!
~
I-'
, II
<Il
:J
o
I
<J
«
.001
.0008
:
.0004
I
/
/ II
.000
/'/
.02
.04
.08 . 1
.2
.4
.8 I
2
4
8 10
kR=~
FIG. 5.2. The acoustical resistance, 1'1, and the acoustical reactance,
load per unit area divided by pc, as a function of kR for the
following radiators: 1. a vibrating piston of radius R set in an
infinite baffle ; 2. a pulsating sphere of radius R; 3. an oscillating
sphere of radius R. Note: The ordinate scale of the characteristics
labeled 3 must be multiplied by one-third. (See Sec. 5.10.)
Xl,
direct radiator loudspeaker. It is also customary to use these characteristics
for the acoustical impedance at the mouth of a finite horn in computing the
throat acoustical impedance.
5,9, Mechanical and Acoustical Impedance Load Upon a Pulsating
Sphere,-The pulsating sphere is a sphere whose radius increases and de­
creases with time. The motion of the air around the sphere will, like the
94
ACOUSTICAL ENGINEERING
motion of the sphere itself, take place only in radial directions and will have
the same velocity in all directions, but will depend upon the distance from
the center of the sphere.
The mechanical impedance, in mechanical ohms, of a pulsating sphere is
_
ZM -
where R
=
p =
k
=
t..
=
c
=
2
47TR pc
++
[(kR)2
jkR]
1
(kR)2 .
5.14
radius of the sphere, in centimeters,
density, in grams per cubic centimeter,
27T/t..,
wavelength, in centimeters, and
velocity of sound, in centimeters per second.
The acoustical impedance, in acoustical ohms, of the air load upon
pulsating sphere is
+
_
pc [(kR)2
j(kR)]
47TR2 1 + (kR)2
ZA -
5.15
The acoustical impedance per unit area is
Zl
=
+
(kR)2
jkR]
pc [ 1
(kR)2
+
5.16
The resistive and reactive acoustical impedance components of the air
load per unit area of a pulsating sphere are shown in Fig. 5.2. It will be
noticed that the load upon a pulsating sphere is practically the same as
that of a vibrating piston.
5.10. Mechanical and Acoustical Impedance Load upon an Oscil­
lating Sphere.-An oscillating sphere is a sphere whose radius remains
constant while the sphere executes a movement of translation as a function
of the time. The mechanical impedance, in mechanical ohms, of the air
load upon an oscillating sphere is
+
+
_ 47TR 2pC[k4R4
j(2kR
k3R3)]
3
4-+k4R4
ZM-
where R
=
p =
k
=
t.. =
c=
5.17
radius of the sphere, in centimeters,
density, in grams per cubic centimeter,
27T/t..,
wavelength, in centimeters, and
velocity of sound, in centimeters per second.
The acoustical impedance, in acoustical ohms, of the air load upon an
oscillating sphere is
ZA
=
+
pc [k4R4
j(2kR + k3R3)]
127TR2
4k4R4
+
5.18
ACOUSTICAL ELEMENTS
95
The acoustical impedance per unit area of an oscillating sphere is
_ PC[k4R4
+ j(2kR + k3R3)]
Zl-"3
4+k 4R4
5.19
The average reactive and resistive acoustical impedance components of the
air load upon an oscillating sphere are shown in Fig. 5.2. The load on an
oscillating sphere is not uniform. In order to compare the radiation charac­
teristics with those of a piston and a pulsating sphere, the ultimate acoustical
resistance has been made the same. However, the average acoustical
impedance per unit area of a vibrating sphere is one-third that of charac­
teristics 3 shown in Fig. 5.2.
The oscillating sphere is an acoustical doublet (see Sec. 2.3). There­
fore, the acoustical resistance component is proportional to the fourth power
of the frequency when the dimensions are small compared to the wave­
length. The oscillating sphere represents the direct radiator loudspeaker
without a baffie.
5.11. Mechanical and Acoustical Impedance Load upon a Pulsating
Cylinder.1~The pulsating cylinder is a cylinder whose radius increases
and decreases with time. The motion of the air around the cylinder will,
like the motion of the cylinder itself, take place in radial directions in planes
normal to the axis of the cylinder and will have the same radial velocity in
all directions but will depend upon the distance from the center of the
cylinder.
The mechanical impedance, in mechanical ohms, of the air load, per unit
length, upon a pulsating cylinder is
ZM =
271Rpc [
(2kR)2 + j2kR]
1 + (2kR)2
5.20
where R = radius of the cylinder, in centimeters,
p = density of air, in grams per cubic centimeter,
k = 271/il
iI = wavelength, in centimeters, and
c = velocity of sound in centimeters per second.
The acoustical impedance, in acoustical ohms, of the air load per unit
length upon a pulsating cylinder is given by
_ ~ [(2kR)2 + j2kR]
271R 1 + (2kR)2
ZA -
5.21
The acoustical impedance per unit area is
+
Zl =
(2kR)2
j2kR]
pc [ 1 + (2kR)2
5.22
The resistive and reactive impedance components of the air load per unit
area of a pulsating cylinder are shown in Fig. 5.3.
12
Rueter and Bolt, "Sonics," John Wiley and Sons, New York, N.Y., 1955.
96
ACOUSTICAL ENGINEERING
I
.8
1--' I-"'"
A
u
a.
//
/
...
\oJ
It:
...
~/
.I
::>
'"
\oJ
CL
1'-'''',
k'r'i
lI,
"- "
""­
.08
l­
i'
"
,,'V
~/ J
.2
-I-
~/
.04
.02
/
/
/1
If1
XI
r.
"
"'­
2.
./
/ r,
\oJ
U
~ .01
~ .008
Q.
:I!
.....J
.004
L
J
.'
U
/
:;; . 002
::>
o
u
.... 001
.0008
/
.0004
.0002
.0001
.01
.02
.04
0.8 J
.2
kR=
2~R
.8 I
.4
, kD
= 2~O
4
e
10
20
FIG. 5.3. The acoustical resistance, YI, and acoustical reactance, Xl,
load per unit area divided by pc as a function of kR or kD, for two
radiators as follows: 1. A pulsating cylinder of infinite length and of
radius R. 2. A vibrating strip of infinite length and of width 2D, set in
an infinite baffle. (1. After Rueter and Bolt.)
5.12. Mechanical and Acoustical Impedance Load upon a Vibrating
Strip,13-The mechanical impedance,14 in mechanical ohms, of the air load,
per unit length, upon one side of an infinitely long vibrating strip set in an
infinite baffle is
_
[(2kD)3/2 (2kD)2/3]
5.23
zM - 2pcD
1 + (2kD)3/2
+
Rueter and Bolt, "Sonics," John Wiley and Sons, New York, N.Y., 1955.
The expressions given in equations 5.23, 5.24, and 5.25 are not the result of an
analytical derivation, but are approximations obtained from numerical integration.
13
14
ACOUSTICAL ELEMENTS
97
where 2D = width of the vibrating strip, in centimeters,
p = density, air grams per cubic centimeter,
c = velocity of sound, in centimeters per second,
k = 27r/).., and
).. = wavelength, in centimeters.
The acoustical impedance, in acoustical ohms, of the air load per unit
length, upon a vibrating strip set in an infinite baffle is
_
pc
2-D
ZA -
+
[(2kD)3/2
(2kD)2/3]
1
(2kD)3/2
+
5.24
The acoustical impedance per unit area of the strip is
_
Zl -
+ (2kD)2/3]
+ (2kD)3/2
[(2kD)3/2
pc
1
5.25
The resistive and reactive acoustical inpedance components of the air load
per unit area of one side of an infinitely long vibrating strip set in an infinite
baffle is shown in Fig. 5.3.
5.13. Mechanical and Acoustical Impedance upon a Vibrating
Piston in the End of an Infinite Tube. 15 ,16-The resistive and reactive
acoustical impedance components of the air load per unit area on the free
space side of a vibrating piston set in the end of an infinite tube is shown in
Fig. 5.4.
The mechanical impedance, ZM, in mechanical ohms, of the air load on the
free space side of a vibrating piston set in the end of an infinite tube is
given by
5.26
ZM = 7rR2Z1 = 7rR2(rl
Xl)
+
where
acoustical impedance per unit area,
acoustical resistance per unit area, Fig. 5.4,
Xl = acoustical reactance per unit area, Fig. 5.4, and
R = radius of the piston, in centimeters.
Zl =
rl =
The acoustical impedance, ZA, in acoustical ohms, of the air load on the
free space side of a vibrating piston set in the end of an infinite tube is given
by
+
Zl
YI
Xl
ZA = 7rR2 =
7rR2
5.27
Where the quantities are the same as in equation 5.26.
An example of a vibrating piston set in a tube is that of a loudspeaker
mechanism set in a completely-enclosed cabinet having front face area not
appreciably larger than the loudspeaker mechanism.
15
16
Levine and Schwinger, Phys. Rev., Vol. 73, No.4, p. 383, 1948.
Beranek, .. Acoustics," McGraw-Hill Book Company, New York, N.Y., 1954.
98
ACOUSTICAL ENGINEERING
Referring to Figs. 5.2 and 5.4, it will be seen that in the region below k = 1
the radiation resistance for a vibrating piston in an infinite bailie is two times
the radiation resistance of a vibrating piston located in the end of an infinite
tube. This agrees with the conclusions of Sec. 2.2 for a point source radiating
2
-.-.
I
.8
.,
/
X,
Q..
.~
I
.08
.,
>-
.04
Z
.02
Q.
\oJ
U
~
.0 I
~ .008
\
"
I
I
"'~
'1
~
/
:>
ffi
~ .'x,
,.
\oJ
a:
I
I
.~
I
.2
u
\
I
.4
.'
Vi
2
I.
/~
r,
Q.
~
~ .004
I
u
;::
I
(/)
..g
u
.002
I
.00 I
.0008
.0004
"
I
I
.0002
.000 I
.01
/
.02
I
I
I
.04
.08 .I
.2
.4
.8 I
2
4
8 10
20
kR= 2~R
FIG. 5.4. The acollstical resistance, rl, and the acoustical reactance, Xl,
load per unit area divided by pC'as a function of kR for the following radia­
tors: 1. a vibrating piston of radius R set in the end of an infinite pipe.
2. a vibrating piston of radius R in free space.
into 27T and 47T solid angles. This is to be expected because in the region
below kR = 1 the piston is essentially a point source.
Computing the end correction17 for a flanged and an unflanged pipe from
the mass reactance in the region below kR = 1 from Figs. 5.2 and 5.4, the
end corrections are .82R and .61R, respectively.
17
Levine and Schwinger, Phys. Rev., Vol. 73, No.4, p. 383, 1948.
ACOUSTICAL ELEMENTS
99
5.14. Mechanical and Acoustical Impedance upon a Vibratin~
Piston in Free Space,18,19-The resistive and reactive acoustical impedance
components of the air load on one side of a vibrating piston in free space is
shown in Fig, 5.4.
The mechanical impedance, ZM, in mechanical ohms, of the air load on
one side of a vibrating piston in free space is given by
ZM = 7TR2zl = 7TR2(rl + Xl)
5.28
where Zl = acoustical impedance per unit area,
rl = acoustical resistance per unit area, Fig. 5.4,
Xl = acoustical reactance per unit area, Fig. 5.4, and
R = radius of the piston, in centimeters.
The acoustical impedance, ZA, in acoustical ohms, of the air load on one
side of a vibrating piston in free space is given by
ZA =
Zl
7TR2 =
+
(rl
Xl)
7TR2
5.29
Where the quantities are the same as in equation 5.28.
An example of a vibrating piston in free space is a loudspeaker mechanism
operating in free space without a baffle, cabinet, etc.
5.15. Acoustical Impedance of a Circular Orifice in a Wall of
Infinitesimal Thickness.-The acoustical impedance of a circular orifice
in a wall of infinitesimal thickness may be considered to be the same as that
of the air load upon a piston of infinitesimal thickness and zero mass set in
the opening. Then the acoustical impedance of a circular aperture in a thin
wall is obtained from equation 5.12 by multiplying by 2.
5.16. Acoustical Impedance of an Open Pipe with Lar~e Flan~es.­
In this case it will be assumed that the mouths of the pipe are fitted with
freely moving massless pistons and that the length of the pipe is small
compared to the wavelength. The acoustical impedance is the sum of the
mass reactance of the air between the pistons and the acoustical impedance
of the air load upon the pistons.
The acoustical reactance, in acoustical ohms, of the column of air between
the two pistons, from equation 5.3, is
pl
XA =
7TR2 w
where p = density of air, in grams per cubic centimeter,
1 = length of the pipe, in centimeters,
R = radius of the pipe, in centimeters,
w = 27r/, and
/ = frequency, in cycles per second.
18
19
Wiener, F. M., Jour. Acous. Soc. Amer., Vol. 23, No.6, p. 697,1951.
Beranek, .. Acoustics," McGraw-Hill Book Company, New York, N.Y., 1954.
5.30
100
ACOUSTICAL ENGINEERING
The acoustical impedance, in acoustical ohms, of the entire system is
2pc [
h(kR)]
. wp
jpl
5.31
ZA = 7TR2 1 J7TR4k3K1(2kR)
7TR2 w
---w- +
+
5.17. Horns.-A horn is an acoustical transducer consisting of a tube
of varying sectional area. Horns have been used widely for centuries for
increasing the radiation from a sound source. The principal virtue of a
horn resides in the possibility of presenting practically any value of acousti­
cal impedance to the sound generator. This feature is extremely valuable
for obtaining maximum overall efficiency in the design of an acoustical
system. As an example, in a horn loudspeaker high effici~cy is obtained
by designing the system so that the driving force works against resistance
instead of inertia of the diaphragm. Employing suitable combination of
horns, directional characteristics which are independent of frequency, as
well as practically any type of directional pattern, may be obtained. The
combination of high efficiency and the possibility of any directional pattern
makes the horn loudspeaker particularly suitable for larger scale sound
reproduction. It is the purpose of this section to consider some of the
factors which influence the characteristics of a horn.
5.18. Fundamental Horn Equation. 20 ,21,22,23,24,25,26,27 ,28, 29,30,31_Con_
sider a tube with a certain rate of flare and with the diameter small compared
to the wavelength of the sound passing through it. Let the axis of the tube
coincide with the x axis. Take an element of volume of the tube defined as
Sllx
5.32
where 5 = cross-sectional area of the tube at x, and
llx = length of the element of volume.
The growth of matter in this volume is the difference between the influx
and efflux of fluid through the faces and may be expressed as
llxo(Sp'u)
ox
where u
=
5 33
.
component of the particle velocity along the axis, and
p' = density of the medium.
Webster, A. G., Jour. Nat!. Acad. Sci., Vol. 5, p. 275, 1919.
Stewart, G. W., Phys. Rev., Vol. 16, No.4, p. 313, 1920.
22 Goldsmith and Minton, Proc. Inst. Rad. Eng., Vol. 12, No.4, p. 423,1924.
23 Slepian and Hanna, Jour. A mer. Inst. Elec. Eng., Vol. 43, p. 393, 1924.
24 Ballantine, G., Jour. Frank. Inst., Vol. 203, No. 1. p. 85, 1927.
25 Crandall, " Vibrating Systems and Sound," D. Van Nostrand Company, Princeton,
N.J., 1926.
26 Stewart and Lindsay, " Acoustics," D. Van Nostrand Company, Princeton, N.J.,
1930.
27 Olson and Massa, .. Applied Acoustics," P. Blakiston's Son and Company, Phila­
delphia, Pa., 1934.
28 Mawardi, Osman K., Jour. Acous. Soc. Amer., Vol. 21, No.4, p. 323, 1949.
29 Lambert, Robert F., Jour. Acous. Soc. Amer., Vol. 26, No.4, p. 1024, 1954.
30 Jensen and Lambert, Jour. Acous. Soc. Amer., Vol. 26, No.4, p. 1029, 1954.
31 Scibor-Marchoki, R. I., Jour. Acous. Soc. Amer., Vol. 27, No.5, p. 939, 1955.
20
21
ACOUSTICAL ELEMENTS
101
The principle of continuity was expressed in Sec. 1.3. Applying the
principle, the difference between the influx and efflux of the fluid into the
element of volume must be equal to the time rate of growth of mass.
~'S~x
ot
=
~xo(Sp'u)
-
ox
5
.34
or
sop'
ot
From equations 1.19 and 1.6
+ o(Sp'u) =
0
ox
- Pi> =
c2.p'
5.35
5.36
From equation 1.11
oq,
u= 5.37
ox
Substituting equations 5.36 and 5.37 in 5.35 the result may be written as
..
oq, 0
o2q,
q, - c 2 ox ox(log 5) - c2 ox 2 0
5.38
Equation 5.38 is the wave equation for the axial motion in a tube of
varying section.
5.19. Infinite Cylindrical Horn (Infinite Pipe).-The equation express­
ing the cross-sectional area as a function of the distance along the axis is
5 = 51
5.39
where 51 = cross section of the pipe, in square centimeters.
The general horn equation for the infinite pipe from equations 5.38 and
5.39 is
..
o2q,
q, - C2 0X2 = 0
5.40
The velocity potential, pressure, and volume current are
q, = A EJk(ct-x)
P=
U
=
kcpA EJk(ct-x)
SlkA EJk(ct-x)
5.41
5.42
5.43
where k = 27TjA,
A = wavelength, in centimeters, and
p = density of the medium, in grams per cubic centimeter.
The real and imaginary components of the acoustical impedance, in
acoustical ohms, at the throat or input end of the pipe are
5.44
5.45
102
ACOUSTICAL ENGINEERING
5.20. Infinite Parabolic Horn. 32-The equation expressing the cross­
sectional area as a function of the distance along the axis is
S = SIX
5.46
The general horn equation for the parabolic horn is
. c2 04>
024>
OX - c2Ox2 = 0
5.47
.;; - x
The velocity potential, pressure, and volume current are
4> = A [Jo(kx) - jYo(kx)] £1wt
5.48
P = - jwpA [Jo(kx) - jYo(kx)] £1wt
5.49
U = ASk [ - J 0' (kx)
jYo' (kx)] £1wt
5.50
The real and imaginary components of the acoustical impedance, in
acoustical ohms, at the throat are
pC
2
5.51
r A = Sl 7Tkxl[h 2 (kXl)
Y 1 2(kxl)]
+
XA =
where J0, h
Yo, Yl
+
pC JO(kXl)h(kxl) + YO(kXl)Yl(kxl)
Sl
h2(kxl) + Y 1 2(kxl)
5.52
Bessel functions of the first kind of order zero and one,
Bessel functions 33 of the second kind of order zero and one,
p = density of the medium, in grams per cubic centimeter,
c = velocity of sound, in centimeters,
Sl = area at Xl, in square centimeters,
Xl = distance of the throat from X = 0, in centimeters,
k = 27Tj>', and
>. = wavelength, in centimeters.
=
=
5.21. Infinite Conical Horn.- The equation expressing the cross­
sectional area as a function of the distance along the axis is,
S = SlX 2
5.53
The general horn equation for the conical horn is
.;; _ 2c 2 04> _ C2024> = 0
5.54
X ox
OX2
The velocity potential, pressure, and volume current are
4> = A
- £j(wt-kx)
5.55
X
P=
_
jwpA £1 (wt-kx)
X
U = _ AS(!
+ jkx) £1 (wt-kx)
X2
82
88
Olson and Wolff, Jour. Acous. Soc. Amer., Vol. 1, No.3, p. 410, 1930.
Jahnke and Erode, .. Tables of Functions," Tuebner, Berlin, 1928.
5.56
5.57
ACOUSTICAL ELEMENTS
103
The real and imaginary components of the acoustical impedance, in
acoustical ohms, at the throat are
pc
k2X12
51 1 k2x12
rA
= -
XA
= 5- 1 1-:------,.~""'
+ k2x12
+
pc
where 51
=
Xl
=
k
=
I.
=
5.58
:;--:--:~"""
kX1
5.59
area at Xl, in square centimeters,
distance of throat from X = 0, in centimeters,
271"/1., and
wavelength, in centimeters.
5.22. Infinite Exponential Horn.-The equation expressing the cross­
sectional area as a function of the distance along the axis
where 51
m
=
=
5 = 5 1 Emx
area at the throat, that is, at X
flaring constant.
5.60
0, and
=
The general horn equation for the exponential horn is
fj 2e/>
c2fjx2 = 0
..
fje/>
e/> - c2mfjx -
5.61
The velocity potential, pressure, and volume current are
e/> = c(m/2)x
p=
U
]
[A c 1V4k'-m'
-2-
x Ejwt
5.62
[A C1--2-X]
V4k'-m'
-
=-
jwpE-(m/2)x
A5
. /4k 2 - --2]
[m2" +:v
m
J
2
Ejwt
m
5.63
v4k'-m'
- -x-j---x+jwt
E2
2
5.64
The real and imaginary components of the acoustical impedance, in
acoustical ohms, at the throat are
rA =
m2
pc)
51 1 - 4k2
pc m
XA =
s;:
2k
5.65
5.66
When m = 2k or 47TJ = me the acoustical resistance is zero. This is
termed the cutoff frequency of the exponential horn.
Below the cutoff frequency the acoustical impedance is entirely reactive
and
XA =
pc(m2k Si
) 1 - 4k2
m2)
5.67
104
ACOUSTICAL ENGINEERING
5.23. Infinite Hyperbolic Horn. 34-The equation expressing the cross­
sectional area along the axis is
5
where T
ex
=
=
x =
Xo =
51
=
51 (cosh ex
=
+ T sinh ex)2
family parameter, in the hyperbolic horn T < 1,
xlxo, dimensionless axial distance,
axial distance from the throat, in centimeters,
reference axial distance, in centimeters, and
area at the throat, in square centimeters, that is, at x
5.68
= O.
The expressions for the velocity potential, pressure, and volume current
are quite complex and will not be considered.
The real and imaginary components of the acoustical impedance, in
acoustical ohms, at the throat are
J1-!
pc
rA
=51
f-L
1 _ T2
5.69
1--­
f-L2
T
pc
XA =
where f-L
=
k =
/0 =
/ =
kxo
s;: 1
f-L
1 - T2
5.70
--py:­
= /1/0,
2'TTI>',
cutoff frequency, and
frequency under consideration.
Below the cutoff frequency, f-L
reactive and
XA
=
=
1, the acoustical impedance is entirely
I -J~f-L2 -1
pc f-L
s;,
1 ­ T2
5.71
1--­
f-L2
5.24. Throat Acoustical Impedance Characteristic of Infinite
Parabolic, Conical, Exponential, Hyperbolic, and Cylindrical Horns.
-The throat acoustical impedance of infinite horns may be computed from
the equations of Secs. 5.18. 5.19, 5.20, 5.21, 5.22, and 5.23 . In order to
compare the throat acoustical impedance characteristics of infinite parabolic,
conical, exponential, hyperbolic, and cylindrical horns, a specific example
has been selected in which the throat area is the same in all horns. In
34
Salmon, V., Jour. Acous. Soc. Amer., Vol. 17, No.3, p. 212, 1946.
ACOUSTICAL ELEMENTS
105
addition, the area at a distance of 100 centimeters from the throat is the
same for the four horns with flare, as shown in Fig. 5.5. The value of T for
the hyperbolic horn is .5. The acoustical resistance and acoustical reactance
frequency characteristics for the five horns are shown in Fig. 5.5.
FIG. 5.5. Throat acoustical resistance rA, and acoustical reac­
tance XA, frequency characteristics of infinite parabolic, conical,
exponential, hyperbolic, and cylindrical horns having a throat
area of 1 square centimeter. The cross-sectional area of the
parabolic, conical, exponential, and hyperbolic horns is 100
square centimeters at a distance of 100 centimeters from the
throat.
5.25. Finite Cylindrical Horn.-The acoustical impedance, in acousti­
cal ohms, at the throat of the finite cylindrical horn of Fig. 5.6 is
ZAI =
where PI
U1
=
=
PI
U1
5.72
pressure at the throat, in dynes per square centimeter, and
volume current, in cubic centimeters per second.
The acoustical impedance, in acoustical ohms, at the mouth of a cylindri­
cal horn is
ZAZ =
where
P2
=
U2 =
PZ
Uz
pressure at the mouth, in dynes per square centimeter, and
volume current, in cubic centimeters per second.
5.73
106
ACOUSTICAL ENGINEERING
From equations 5.52 and 5.53 the expressions for the pressures and
volume currents at the throat and mouth are given by
5,
52
At x = 0,
PI = kcpA,}kct
5.74
Ul = SlkA€1kct
5.75
P2
=
kcpA€1k(ct-l)
5.76
At x = l,
CYLINDRICAL
5.77
U 2 = SlkA€1k(ct-l)
From equations 5.72, 5.73, 5.74, 5.75, 5.76, and
5.77 the expression for the acoustical impedance,
Z A ' r - - - - - : A2
ZAl, at the throat in terms of the length and
0,
cross-sectional area of the horn and the acousti­
CONICAL
cal impedance, ZA2, at the mouth is
~2
L==£
~2
+
+
ZAI = ~ (~IZA2 co~ (kl)
jpc sin (kl))
5.78
51 JSIZA2 SIn (kl
pC cos (kl)
where p = density of the medium, in grams per
~l
'I
cubic centimeter,
EXPONENTIAL
k = 2rrjA,
FIG. 5.6. Finite cylindrical,
A
= wavelength, in centimeters,
conical,
and exponential
c = velocity of sound, in centimeters per
horns. Z..tl = input acousti­
cal impedance at the throat.
second,
Sl = cross-sectional area at
51 = cross-sectional area of the pipe, in
the throat, in square centi­
square centimeters,
meters. X..t2 = terminating
l = length of the pipe, in centimeters, and
acoustical impedance at the
throat. S2 = cross-sectional
ZA2 = acoustical impedance at the mouth,
area at the mouth, in square
in acoustical ohms.
centimeters.
1 = length, in
centimeters.
The throat acoustical impedance character­
istics of a finite cylindrical horn or pipe are
shown in Fig. 5.7. The mouth acoustical impedance is assumed to be the
same as that of a piston in an infinite baffle. In this case the mouth
acoustical impedance, ZA2, is given by equation 5.12. It will be seen that the
variations in the acoustical resistance and acoustical reactance components
are quite large at the low frequencies where the mouth acoustical resistance
is small.
5.26. Finite Conical Horn.-The acoustical impedance at the throat of a
finite conical horn of Fig. 5.8 may be obtained in a manner similar to the
procedure for the finite cylindrical horn in the preceding section by employ­
ing the equations for the pressure and velocity in an infinite conical horn
and applying the proper boundary conditions. The expression for the
acoustical impedance, ZAl, at the throat in terms of the dimensions of the
horn and the acoustical impedance, ZA2, at the mouth is
ZA'~A2
ZAI
pC [
= St
.
sin k(l - 82)
JZA2
. k8 2
SIn
+
+ 5pC . kl
SIn
2
]
+
sin k(l
81 - 82)
jpc sin k(l
81 )
ZA2 sin k81 sin k8 2 - 52
sin k81
5.79
107
ACOUSTICAL ELEMENTS
5,
Z"-S~~CM.
ZA.
f-----Z-S-C-M-.-~""'i
ZO
18
6
~14
Ii..
;; 12
x
10
t:
z
8
"
;§ 6
w
~
4
..J
Z
0(
!! 0
~ -2
80( -4
I
I 1
----
l
[)
/,
"r1
.' I
"/"
l
~'-:
-:~
....
;-
I
I XA,
-6
II
-8
"
-,0,02
]
5
fREQUENCY
6
IN
7
8 'loJ
CYCLES
2
PER
J""
5
&
7
8 • 1041
SECOND
FIG. 5.7. The throat acoustical resistance and acoustical reactance fre­
quency characteristics of a finite cylindrical horn. 1'.tIl = acoustical
resistance. XA1 = acoustical reactance. Note: The characteristics shown
are the throat acoustical resistance and acoustical reactance multiplied by
S1 and divided by pc.
where 51
area of the throat, in square centimeters,
S2 = area of the mouth, in square centimeters,
1 = length of the horn, in centimeters,
k(h = tan-1 kX1,
k8 2
=
= tan-l
kX2
Xl
= distance from the apex to the throat, in centimeters,
X2
= distance from the apex to the mouth, in centimeters,
k = 2rr/A,
A = wavelength, in centimeters,
c = velocity of sound, in centimeters per second,
p = density of air in grams per cubic centimeter,
ZA2
= acoustical impedance at the mouth, in acoustical ohms.
The throat acoustical impedance characteristics of a finite conical hom
are shown in Fig. 5.8. The acoustical impedance at the mouth of the horn
is usually assumed to be the same as that of a piston in an infinite baffle.
In this case the mouth acoustical impedance, ZA2, is given by equation 5.12.
108
ACOUSTICAL ENGINEERING
1.4
v
."
'1.2
vi
1.0
'"z
rAJr ~
u
.
..
o
'"
:I
,11
.6
..J
u
~
::>
"
~
.2
o
,
o
,
"
_...----- /
102
,tV
,
, - , X..,
\
'"\
~.
r ~V
.
.5
FREQUENCY
V
V"
.. .8
6 7. 101
IN CYCLES
-....
.....
1
PER
.....
"
..... .....
5 I
1 •
10•
SECOND
FIG. 5.8. The throat resistance and acoustical reactance fre­
quency characteristics of a finite conical horn. Y.41 = acoustical
resistance. XAI = acoustical reactance. Note: The charac­
teristics shown are the throat acoustical resistance and acoustical
reactance multiplied by 5 1 and divided by pc.
5.27. Finite Exponential Horn. 3s-The acoustical impedance at the
throat of a finite exponential horn of Fig. 5.6 may be obtained in a manner
similar to the procedure for the finite cylindrical horn in the preceding
section by employing the equations for the pressure and velocity in an in­
finite exponential horn and applying the proper boundary conditions. The
expression for the acoustical impedance, ZAl , at the throat in terms of the
length and flare constant of the horn and the acoustical impedance, ZA2,
at the mouth IS
ZAI
=
where
8=
a=
area of the throat, in square centimeters,
area of the mouth, in square centimeters,
length of the horn, in centimeters,
acoustical impedance of the mouth, in acoustical ohms,
tan-1 a/b,
m/2, and
b=
tv'4k 2
S1 =
S2 =
I=
ZA2 =
3S
+ +
+
~r~2ZA2 Ico.s (bl
8)J
jpc [sin (bl)J]
[sm (bl)J
pc [cos (bl - 8)J
S1US2ZA2
-
m2.
Olson, H. F ., RCA Review, Vol. L p. 68,1937.
5.80
ACOUSTICAL ELEMENTS
109
For b = 0, equation 5.80 is indeterminate. To evaluate, take the de­
rivative of the numerator and denominator with respect to b and set b = 0.
Then the expression for the throat acoustical impedance becomes
+ j S2
pc 1m J
2
~ jZA/!!! + ~(l + m1)
pc
ZAI
=
~
ZA2(1 - m1)
2
2
S2
2
5.81
I
Below the frequency range corresponding to bl = 0, bl is imaginary. For
evaluating this portion of the frequency range the following relations are
useful:
tan- I jA = j tanh- I A = t j[logE (1 + A) - logE (1 - A)]
5.82
logE (- 1) = ± j7T (2K + 1), K = any integer
5.83
cos (A ± jB) = cos A cosh B =f j sin A sinh B
5.84
sin jA = j sinh A
5.85
The resistive and reactive components of the acoustical impedance of a
finite exponential horn are shown in Fig. 5.9. The acoustical impedance,
ZA2, at the mouth was assumed to be that of a piston in an infinite baffle
as given by equation 5.12. An examination of the acoustical resistance
characteristic of Fig. 5.9 shows that there is a sudden change in acoustical
1.4
2
II
f\'
x
VV
1.0
bJ
U
~
a
,
.8
bJ
.6
I
I
I
I.'
~,/
-'
010 2
/
,~
2
/
­
'-'
\
f
"­
::;
11II \
I\
II
\
\
,
,..XAI
\
, , I, ,
"
'-' ,
I
• 4• 6 7 ' 10'
FREQUENCY IN CYCLES
\
I
\~,
2
PER
'
\
\., \, ... ­
'4
SECOND
S
6 7 8 104
FIG. 5.9. The throat acoustical resistance and acoustical reac­
tance frequency characteristics of a finite exponential horn.
1'.... 1 = acoustical resistance.
.1'.... 1 = acoustical reactance. Note:
The characteristics shown are the throat acoustical resistance
or acoustical reactance multiplied by 51 and divided by pc.
110
ACOUSTICAL ENGINEERING
impedance in the frequency region, j = mc/47T. Above this frequency the
acoustical resistance multiplied by 5 1/ pc approaches unity, below this region
the acoustical resistance is relatively small. In the finite exponential
horn the acoustical resistance is not zero below the frequency, j = mc/4TT,
the flare cutoff frequency, which means that the horn will transmit below
this frequency. In the case of the finite conical horn, Fig. 5.8, there is no
sudden change in the acoustical resistance. On the other hand, the exponen­
tial horn shows a larger ratio of acoustical resistance to acoustical reactance.
This, coupled with the more uniform acoustical resistance characteristic,
makes the exponential horn more desirable and accounts for its almost
universal use. In view of its widespread use it is interesting to examine
some of the other characteristics of exponential horns.
5.28. Throat Acoustical Impedance Characteristics of Finite
Exponential Horns. 36-The throat acoustical impedance characteristic
as a function of the mouth area, with the flare and throat kept constant,
is of interest in determining the optimum dimensions for a particular applica­
tion. The acoustical impedance characteristics of four finite horns having a
cutoff of 100 cycles, throat diameter of 1 inch and mouth diameters of
10, 20, 30, and 40 inches and the corresponding infinite horn are shown in
Fig. 5.10. These results may be applied to horns of a different flare by
multiplying all the dimensions by the ratio of 100 to the new flare cutoff
frequency (see Sec. 1.13). The flare cutoff frequency of an exponential
horn is given by
2w =mc
5.86
where w = 27Tj,
j = frequency, in cycles per second, and
c = velocity of sound, in centimeters per second.
The acoustical radiation resistance of a mouth 10 inches in diameter is
relatively small below 500 cycles. The large change in acoustical impedance
in passing from the mouth to the free atmosphere introduces reflections at
the mouth and as a result wide variations in the acoustical impedance
characteristic as shown in Fig. 5. lOA. For example, the first maximum in
the acoustical resistance characteristic is 150 times the acoustical resistance
of the succeeding minimum.
By doubling the diameter of the mouth the maximum variation in the
acoustical resistance characteristic is 7.5, Fig. 5.lOB.
Fig. 5.10C shows the acoustical impedance characteristic of a horn with
a mouth diameter of 30 inches. The maximum variation in the acoustical
resistance characteristic of this horn is 2.
The acoustical impedance characteristic of a horn with a mouth diameter
of 40 inches, Fig. 5.lOD, shows a deviation in acoustical resistance of only
a few per cent from that of the infinite horn of Fig. 5.IOE.
These results show that as the change in acoustical impedance in passing
from the mouth to the free atmosphere becomes smaller by employing a
36
Olson, H. F., RCA Review, Vol. 1, No. 4, p. 68, 1937.
111
ACOUSTICAL ELEMENTS
mouth diameter comparable to the wavelength, the reflection becomes cor­
respondingly less and the variations in the acoustical impedance charac­
teristic are reduced.
The throat acoustical impedance characteristic as a function of the
A
..I~. 5
« III
u )(
8
12
7
i= wI.
III U
:>z
6
I
o «
I
5
I
I
.
I
I) VI b?i l\ "
,/
2
I'
3
""
4
.,
;:X A1
""
..I
«
400
1000
FREQUENCY
u
___
..L~
B
i=w
uo
«
~o
.75
"
f'VJ \
\
~
-,
~.25
zoo
..I~.OO , ,,
~
IIlU
5 ~.50
" •75
t- w
A-
,
•
I
••
I
Al
u8
« 11. .25
"
I
100
~A'
...
~
--
400
fREQUENCY
: ' ,r~ r,,'A. no'
"'",
/\ VJ' yJ".
'x ~ / \ 1'\ 1'.. . ...
~
~~
~
E
.
[V'-'
rV
1000
20"
I:
lIlu
:>z I,
0..:
79'_
rA'
-L
~
..I Iff2
\..,., , f-f--­
1;;r~T
3
~)(
'-;
~,
,J
1 I
S
~.50
Uo
5.
J
~.oo "
«Ill
~ )(
lilt!
200
"­
~X
1.25
I
100
"
JnVJ\\ 1/ \..
D
I
• I
.t\
o
11!\ J
: \
~
rAJ
,\
100
1\
"
'-;J.\t
o
•
I
11.
I
,,
2
r;].5
~
,.
'.
3
~
rAI
,:
4
c
I
, )l
/'-,
­
,
,~
X~I
~
zoo
400
FREQUENCY
1000
200
400
-- 1000
fREQUENCY
FIG. 5.10. The throat acoustical resistance and acoustical reactance
frequency characteristics of a group of exponential horns, with a flare
cutoff of 100 cycles and a throat diameter of 1 inch, as a function of the
mouth diameter. S1 = the throat diameter in square centimeters.
rA1 = acoustical
resistance. XA1 = acoustical reactance. Note: The
characteristics shown are the throat acoustical resistance or acoustical
reactance multiplied by S1 and divided by pc.
throat size with the mouth and flare held constant is of interest in determin­
ing the optimum length and a suitable matching impedance for the driving
mechanism. The acoustical impedance characteristics of four horns hav­
ing a cutoff of 100 cycles, mouth diameter of 20 inches, and throat diameter
of 1, 2, 4, and 8 inches are shown in Fig. 5.11. A consideration of these
112
ACOUSTICAL ENGINEERING
characteristics shows that the throat size has no appreciable effect upon
the amplitude of the variations in the acoustical impedance characteristics.
However, the separation in frequency between successive maxima is in­
creased, as the throat becomes larger, due to the decreased length of the
horn. The frequency at which the first maximum in the acoustical resis­
tance characteristic occurs becomes progressively higher as the length is
decreased.
The characteristics in Figs. 5.10 and 5.11 cover the range from 100 to
1000 cycles, the lower value being the flare cutoff frequency. The finite
.
c
4
u
~
~
"'-z
<.1 )(
j:w I
"'0
5~
00
«
~o
::::Ii
"
:I,
~ 1\
VlV\ V\
A
rA.
1\
~"
,,
'XAl. 1\
11;:".. /\I'
~,lr.
.'
,
zoo
100
• ,
•
v'
iI
-I
rA •
/11\
/\
400
1000
fREQUENCY
I
100
zoo
~
IA
~
i'"
\ "
\
400
1000
fREQUENCY
.....L~T
8#
B
20#
TI____ .L
D
1+ 19 5".1
,.•
I
••
) !"'P \~
I
, I
I
1/\
'N.!
\ '
/\
I
\
"
"\
y\ ';
,-
\0..
r;;.;
~,
""­
\~et,
v
"
I
100
,II
rAI
,./\
200
400
fREQUENCY
1000
-I
100
zoo
400
FREQUENCY
-,
, '
1000
FIG. 5.11. The throat acoustical resistance and acoustical reactance fre­
quency characteristics of a group of exponential horns, with a flare cutoff
of 100 cycles and a mouth diameter of 20 inches, as a function of the throat
mouth diameter. Sl = the .throat diameter, in square centimeters.
r.H = acoustical resistance. XA1 = acoustical reactance. Note:
The
characteristics shown are the throat acoustical resistance or acoustical
reactance multiplied by Sl and divided by pc.
horn, of course, transmits below this frequency because the acoustical
resistance is not zero. However, save for the case where the throat is
comparable to the mouth, as for example, Fig. S.l1D, the value of the
acoustical resistance, at and below the flare cutoff frequency, is quite small.
5.29. Exponential Connectors.-A transformer is used in electrical
circuits to transfer between two acoustical impedances of different values
113
ACOUSTICAL ELEMENTS
without appreciable reflection loss. In acoustical systems a horn may be
used to transfer from one acoustical impedance to another. As a matter
of fact a horn may be looked upon as an acoustical transformer, transform­
ing large pressures and small volume currents to small pressures and large
volume currents. I t is the purpose of this section to show how an exponen­
tial horn or connector may be used to transfer from one acoustical impedance
to another.
Fig. 5.10 shows an exponential horn coupled to an infinite tube. The
acoustical impedance of an infinite tube is
pc
5.87
S2
where p
=
density, in grams per cubic centimeter,
c = velocity of sound, in centimeters per second, and
S2
=
cross-sectional area of the infinite tube, in square centimeters.
5,
ZAI .......
-~---
~16.7CM~
r
.....­
2
-­ - ­
~'
,
/\
,...
lA,
v
, .f.
,, \\ X
•, ,
V
"
[1"\
100
fI
1000
"'
\
10000
-- ­
I
\Y
'00
FREQUENCY
FREQUENCY
'\rl"
1000
10000
FIG. 5.12. The throat acoustical resistance and acoustical reactance frequency charac­
teristics of two exponential connectors with a flare cutoff of 100 cycles. The mouth of
the horn is connected to an infinite pipe. r A1 = acousticalresistance. XA1 = acoustical
reactance. Note: The characteristics shown are the acoustical resistance or acoustical
reactance multiplied by 51 and divided by pc.
Equation 5.87 is the mouth acoustical impedance of the exponential
horn. Equation 5.80 then becomes
_ f!!: [ cos (bl
ZAI -
+ + j sin (bl)]
+ j sin (bl)
8)
SI cos (bl _ 8)
5.88
For b = 0, equation 5.88 is indeterminate. To evaluate take the deriva­
tive of the numerator and denominator with respect to b and set b = O.
Then the expression for the throat acoustical impedance becomes
pcrl + j~ -
ZAI =
~
I
~l
+ l~ + j l~_
5.89
114
ACOUSTICAL ENGINEERING
Below the frequency corresponding to b = 0, b is imaginary. This por­
tion of the range may be evaluated by employing equations 5.82, 5.83,
5.85, and 5.89.
The acoustical impedance characteristics of two exponential connectors
with a flare cutoff of 100 cycles (that is b = 0 at 100 cycles) is shown in
Fig. 5.12. Below 100 cycles the throat acoustical impedance is the same
as that of the infinite pipe. However, at the high frequencies the throat
acoustical impedance is the same as the surge acoustical impedance of a
pipe of the diameter of the throat. In order to effect a constant transfer
of acoustical impedance with respect to frequency over a certain frequency
range the cutoff of the connector must be placed below the low-frequency
limit of the frequency range .
,~~~
,:Jtr~:~1
1.2r-,--,-rrTTrr-'--'-""-'---'--'''''TTr--'-''''-'--'--'rTl"TT---'
1.I1-++_++t+f-t---+-H-+-Ht+I--+_-+-+--Hm%----j
l.ol-+_+--i---t-1f-ttt---+_-+-++-+++++---+---t-t-+hVH++\-\v-T-I
uilg,gl--t-++++++t---t--t-++++++I----+-+-+-++-f-ttt------1
1
x .8t---~+-r+++H_--+-~-+~++t----+--+-+~-H+r-~
~ ~1-++_++t+f-t---+-H-+-Ht+I--+_-+~/__H-H~----j
~I~
z
8·61--1--=:~:~::::~====:==~~~:~::::~===:~~~~+L-~r-~~~+~~+-_-_-_~~
.5
Q.
~
,
/
~t__"mr+Hr+++H---+-l-v~+-+~+I------t--+-r,~,~H++--~
~.3~,fl+~,I--H~~~-~~~-I--+-+~+I------tX~~,~-I--~H+-~
~.2 ,! \:
6 . j I,
1
~
0
-1 20
I
,
~,
I
;:
,,"
\
l\._, .. _ .-,'
,
",,'
,-... ,
'J
"
,,
\:
100
FREQUENCY
1000
IN
CYCLES
PER
10000
SECOND
FIG. 5.13.
The throat acoustical resistance and acoustical
reactance frequency characteristics of a multiple flare exponential
horn of three sections. The cutoffs due to flare of the three
horns are 25, 100, and 1400 cycles. rA1 = acoustical resistance.
XA1 = acoustical reactance.
Note: The characteristics shown
are the throat acoustical resistance or acoustical reactance multi­
plied by 51 and divided by pc. 51 = area at the throat of the
small horn in centimeters.
5.30. A Horn Consistin~ of Manifold Exponential Sections. 37-The
efficiency of a horn loudspeaker is governed, among many other factors, by
the throat acoustical resistance. To obtain the maximum efficiency at any
frequency the effective acoustical reactance of the entire vibrating system
should be equal to the effective acoustical resistance. This, in general,
means that to obtain maximum efficiency the throat acoustical resistance
37
Olson, H. F., Jour. Soc. Mot . Pic. Eng., Vol. 30, No. 5, p. 511, 1938.
115
ACOUSTICAL ELEMENTS
of the horn should be proportional to the frequency, since the acoustical
reactance is primarily an inertive reactance and, therefore, proportional
to the frequency. Practically any throat acoustical impedance frequency
characteristic may be obtained by employing a horn consisting of manifold
exponential sections.
A horn consisting of three rates of flare is shown in Fig. 5.13. The
acoustical impedance characteristic at the throat of the small horn is
obtained in stages. First, the throat acoustical impedance characteristic
for the large horn is obtained by using equation 5.80. The throat acoustical
impedance obtained for the large horn now becomes the mouth acoustical
impedance of the intermediate horn. The acoustical impedance of the
throat of the intermediate horn is obtained by employing equation 5.80.
For the frequency corresponding to b = 0 of the intermediate horn the
acoustical impedance at the throat of the intermediate horn becomes
indeterminate. The expression can be evaluated as shown in Sec. 5.27 on
the finite exponential horn. N ext, the throat acoustical impedance at the
throat ·of the small horn is obtained by again employing equation 5.80.
The mouth acoustical impedance of the small horn is the throat acoustical
impedance just obtained for the intermediate horn. The acoustical
impedance characteristic of Fig. 5.13 shows three distinct steps depicting
the surge acoustical impedance of each section.
5.31. Closed Pipe with a Flange.-The acoustical impedance of a pipe
closed at one end and equipped with a flange at the open end may be con­
sidered to be the sum of the acoustical impedance of the pipe and the end
correction. It will be assumed that the open end of the pipe is equipped
with a massless piston.
The input acoustical impedance to the pipe at the massless piston may
be obtained from equation 5.74 by setting ZA2 = 00. The input acoustical
impedance of the pipe closed at the far end is
ZAO =
where l
R
=
=
p =
c=
k
=
A=
-
jpc
7I"R2 cot kl
5.90
length of the pipe, in centimeters,
radius of the pipe, in centimeters,
density, in grams per cubic centimeter,
velocity of sound, in centimeters per second,
2/7I"A, and
wavelength, in centimeters.
The above equation is the acoustical impedance of a closed pipe when
there is no end correctiQn, as for example, when the pipe is used in a closed
system.
When the open end is free and terminated in a large baffle the total
acoustical impedance is the sum of equations 5.12 and 5.90.
ZAT =
pc [
h(kR)]
7I"R2 1 -
--w- + J. 271"Rwp4k3 Kl(2kR)
. pc
- J 7I"R2 cot kl
5.91
116
ACOUSTICAL ENGINEERING
The ratio of the pressure at the closed end of the tube to the free space
pressure is useful in predicting the performance of pipes and cavities.
The ratio of the pressure at the closed end to that in free space is
i!!! = J[COS kl P
7TR2 XA sin kl]2
~
+ (7TR2)2 rA 2 sin 2 kl
5.92
~~2
where P = pressure at the closed end, in dynes per square centimeter,
Po = pressure in free space, in dynes per square centimeter,
r A = acoustical resistance, in acoustical ohms, equation 5.12, and
XA = acoustical reactance, in acoustical ohms, equation 5.12.
«=f
3.4
3.2
llil
3.0
2.8
I
2.6
2,4
0(=1
I
1\
\
22
)-...
4 2D
...../.
/«=1. 'I'~
/.8
.,A< =2 V\
0 , •6
I'
f:: 1.4
~
/
_v__ V
e::::;..
1.2
1.0
/
,/
.....-:
l\
.....,;,3...k­ ~
. . . fi(=s.l .n
...
\
.8
, ""- tz....
\
....... f-"""
.6
....,.
~
jJ
.2
~,
.2
.3
.4
.S.6
.8
1.0
2
3
4
.S
kR=~
FIG. 5.14.
Ratio of the pressure at the bottom of a cylindrical
cavity to the free space pressure in the incident sound wave as a
function of kR.
The characteristics of Fig. 5.14 depict the ratio of the pressure at the
closed end of a cylindrical cavity to the pressure in free space as a function
of the dimensions of the cavity and the wavelength of the sound.
5.32. Sound Transmission in Tubes. 38 ,39,4o,41-The effect of viscosity
upon the characteristics of small holes and slits was considered in Sees. 5.3
38 Crandall, .. Vibrating Systems and Sound," D. Van Nostrand Company, Princeton,
N.J., 1926.
39 Lamb, .. Theory of Sound," E. Arnold, London, 1931­
40 Rayleigh, .. Theory of Sound," Macmillan and Company, London, 1926.
41 Mason, W. P., Phys. Rev., Vol. 31, No.2. p. 283, 1928.
117
ACOUSTICAL ELEMENTS
and 5.4. The transmission loss in tubes of circular section is of interest in
problems in acoustics involving the use of tubes. The equation 41 expressing
the sound transmission in a tube is
A
where A
=
a =
Aoc ax
=
5.93
amplitude (pressure or volume current) at a distance x centi­
meters from the amplitude A o,
;~J~:,
R = radius of the tube, in centimeters,
c = velocity of sound, in centimeters per second,
w = 27TJ,
f = frequency, in cycles per second,
JL = viscosity coefficient, 1.86 X 10- 4 for air,
p = density, in grams per cubic centimeters,
y' = 1 + 1.58 (y1/2_ y-1/2), and
y = ratio of specific heats, 1.4 for air.
The attenuation characteristics of tubes of various diameters as a function
of the frequency are shown in Fig. 5.15.
10
•
•
·
V
•
•
·
··
.-
2
I
'-
..2
~
I­
~
0
- ­--­
- ­
-.-
V
2$"
L.-­
..
i--'
l.--­
V3
f.-
I--"'"
2
.0I.
~
~
~
-
2
f-"
•
$
e
7
a
IJI
10 2
2
rREQUENCY
3
4
IN
CYCLES
5
& 7
a
II
PER
/0 3
2
3
....
5
e 7ee 10'"
SECOND
FIG. 5.15.
The attenuation of a sound wave, in decibels per foot, as a
function of the frequency, in cycles per second, in pipes of various diameters
and filled with dry air at 20° Centigrade.
5.33. Transmission from One Pipe to Another Pipe of Different
Cross-sectional" Area. 4 2--Consider two pipes of cross sections S1 and S2
joined as shown in Fig. 5.16. Assume that sound travels from pipe Sl to
pipe S2.
42
Stewart and Lindsay, " Acoustics," D. Van Nostrand Company, Princeton, N.J.,
1930.
118
ACOUSTICAL ENGINEERING
The boundary conditions are
1. Continuity of pressure,
2. Continuity of volume current.
The condition for pressure may be written
5.94
h+h' =P2
where h = incident pressure in pipe 51, in dynes per square centimeter,
h' = reflected pressure in pipe 51, in dynes per square centimeter, and
P2 = transmitted pressure in pipe 52, in dynes per square centimeter.
The condition for volume current may be written
UI-Ul' = U2
where Ul
=
5.95
incident volume current in 51, in cubic centimeters per second,
Ul ' = reflected volume current in 51, in centimeters per second,
U 2 = transmitted volume current in 52, in centimeters per second.
The acoustical resistance of the first pipe 51 is
rAl =
pc
-
PI
=-
5.96
51
Ul
where p = density of the medium, in grams per cubic
centimeter,
c
=
velocity
of sound in the medium, in centi­
FIG. 5.16. Two con­
nected pipes of cross­
meters per second, and
sectional areas S 1 and
51 = cross-sectional area of the first pipe, in
52.
square centimeters.
The acoustical resistance of the second pipe 52 is
P2'
pc
-=5.97
52
U2
where 52 = cross-sectional area of the second pipe, in square centimeters.
Expressing equation 5.91 in terms of pressure
rA2=
h51 - h'51 =
P25 2
or
2
PL -P'1 -_P25
51
5.98
Eliminating h' from equations 5.94 and 5.98
P2 = h(251)
51
+ 52
=
~
1
52
+
5.99
51
Expressing in terms of volume current,
U2=~
1
+ 51
52
5.100
ACOUSTICAL ELEMENTS
119
Equations 5.99 and 5.100 show that the pressure and volume current
of the transmitted wave in pipe 52 is always in phase with the pressure and
volume current of the incident wave in pipe 51.
The reflected pressure in terms of the incident pressure is
PI'
=
(~~ ~ ~:)PI
5.101
The reflected volume current in terms of the incident volume current is
5.102
Equations 5.101 and 5.102 show that if 51 < 52 the reflected pressure or
volume current are in phase with the incident pressure or volume current. If
51> 52, the reflected pressure or volume current are opposite in phase with the
incident pressure or volume current. If 51 = 52, there is no reflected wave.
The ratio of the transmitted power to the incident power is
5.103
5.34. Transmission Through Three Pipes. 43-Consider three pipes of
cross sections 51, 52, and 53 as shown
S,
A
in Fig. 5.17. Assume that sound
B
travels from pipe 1 to pipe 3. Let the
boundary between 51 and 52 be de­
noted by A and between 52 and 53
by B.
The boundary conditions are
1. Continuity of pressure,
2. Continuity of volume current.
At the boundary A the conditions for
the pressure may be written
5.104
b PI' = Pz Pz'
+
where PI
+
FIG. 5.17.
Three pipes of cross-sec­
tional areas 51, 52, and 53. The pipe
52 is of finite length l.
incident pressure in 51, in dynes per square centimeter,
PI' = reflected pressure in 51, in dynes per square centimeter,
P2 = transmitted pressure in 52, in dynes per square centimeter, and
P2' = reflected pressure in 52, in dynes per square centimeter.
At the boundary A the conditions for the volume current may be written
=
Ul
where Ul
Ul '
-
Ul '
U2 -
U2'
5.105
incidel).t volume current in 51, in cubic centimeters per second,
= reflected volume current in 51, in cubic centimeters per second,
U 2 = transmitted volume current in 52, in cubic centimeters per
second, and
U 2' = reflected volume current in 52, in cubic centimeters per second.
=
43 Stewart and Lindsay, " Acoustics," D. Van Nostrand Company, Princeton, N.].,
1930.
120
ACOUSTICAL ENGINEERING
At the boundary B the conditions for the pressure may be written
where P2
=
P2'
=
Pa
=
I =
P2clkl + h' Elkl = Pa
5.106
incident pressure in 52, in dynes per square centimeter,
reflected pressure in 52, in dynes per square centimeter,
transmitted pressure in 5 a, in dynes per square centimeter,
length of pipe 52, in centimeters,
k = 21T/A,
A = wavelength, in centimeters.
At the boundary B the conditions for the volume current may be written
U2clkl - U2'Elkl = Ua
5.107
where U 2
transmitted volume current in 52, in cubic centimeters per
second,
U 2 ' = reflected volume current in 52, in cubic centimeters per second,
and
Ua = transmitted volume current in 5 a, in cubic centimeters per
second.
From equations 5.104,5.105,5.106, and 5.107,
=
PI
=
~a[(~ + 1) cos kl + j(~: + ~~) sin kl]
5.108
The ratio of the power transmitted in 5 a to the incident flow of power
in 51 is
PIa =
(~+
1)2 cos2 kl + (~ + ~)2 sin2 kl
51
51 52
5.109
If kl is small, the transmission is independent of the cross section of the
channel 52. If sin kl = ±1, the power transmission is
4~
PIa =
(~
51
51
+ ~)2
=
45 a5 225 1
(5 22 + 515 a)2
5.110
52
Equation 5.105 shows that P = 1, if 5 22 = 515 a. That is, if sin kl = ±1
and providing the area of 52 is a geometric mean of 51 and 5 a, the trans­
mission is unity..
5.35. Transmission from One Medium to Another Medium. 44- The
problem of transmission from one medium to another medium as shown
in Fig. 5.18 is the same as the problem transmission from one pipe to another
44 Stewart and Lindsay, "Acoustics," D. Van Nostrand Company, Princeton, N.J.,
1930.
121
ACOUSTICAL ELEMENTS
pipe of different cross-sectional area. The boundary between the two media
is assumed to be plane and parallel to the wavefront which is also assumed
to be plane.
The ratio of the power transmitted in the medium 2 to incident flow of
power in the medium 1 of Fig. 5.18 is
P12
=
4rAIrA2
5.111
+
(1' Al
l' A2)2
where l' Al = acoustical resistance of medium 1, and
l' A2 = acoustical resistance of medium 2.
FIG. 5.19. Three media of acoustical
resistances rAl, rA2, and rA3. The
length of the medium 2 is 1.
FIG. 5.18. Two media of
acoustical resistances r A 1 and
YA2·
5.36. Transmission Throu~h Three Media. 45-The problem of trans­
mission through the three media of Fig. 5.19 is the same as that through the
three pipes. The ratio of the power transmitted in the medium 3 to the
incident flow of power in the medium 1 is
PI3 =
( rAI
l' A3
where
l' Al
=
l' A2 =
l' A3
=
l
=
r
+1
cos2kl
{AI
+rAtrAI + rA2
acoustical resistance of the
acoustical resistance of the
acoustical resistance of the
length of the medium 2, in
l' A2
l' A3
r
5.112
sin 2 kl
medium 1,
medium 2, between 1 and 3,
medium 3,
centimeters,
k = 27T/A,
A = waveleng~h in the medium 2, in centimeters.
5.37. Tubes Lined with Absorbin~ Material.-In ventilator and ex­
haust systems it is desirable to provide a high degree of attenuation for
audio-frequency waves while offering low resistance to continuous flow of
alr. For that purpose, the most satisfactory systems are ducts lined with
45
Stewart and Lindsay, "Acoustics," D. Van Nostrand Company, Princeton, N.J.,
1930.
122
ACOUSTICAL ENGINEERING
absorbing material. Longitudinal isolation of the walls of the duct should
be provided to prevent longitudinal transmission of sound by the walls of
the duct. This can be accomplished by the use of rubber connectors at
regular intervals. The walls of the duct should be rigid so that air-borne
sounds are not transmitted through the walls. Very high attenuation
can be obtained in ducts of this type.
The attenuation, in decibels per foot, in a square or rectangular conduit
lined with absorbing material may be obtained from the following empirical
formula,46
db
P
5.113
Tt = 12.6a1.4 A
where P
A
perimeter, in inches,
= cross-sectional area in square inches, and
a = absorption coefficient of the material used for lining the duct.
Equation 5.113 holds for square ducts and rectangular ducts in which
the ratio between the two sides is not greater than two.
The general subject 47 ,48 of tubes lined with absorbing material, with
both rigid and vibratile walls, has been considered theoretically and experi­
mentally.
5.38. Response of a Vibrating System of One Degree of Freedom.­
Consider the electrical circuit, consisting of inductance, electrical resistance,
and electrical capacitance and a voltage connected in series, as shown in
Fig. 5.20. The resonant frequency, in cycles per second, is given by
1
5.114
/r = 21T VLC E
=
where L = inductance, in henries, and
CE = electrical capacitance, in farads.
The current in the circuit is given by
e
i =
YE
where
1
+ j(wL + jwCE)
5.115
electrical resistance, in ohms,
e = driving voltage, in volts, and
i = current, in amperes.
The quantity Qr is given by
YE =
Qr _
where Wr
=
wr L
YE
21T/r.
Sabine, H. J., Jour. Acous. Soc. Amer., Vol. 12, No. 1. p. 53, 1940.
47 Sivian, L. J., Jour. Acous. Soc. Amer., Vol. 9, No.2, p. 135, 1937.
48 Molloy, C. T., Jour. Acous. Soc. Amer., Vol. 16, No.1, p. 31, 1944.
46
5.116
ACOUSTICAL ELEMENTS
L
CE
123
To
~=:Ji
ELECTRICAL
VALUES
CIRC UIT
OF
Qr
1.0
.8
1 - '-f-­
.2
I
.08
w .04
f/)
z
o
II.
f/)
w
.02
II:
7
~
l--'
2f- f-~
I......
~
~
./
V
I l\
V
~
.004
/"
V
./
.0 I
.008
'-'"
~
,,,
7
7
/ ,I
V
V
l--'
.00.I 5
V
.6
.7
" r-.....
~
"­
'"
/
.8
.9
1.0
1.1
5
~
......
10
I---t­
­
0 - 1-­
\..
I \~
II \ \
./
.002
I--­
I'-.
~
V / \\ '\r--... I'-.I'-.
~
~
......
\ !'\.
//
.........
I'-. r-...
""- so ......
~
00
K
r-...
ff
..........
r-.
I-f-
1---1--­
1.2 1.3 IA 1.5 1.6 1.71.81.9 2.0
f+fr
FIG. 5.20. The current response characteristics of a simple series
circuit as a function of the ratio f -:- fr. where fr = the resonant
frequency, and f = the frequency under consideration. The
numbers of the characteristics refer to the value of Qr. Qr =
27rfrL/YE. The above characteristics are applicable to acoustical
and mechanical systems by the substitution of the elements and
quantities which are analogous to the electrical system.
The current response characteristics as function of the ratio I -:- Ir for
various values of Qr are shown in Fig. S.20.
The above characteristics are applicable to acoustical and mechanical
systems by the substitution of the elements and quantities which are analo­
gous to the electrical system (see Chapter IV).
6
DIRECT RADIATOR LOUDSPEAKERS
6.1. Introduction. 1-A loudspeaker is an electroacoustic transducer
designed to radiate acoustical energy into a room or open air. There
are two general types of loudspeakers in use today, namely: the direct
radiator and the horn type loudspeaker. The diaphragm of the direct
radiator loudspeaker is coupled directly to the air. The diaphragm of the
horn loudspeaker is coupled to the air by means of a horn. The direct
radiator loudspeaker will be considered in this chapter and the horn loud­
speaker will be considered in the following chapter.
The almost universal use of the direct radiator loudspeaker is due to
the simplicity of construction, small space requirements, and the relatively
uniform response characteristic. Uniform response over a moderate fre­
quency band may be obtained with any simple, direct radiator dynamic
loudspeaker. However, reproduction over a wide frequency range is
restricted by practical limitations. The two extreme ends of the audio­
frequency band are the most difficult to reproduce with efficiency comparable
to that of the mid-audio-frequency range. Inefficiency at the low frequencies
is primarily due to the small radiation mechanical resistance. There are a
number of means available for increasing the radiation mechanical resistance
at the low frequencies. A large radiation mechanical resistance may be
obtained by using a large cone. A phase inverter consisting of a completely
enclosed cabinet with ports provides a means for extending the low-frequency
range. A horn may be used for presenting a large radiation mechanical
resistance to a diaphragm at the low frequencies. The efficiency of a direct
radiator loudspeaker at the high frequencies is limited by the mechanical
mass reactance of the vibrating system. There are a number of arrange­
ments suitable for reducing the mass of the vibrating system at the high
frequencies. Two or more separate loudspeaker mechanisms may be used,
each designed to reproduce a certain portion of the range. Multiple cones
driven by a single voice coil may be arranged so that the mass of the system
decreases at the high frequencies. The voice coil may be sectionalized to
decrease the mass and inductance at the high frequencies and thereby
increase the high-frequency range. Multiple coils and multiple cones
1
Mawardi, Osman K ., Jour. Acous. Soc. Amer., Vol. 26, No. L p. L 1954.
124
DIRECT RADIATOR LOUDSPEAKERS
125
combined into a single mechanism may be designed to yield uniform response
to the upper limit of audibility.
It is the purpose of this chapter to outline the factors which influence
the performance of the conventional, direct radiator loudspeaker, to illustrate
systems for controlling the response with respect to frequency and to describe
several means for decreasing the effective mass of the vibrating systems at the
high frequencies and for improving the efficiency at the low frequencies.
6.2. Single-Coil, Single-Cone Loudspeaker.-The simple dynamic
loudspeaker consists of a paper cone driven by a voice coil located in a
magnetic field. A cross-sectional view, the voice coil circuit and the
mechanical circuit of a dynamic loudspeaker are shown in Fig. 6.1. The
VOICE
COIL
MECHANICAL
OF
ELECTRICAL
CIRCUIT
CIRCUIT
THE
MECHANICAL
SYSTEM
CROSS-SECTIONAL
VIEW
FIG. 6.1.
Cross-sectional view of a single-coil, single-cone, direct radiator,
dynamic, loudspeaker mechanism mounted in a baffle. In the voice coil
circuit, e = the internal voltage of the generator. rEa = the internal electrical
resistance of the generator. rEO and L = the electrical resistance and induc­
tance of the voice coil. ZE,U = the motional electrical impedance. In the
mechanical circuit, me = the mass of the cone and voice coil. eMS = the com­
pliance of the suspension system. rMS = the mechanical resistance of the
suspension system. mA = the mass of the air load. rMA
the mechanical
resistance of the air load. hI = the mechanomotive force in the voice coil.
ZME = the mechanical impedance due to the electrical circuit.
IMo = the
mechanomotive force of the mechanical generator.
total mechanical impedance, in mechanical ohms, of the vibrating system
at the voice coil is
J
WCMS
where
6.1
mechanical resistance of the suspension system, in mechanical
ohms,
= mechanical resistance of the air load, in mechanical ohms,
rMS =
rMA
me
=
mass of the cone and the voice coil, in grams,
mA = mass of the air load, in grams, and
CMS =
compliance of the suspension system, in centimeters per dyne.
126
ACOUSTICAL ENGINEERING
Equation 6.1 may be written as follows
ZMT =
where
rMS =
rMA =
XMC =
XMA =
XMS =
rMS
+ rMA + jXMC + jXMA
-
6.2
jXMS
mechanical resistance of the suspension system, in mechanical
ohms,
mechanical resistance of the air load, in mechanical ohms,
wmc = mechanical reactance of the voice coil and cone, In
mechanical ohms,
wmA = mechanical reactance of the air load, in mechanical
ohms, and
1
-C
w
MS
=
mechanical reactance of the suspension system, in
mechanical ohms.
The mechanical resistance and mechanical reactance of the air load may
be obtained from Sec. 5.10 and Fig. 5.2.
The motional electrical impedance,2 in abohms, of the mechanical system
is
(Bl) 2
ZEM=-­
ZMT
where B
l
=
=
ZMT =
6.3
flux density in air gap, in gausses,
length of the conductor in the voice coil, in centimeters, and
total mechanical impedance of the mechanical system, in
mechanical ohms.
The efficiency of the loudspeaker is the ratio of the sound power output
to the electrical power input. The efficiency, in per cent, may be obtained
from the voice coil circuit of Fig. 6.1 and expressed as follows,
JL =
rEC
where
rER =
rEM =
rEC =
rER
X
rEM
+
100
6.4
component of the motional electrical resistance due to the
radiation of sound, in abohms,
total motional electrical resistance, in abohms, and
damped electrical resistance of the voice coil, in abohms.
The components rER and rEM may be obtained from equations 6.2 and
6.3.
From equations 6.2, 6.3, and 6.4, the efficiency, in per cent, of the loud­
speaker is
I'- =
(Bl)2rMA
(Bl)2(rMS
+ rMA) + rEC[(rMc + rMA) 2 + (XMA +XMC -XMS) 2]
X
100 6.5
2 Olson, .. Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J.,
1943.
DIRECT RADIATOR LOUDSPEAKERS
127
To simplify the discussion assume that the mechanical reactance and
mechanical resistance of the suspension system are zero. The mechanical
impedance characteristics of the mechanical system are shown in Fig. 6.2.
Since YMA is small compared to XMA and XMC, equation 6.5 becomes
+
SYSTEM
DIAMETER INCHES
MASS Of CONE GRAMS
MASS Of VOICE COIL GRAMS
COMPLIANCE SUSPENSION
SUSPENSION
MECH. RES.
VOICE COIL MATERIAL
AIR GAP FLUX GAUSSES
A
<!
r
'7
'"zu
·fu •
C3
,~
'" I
Q.
"
'"2
\ x....
40
4
I
.35
.015
.015
3.2 X 10""7 e.oxl()",7 5.3XIO-7
ZOO
CU.
110
CU.
10000
10000
10000
2400
/
,
~ 10'
:!!
/ ..L
,
//
..J
'XV
<:
u
\
~Io'
Xw:'
'/ ~
/
V
v
air!
2
""
8103 2
""
Buf Z
o.
•
fREQUENCY
,
100
•10'
I~
X MS
,
/
1,1 /17<
" 102
2
/
//
/ / X"" /~A
"
102
/
,
J:
"
J
"
!oJ
XMS
[Il'MA
4
"
u
'
/
4
o
/
c/
,
2
u
z
<:
x, 10"
<'i
Q.
:;::,
I
lri'
'"
/
II
10
AL.
c
Iri'
u
'\
X..a
'""
C
I
10'
7 "
..J
<:
II
<: I
J:
u
B
4
A
u
/
2
16
B
r!
I~
6.6
I
I/­
0>
100
(Bl)2YMA
X
YEC(XMA
XMC)2
J1, =
4
•
10 2 4
2
FREQUENCY
r/
,
V
,
.,0' 2 4 "10' 2
02'
•
fREQUENCY
100
100
,
}J
,. 10
0
!:1
";;:'
..'"u
I
~
II',
,
1;
U
I
'\
I
'0. 2
4
"10>
2.
•
1()3 2
fREQUENCY
4
•10' •
...;;:
'"
I\,
I
I
0.I,
lu.
,
0
zw
I
I
I
I.b
"10>
2
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10'
fREQUENCY
•
4
•10' 2
FIG. 6.2. The mechanical impedance frequency characteristics of three direct radiator
loudspeakers having I-inch, 4-inch, and I6-inch diameter cones. X.lfC = the mechanical
reactance due to the cone and coil. XMS = the mechanical reactance due to the suspen­
sion system. XMA = the mechanical reactance due to the air load. 'YMA = the
mechanical resistance due to the air load. The efficiency characteristics shown are for
the constants as shown in the table and the graphs of the mechanical impedances. In
the efficiency characteristics, /-'1 = the efficiency for XMS equal to zero. /-'2 = the
efficiency for XM8 as indicated by the graph.
128
ACOUSTICAL ENGINEERING
In terms of the resistivity and density of the voice coil, equation 6.6
becomes
6.7
where
ml =
p =
Kr
=
mass of the voice coil, in grams,
density of the voice coil conductor, in grams per cubic centi­
meter, and
resistivity of the voice coil conductor, in microhms per centi­
meter cube.
The density, resistivity, and density-resistivity product of various ele­
ments are shown in Table 6.1.
TABLE 6.1.
DENSITY p. in GRAMS PER CUBIC CENTIMETER; RESISTIVITY K" IN MICROHMS
PER CENTIMETER CUBE AND DENSITY-RESISTIVITY PRODUCT pKT OF VARIOUS ELEMENTS;
TEMPERATURE, 20° C.
Element
Sodium
Lithium
Potassium
Calcium
Aluminum
Magnesium
Titanium
Copper
Silver
Chromium
Beryllium
Barium
Manganese
Caesium
Zinc
Gold
Molybdenum
Cadmium
Nickel
Iron
Cobalt
Tin
Tungsten
Iridium
Platinum
Lead
Antimony
Bismuth
Mercury
p
,
.97
.53
.87
1. 55
2.70
1.74
4.5
8.89
10.5
6.93
1.8
3.5
7.2
1.9
7.14
19.3
10.2
8.6
8.8
7.9
8.7
7.3
19.0
22.4
21.3
11.0
6.6
9.7
13.5
KT
4.6
9.4
7.1
4.6
2.82
4.6
3.2
1.72
1.63
2.6
10.1
9.8
5.0
21.2
5.9
2.44
5.7
7.4
7.8
9.8
9.7
11.5
5.5
6.5
9.8
22.0
41. 7
119.0
95.7
pKT
4.5
5.0
6.2
7.1
7.6
8.0
14.4
15.2
17.1
18.0
18.2
34.0
36.0
40.2
42.0
47.0
58.0
64.0
69.0
78.0
84.0
84.0
105.0
146.0
208.0
242.0
275.0
1116.0
1290.0
The relation between the efficiency and the ratio of the mass of the voice
coil to the mass of the cone and the air load may be obtained from equation
6.7 and is depicted in Fig. 6.3. The maximum efficiency occurs when the
mass of the voice coil is equal to the mass of the cone and air load.
129
DIRECT RADIATOR LOUDSPEAKERS
~o
li
w­
a
/
I
~
-2
V
III
'-I"
o
/
-3
/
/
-.......
Vi-'
w
In
V
"­
"\
\
'\
\
/
.2
.8
.4
I
2
1\
8
4
10
m,
mo+m"
FIG.
6.3.
The efficiency loss in a direct radiator loudspeaker as a
function of the ratio
mn
mr
+ mA
, where mr = the mass of the voice
coil, mn = the mass of the diaphragm, mA = the mass of the air
load. The maximum efficiency is arbitrarily depicted as db,
°
In general, in commercial loudspeakers 3 ,4 it is not practical to make the
cone mass equal to the voice-coil mass. As a matter of fact, the cone mass
is usually several times the voice-coil mass. A consideration of equation
6.7 shows that the efficiency can be increased by the use of a light-weight
9
m
06
~
t­
:::J
Q.
t-3
:::J
o
o
----- ---LO
1.5
r-­
2.0
2.5
3.0
RELATIVE CONE MASS
FIG. 6.4.
Output of a typical direct radiator
loudspeaker as a function of the mass of the
cone.
cone. The relative output of a typical direct radiator loudspeaker as a
function of the weight of the cone is shown in Fig. 6.4. In this example, the
weight of the permanent magnet was kept constant. However, the mass
of the voice coil and the air gap were selected to obtain maximum output.
3
4
Olson, H. F., Audio Engineering, Vol. 34, No. 10, p. 5, 1950.
Olson, Preston, and May, Jour. Aud. Eng. Soc., Vol. 2, No.4, p. 219, 1954.
130
ACOUSTICAL ENGINEERING
There is a limit to the extent to which the reduction in mass of the cone can
be carried because, as the cone mass is reduced, the strength of the cone is
reduced and as a consequence the nonlinear distortion is increased due to
overload of the material of the cone. High sensitivity and low distortion
are not compatible. In order to obtain low nonlinear distortion, a relatively
heavy cone must be used. The subject of nonlinear distortion and cone
weight will be discussed in Sec. 6.26.
The mechanical impedance and corresponding efficiency characteristics
assuming the mechanical reactance due to the suspension to be zero are
shown in Fig. 6.2. The air load mechanical resistance and mechanical
reactance are assumed to be the same as those on two sides of a vibrating
piston with the diameter equal to the cone diameter (see Sec. 5.8). The
weights of the cones and voice coils are typical of loudspeakers in actual
use today. It will be seen that the efficiencies of all three systems are
practically the same. Of course, the power-handling capacity of the smaller
cones is very small at the lower frequencies.
In the preceding considerations the mechanical reactance due to the
suspension system was assumed to be zero. The efficiency in which all
the elements of the vibrating system are included may be obtained from
equation 6.5. The mechanical, resistance rMC, due to the suspension system
is also a factor in the efficiency in the region of resonance. Typical values
of rMC for 16-, 4-, and I-inch cones are shown in Fig. 6.2. The efficiency
characteristics under these conditions are shown in Fig. 6.2. It will be
noted that the efficiency is high at the resonant frequency. However,
when coupled to a vacuum tube driving system the motional electrical
impedance is also increased which reduces the power input to the voice
coil. For this reason the response is not accentuated to the degree depicted
by the peak in the efficiency characteristic. It will be seen that the
efficiency decreases very rapidly below the resonant frequency. There­
fore, in a direct radiator l~)Udspeaker the limit at the low-frequency end
of the frequency range is determined by the resonant frequency of the
system.
The motional electrical impedance of a dynamic loudspeaker is given
by equation 6.3. The normal electrical impedance, in abohms, of voice
coil is given by
6.8
ZEN = ZEM
ZED
+
where
ZEM =
ZED =
motional electrical impedance, in abohms, and
electrical impedance of the voice coil in the absence of motion,
that is blocked, in abohms.
The components of the motional electrical impedance are shown in
Fig. 6.5. At the resonant frequency the motional electrical impedance is
large because the mechanical impedance is small. The current in the voice
coil circuit may be determined from the voice coil electrical circuit, the
driving voltage and the electrical resistance of the generator.
DIRECT RADIATOR LOUDSPEAKERS
131
The mechamotive force or driving force,5 in dynes, applied to the mechan­
ical system is
6.9
1M = Eli
where B
=
l =
i =
flux density in the air gap, in gausses,
length of the conductor, in centimeters, and
current in the voice coil circuit, in abamperes.
This is the driving force,1M, applied to the mechanical system as shown in
Fig. 6.1.
140
100
Ul
80
1\
:::;;
a::
0
ZE...
JrEM +jX~ ...
5
60
I"E...
!!:
..,
40
20
u
z
«
0
..,
0
\
J
~/
\
w
102
f\
80
\ ZEN
60
\
u
'-....
Z
«
~
w
Q.
~
2
40
0
1I
a
ZEN 1= ZED+tEM
100
~
XE,
~-20 -
120
Ul
:::;;
20
8
4
\
V
./
ZED
102
'"
2
4
8 1()3
FREQUENCY
FREQUENCY
FIG. 6.5. The electrical impedance characteristics of the voice coil in a direct
radiator loudspeaker. ZEN = the normal electrical impedance. ZED = the
damped electrical impedance. ZEM = the motional electrical impedance.
rEM = the resistive component of the motional electrical impedance.
XEM =
the reactive component of the motional electrical impedance.
The mechanical impedance,6
circuit is
111
mechanical ohms, due to the electrical
(El)2
where
ZET =
rEG =
L
=
rEG =
+
+
ZME = - ­
ZET
6.10
rEC
jwL
rEG,
damped electrical resistance of the voice coil, in abohms,
damped inductance of the voice coil in abhenries, and
electrical resistance of the generator, in abohms.
This mechanical impedance appears in the mechanical system as shown in
Fig. 6.1. In calculating the steady state performance the driving force,
1M, applied to the mechanical system is used and the mechanical impedance
due to the electrical system need not be considered. However, in comput­
ing the transient response of the system, the damping constant, etc., the
5 Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J.,
1943.
6 Ibid.
132
ACOUSTICAL ENGINEERING
mechanical impedance due to the electrical circuit must be included. The
driving force of the generator in the mechanical system which will produce
a force, /M, across the mechanical system is
6.11
/MO =/M +/MZME
ZMT
The increase of electrical impedance of the voice coil, with frequency, in
combination with the existing vacuum tube driving system, is another
factor which reduces the response of a dynamic loudspeaker at the higher
frequencies. The electrical impedance characteristics of the vacuum tube
power amplifiers are generally designed so that the voltage across the loud­
speaker, for constant voltage applied to the input of the power stage, is
independent of the frequency. Therefore, the current in the voice coil
decreases with frequency as the electrical impedance increases with fre­
quency. The electrical impedance frequency characteristics of several
voice coils are shown in Fig. 6.6. In the case of a large, heavy voice coil
40
y
1/
;:'~T
/
..
~
~ 10
I-
~
..,.J
. ..
o
1 ••
I
j
2
3
1--"---
.
oS
6
V
7 8 •
rREQUENCY IN CYCLES
10'
10T"
~
/'"
­
r------
.J
<)
~
/
2
f--'-"'
~
3
4
!1e7S'
10'
2
PER SECOND
FIG. 6.6. The electrical impedance characteristics of 1!-inch diameter voice coils of 140,
70, a nd 18 turns and all having 10 ohms d-c resistance.
the rapid increase of the electrical impedance at the higher frequencies
causes a corresponding reduction in the driving force. To maintain the
driving force at the higher frequencies requires a relatively low ratio of the
inductive electrical reactance to the electrical resistance which for a con­
stant value of the electrical resistance is equivalent to a reduction in the
mass of the voice coil.
The response of a loudspeaker is a measure of the sound pressure produced
at a designated position in the medium with the electrical input, frequency,
and acoustic conditions specified. In general, the response is obtained on the
axis of the cone. If the loudspeaker were nondirectional, the efficiency
characteristic would also be the response frequency characteristic. The
system is not nondirectional but is similar to that of a vibrating piston, in
DIRECT RADIATOR LOUDSPEAKERS
133
that the directional becomes sharper with increase in frequency. However,
the piston directional pattern cannot be used because there is considerable
deviation from piston action in a cone loudspeaker. Measured directional
characteristics of direct radiator loudspeakers, having the constants given
.5
45
FIG. 6.7. Directional characteristics of a dynamic, direct radiator loudspeaker, with a
110° cone 4 inches in diameter, mounted in a large baffle.
45
FIG. 6.8. Directional characteristics of a dynamic, direct radiator loudspeaker with a
110° cone 16 inches in diameter, mounted in a large baffle.
in Fig. 6.2, are shown in Figs. 6.7 and 6.8. Employing the mechanical
circuit and the electrical circuit of Fig. 6.1 and the data of Fig. 6.2, the total
output of the loudspeaker may be determined as outlined in this section.
It is quite obvious that the response on the axis will be accentuated at the
high frequencies due to the sharpening of the directional pattern.
134
ACOUSTICAL ENGINEERING
The power output of a loudspeaker may be obtained from the direc­
tional pattern and the response frequency characteristic by considering the
sound flow through a spherical surface in which the loudspeaker is located
at the center (see Sec. 10.3Dl). The surface is divided into incremental
areas and the power transmitted through each area is determined from the
sound pressure. The total power is equal to the summation of the in­
cremental areas and may be expressed as
6.12
where P
=
p =
c=
p
=
dS =
total power, in watts,
density of the medium, in grams per cubic centimeter,
velocity of sound in the medium, in centimeters per second,
root mean square sound pressure over the element of area dS,
in dynes per square centimeter, and
element of area on the spherical surface, in square centimeters.
In the case under consideration the power output, P, as a function of
the frequency may be determined from equation 6.5 and the electrical
input. The directional patterns for the cones having diameters of 4 and
16 inches are shown in Figs. 6.7 and 6.8. From these data, the pressure
on the axis may be determined from equation 6.12. The computed re­
sponse frequency characteristics of the loudspeakers of Fig. 6.2 are shown
in Fig. 6.9. These characteristics are quite similar to the actual response
frequency characteristics.
Another factor of interest in a direct radiator is the power handling
capacity. The sound power output, in watts, is given by
6.13
where
rMA
=
x=
mechanical resistance, in mechanical ohms, obtained from
Sec. 5.8, and
root-mean-square velocity of the piston, in centimeters per
second.
Equation 6.13 may be used to compute the power output of a direct
radiator loudspeaker in the region were all parts of the cone move in phase.
In general, the output is limited by the permissible amplitude. The greatest
amplitude occurs at the low frequencies where the action is essentially that
of a piston. Therefore, piston action may be assumed.
The peak amplitude characteristics of a 16-inch, a 4-inch, and a I-inch
piston mounted in an infinite baffle for 1 watt of sound output are shown
in Fig. 6.10. These characteristics show that for practical amplitudes
a relatively large piston is required to deliver adequate power at the low
frequencies.
The directional pattern of a vibrating paper cone depends on three
principal factors: the cone diameter, the cone angle, and the frequency.
135
DIRECT RADIATOR LOUDSPEAKERS
Other factors, such as the paper pulp, the processing, the corrugations,
the voice coil diameter, and the suspension also influence the directional
pattern, but in a lesser degree. The directional patterns for various fre­
quencies of 110° cones having diameters of 4 and 16 inches are shown in
A
50
.,40
o
~ 30
~30
1"\
...
810 Z 2
A
8'012
4
~20
<II
"' 10
o
l-
i
z
1\
10
10 40
l.r..
o
.....+-.
~20
°2
50
10 40
h
z
o
"''"
c
B
0
~30
\
:};20
7
°2
...
81022
~REQUENCY
4
8'0,2
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o. •
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~REQUENCY
1'\
......
/
'"a: 10
a:
1'042
/
z
o
V
810 "
•
"10"
•
"104'
,REQUENCY
FIG. 6.9. Pressure response frequency characteristics of the loudspeakers of Fig. 6.2
having cone diameters of 1 inch, 4 inches, and 16 inches.
8
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1\:"
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fREQUENCY IN CYCLES PER SECOND
FIG. 6.10. The amplitude frequency characteristics of vibrating pistons
of various diameters, mounted in an infinite wall, for I-watt output on
one side.
Figs. 6.7 and 6.8. It will be see that the directional pattern becomes
sharper with increase in frequency. However, the pattern is broader than
that of a vibrating piston of the same diameter due to the relatively low
velocity of propagation of sound in the paper cone. The directional patterns
136
ACOUSTICAL ENGINEERING
of 130° and 100° cones 4 inches in diameter are shown in Fig. 6.11. It
will be seen that the directional pattern at the high frequencies becomes
broader as the cone angle is increased. This is to be expected because the
velocity of propagation of sound in the paper cone is about two times
the velocity of sound in air. Under these conditions the delay between the
sound emitted from the outside and the center of the cone will increase as
the angle of the cone is increased. As a result the directional pattern will
be broadest for the cone with the widest angle. The preceding observations
..
..
..
FIG. 6.11. Directional characteristics of dynamic direct radiator loudspeakers with
cones 4 inches in diameter for two different cone angles. Row A. 130 0 cone. Row B.
100 0 cone.
with regard to cone type vibrators may be substantiated by theoretical
considerations as outlined in Sec. 2.21.
The characteristics of Figs. 6.2 and 6.9 show that the low-frequency
efficiency may be maintained to the higher frequency ranges by employing
a small and relatively light weight cone and voice coil. On the other hand,
to obtain adequate power handling capacity at the low frequencies with
tolerable excursions of the vibrating system requires a cone of relatively
large area. To insure operation below the elastic limits of the materials,
a cone of large area must be of sturdy construction. Equation 6.7 and Fig.
6.3 show that a large heavy cone also requires a relatively large voice coil
in order to maintain a tolerable efficiency. The efficiency of this system
is low in the high-frequency range. Furthermore, the directional pattern
of a large cone becomes quite narrow in the high-frequency range. Where
the frequency range is confined within the limits of from 80 cycles to
4000 cycles, satisfactory efficiency, response, and directional characteristics
can be obtained from a single-cone, single-coil loudspeaker. The above
discussions show that to obtain adequate power handling capacity and
uniform response over a wide frequency range (greater than 80 to 4000
DIRECT RADIATOR LOUDSPEAKERS
137
cycles) requires a relatively large diameter, heavy diaphragm, and large
coil at the lower frequencies, and a relatively light diaphragm and coil to
obtain good efficiencies at the higher frequencies. There are a number of
direct radiator loudspeaker systems which may be built to satisfy these
conditions. I t is the purpose of the sections which follow to consider a
number of these systems.
6.3. Multiple Single-Cone, Single-Coil Loudspeaker. - Several
arrangements for obtaining uniform response, broad directional pattern,
adequate power handling capacity, and tolerable efficiency are shown in
Fig. 6.12.
The systems of Fig. 6.12A, C, and D consist of a large diameter heavy
cone driven by large voice coil for the low-frequency range and a small
L--------'B
L---------'c
L--------~D
FIG. 6.12. Multiple single-cone, single-coil, direct radiator, dynamic, loud­
speaker systems. A, C, and D. Large low-frequency unit, small high­
frequency unit, and filter system. B. Seven small units connected in parallel.
diameter light cone and small voice coil for the high-frequency range and
a filter system for allocating the power in the high- and low-frequency ranges
to the respective low- and high-frequency units. The filter system consists
of an inductance in series with the low-frequency unit and a condenser in
series with the high-frequency unit. Due to the large inductance of the
large voice coil, as shown in Fig. 6.6, it has been found that for most applica­
tions the inductance in series with the low-frequency unit may be omitted.
On the other hand, if a more elaborate filter system is required, the circuit
of Fig. 7.16 may be used.
138
ACOUSTICAL ENGINEERING
In Fig. 6.12A the low- and high-frequency units are separated by a
relatively large distance. In the overlap frequency region this distance
may be more than 1 wavelength. The directional patterns of two SOUIces
shown in Fig. 2.3 are applicable to this system. These characteristics
show that two separated sources exhibit directional patterns with one or
more lobes with very low response between the lobes. The result is fre­
quency discrimination, for points removed from the axis, in the overlap
region. This condition is reduced in Fig. 6.12C but is not eliminated.
However, a disadvantage of the system of Fig. 6.12C is that sound diffracts
around the high-frequency unit and is reflected from the large cone causing
a ragged response due to interference between the direct and reflected sound.
The obj.ectional features of Fig. 6.12A and C referred to above have
been eliminated in Fig. 6.12D. In this system? the large cone is geo­
metrically a continuation of the small cone. Therefore, in the overlap
15" DIAMETER CONE
o
2f DIAMETER CONE
FIG. 6.13. Directional characteristics of direct radiator loudspeakers with
cone diameters of 15 inches and 2.5 inches.
region the two cones vibrate together as a single cone. In this way phase
and diffraction effects are eliminated.
In a two-unit loudspeaker, employing a large cone for the reproduction of
the low-frequency range and a small cone for the reproduction of the high­
frequency range, a uniform directivity pattern can be obtained over the
entire audio-frequency range. This has been described in connection with
Figs. 6.6 and 6.7. This is illustrated further in Fig. 6.13 in which the
directivity patterns of IS-inch and 2t-inch cone loudspeakers are compared
for a six to one ratio of frequency, that is, for a constant ratio of diameter
to wavelength. Fig. 6.13 shows that the directivity pattern of a IS-inch
loudspeaker at 200 to 1000 cycles corresponds to that of a 2t-inch
7
Olson and Preston. RCA RBview. Vol. 7. No.2. p. 155. 1946.
DIRECT RADIATOR LOUDSPEAKERS
139
loudspeaker at 1200 and 6000 cycles. These relationships were used in
designing the two units of the system shown in Fig. 6.12D.
In the loudspeaker8 ,9 shown in Fig. 6.12D, small cones may be attached
to the large cone to reduce the velocity of wave propagation in the large
cone. Fig. 6.14. This broadens the directivity pattern of the low-frequency
cone. In the high-frequency range, the conical domes attached to the
surface of the low-frequency cone improve the performance in three ways:
by decreasing the angle into which the high-frequency cone feeds, thereby
increasing the output of the high­
frequency cone; by diffusely reflecting
some of the sound emitted by the high­
frequency cone, thereby eliminating
discrete reflections; and by diffracting
some of the sound emitted by the high­
frequency cone, thereby broadening the
directivity pattern.
The angles into which the high­
frequency cone feeds, without and with
the conical domes applied to the low­
frequency cone of Fig. 6.14, are desig­
nated as 7>1 and 7>2 in Fig. 6.15A and
Fig. 6.15B. Since 7>2 is smaller than 7>1,
the acoustic radiation load upon the
cone is greater with the conical domes
than without them. When the acoustic
radiation load upon a direct radiator
loudspeaker is increased, the sound
power output is increased. Thus, it will
be seen that the conical domes increase
the high-frequency sound radiated by
the high-frequency cone. In other FIG. 6.14. A perspective view of a
duo-cone loudspeaker with domes
words, the high-frequency efficiency is attached to the low-frequency cone.
(After Olson, Preston, and May.)
improved.
Some of the sound emitted by the
high-frequency cone is diffusely reflected by the conical domes, as shown in
Fig. 6.16. Without the domes, there would be many similar reflections
which would lead to reinforcements and cancellations with the direct radia­
tion. The result would be corresponding peaks and dips in the response of
the high-frequency cone. With the domes, the symmetry of the low­
frequency cone is upset and there are many reflections in different directions
and of different path lengths. The reflections, therefore, cancel out and the
net result is a smooth response-frequency characteristic.
Some of the sound emitted by the high-frequency cone is diffracted by
the conical domes as shown in Fig. 6.17. By diffraction is meant the bending
8
9
Olson, H. F., Radio and Television News, Vol. 51, No.2, p. 69, 1954.
Olson, Preston, and May, Jour. Aud. Eng. Soc., Vol. 2, No.4, p. 219, 1954.
140
ACOUSTICAL ENGINEERING
of the sound around an obstacle. The pencils of sound designated 1 and 2
in Fig. 6.17 are diffracted. The pencils of sound designated 3 to 7, inclusive,
are radiated directly from the high-frequency cone. It will be seen that the
effect of the diffracted sound is to increase the curvature of the wavefront
in the direction of 1, 2, and 3. As a result, the directivity pattern is
broadened.
I
I-'~ANGLE
HIGH
FREQUENCY
CONE
I
SECTIONAL VIEW
A
CONICAL DOME
HIGH
FREQUENCY
CONE
HIGH
FREQUENCY
CONE
SECTIONAL VIEW
B
FIG. 6.15. A. Duo-cone loudspeaker with a
plain low-frequency cone. B. Duo-cone loud­
speaker with domes attached to the low-fre­
quency cone.
Referring to Figs. 6.2 and 6.9 it will be seen that uniform response may
be obtained over a wide frequency range by means of a light cone driven
by a light coil and resonant at the lower limit of the frequency "range.
Of course, the power handling capacity of a single unit of this type is in­
adequate and a multiple set of units must be employed. The number of
units required may be determined from the required power output and the
allowable excursion together with equation 6.13 and Fig. 6.10. An arrange­
ment of seven small loudspeaker units mounted in a flat baffle with the voice
coils connected in parallel is shown in Fig. 6.12B. The voice coils of the
loudspeakers may, of course, be connected in parallel, series, or series-parallel.
In order to obtain better high-frequency spatial distribution the units may
be inclined at various angles, for example, the units may be mounted so that
the resulting vibrating surface approximates a spherical surface (see
Sec. 2.20).
DIRECT RADIATOR LOUDSPEAKERS
141
CONICAL
DOME
HIGH
FREQUENCY
CONE
REFLECTED
PENCILS
OF SOUND
LOW
FREQUENCY
CONE
4
REFLECTED
PENCILS
OF SOUND
CONICAL
DOME
2
HIGH
FREQUENCY
CONE
LOW
FREQUENCY
CONE
SECTIONAL VIEW
FIG. 6.16. Diffuse reflections of the sound emitted by
the high-frequency cone by the domes attached to the
low-frequency cone.
The frequency range of a direct radiator loudspeaker may be increased
by sectionalizing the coil or cone or both and thereby reducing the mechanical
impedance and electrical impedance or both at the higher frequencies.
These systems will be considered in the sections which follow.
6.4. Single-Coil, Double-Cone Loudspeaker. lO--A typical single-coil,
double-cone loudspeaker, Fig. 6.18B, consists of a single coil coupled to two
cones. In this system an increase in frequency range is obtained by reducing
the mechanical impedance of the diaphragm by coupling a smaller cone to
10
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 10, No.4, p. 305, 1939.
142
ACOUSTICAL ENGINEERING
DIRECT PENCILS
OF SOUND
HIGH
FREQUENCY
CONE
DIFfRACTED
PENCILS
Of SOUND
LOW
FREQUENCY
CONE
PENCILS
OF SOUND
CONICAL
DOME
LOW FREQUENCY CONE
SECTIONAL VIEW
6.17. Diffraction of the sound emitted by the
high-frequency cone by the domes attached to the low­
frequency cone.
FIG.
the voice coil at the high frequencies. The two cones are separated by a
compliance. At low frequencies the mechanical reactance of the compliance,
eM, is large compared to the mechanical impedance, ZMl, of the large cone
and consequently the entire system moves as a whole. At high frequencies
the mechanical reactance of the compliance, eM, is small compared to the
mechanical impedance, ZMl, of the large cone, and the small cone, ZM2 moves
while the large cone, ZMl, remains stationary. By means of this reduction
in cone mechanical impedance the range may be extended almost a full
octave, depending upon the mass and electrical impedance characteristics
of the voice coil. The response characteristics of a single-coil, single-cone
DIRECT RADIATOR LOUDSPEAKERS
143
loudspeaker is shown in Fig. 6.18A. The voice coil and large cone of Fig.
6.l8B is the same as that of Fig. 6.18A. The high-frequency range has been
extended about one-half octave without any sacrifice of power handling
capacity by the addition of the small cone.
6.5. Double-Coil, Single-Cone Loudspeaker.ll-The double-coil,
single-cone loudspeaker, Fig. 6.18C, consists of a voice coil, divided into
two parts separated by a compliance, coupled to a single corrugated cone.
The inductance and electrical resistance of the larger portion of the voice
coil, LI, rEI, is shunted by an electrical capacitance, CEo At low frequencies
the electrical reactance of the electrical capacitance is large compared to
the electrical impedance of the larger portion of the voice coil, LI, rEI,
and the mechanical reactance of the compliance, CM, separating the two
portions of the voice coil is large compared to the mechanical mass react­
ance of ml, and the mechanical impedance ZMI. Therefore, in the low­
frequency range the action is the same as that of a single-coil loudspeaker.
At high frequencies the reactance of the electrical capacitance, CE, is small
compared to the electrical impedance of LI, rEI or L 2, rE2; and the mechani­
cal reactance of the compliance, CM, is small compared to the mechanical
reactance of large coil, mI. The cone is driven by the lighter portion, m2,
of the voice coil and the heavy coil, ml, remains stationary. In the mid­
range there is a phase difference between the currents in the two portions
of the voice coil. A corresponding phase shift occurs in the mechanical
system. As a consequence, a smooth overlap is obtained in going from two­
coil operation at the low frequencies to a single-coil operation at the high
frequencies. Above the frequency of ultimate resistance the radiation
resistance is a constant. In order to obtain uniform output in this range
the mechanical impedance of the system must be independent of the fre­
quency. This may be accomplished by embossing suitable corrugations
in the cone which reduce the effective mass reactance. The double-coil
system reduces the effective mass reactance of the voice coil as compared
to a single coil, as well as the electrical impedance at the higher frequencies.
A typical response characteristic of this loudspeaker is shown in Fig. 6.18C.
6.6. Double-Coil, Double-Cone Loudspeaker.12-The double-coil,
double-cone loudspeaker, Fig. 6.18D, consists of a light coil coupled to a
small cone, connected by a compliance to a heavy coil and large cone. In
this system an increase in range is obtained by reducing the impedance of
both the coil and the diaphragm at the higher frequencies. At low fre­
quencies the electrical reactance of the capacitance, CE , is large compared to
the electrical impedance of the large portion of the voice coil, LI, rEI, and
the same current flows in both coils. The mechanical reactance of the
compliance, CM, separating the two portions of the coil is large compared
to the mechanical impedance of ml, plus ZMI. Therefore, at low frequencies
the system behaves as a single-coil, single-cone loudspeaker. Both parts
11
12
Olson, H. F .. Proc. Inst. Rad. Eng., Vol. 22, No. 1, p. 33, 1034.
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 10, No.4, p. 305, 1939.
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circuit. m = the mass of the voice coil. ZMl = the mechanical impedance of the cone at the
in the voice coil. The graph shows the pressure response frequency. B. Cross-sectional vi
speaker with the voice coil electrical circuit and mechanical network of the mechanical syste
e = the internal voltage of the generator.
'YEG = the internal electrical resistance of the g
resistance and inductance of the voice coil. In the mechanical network, m = the mass of
mechanical impedance of the large and small cones. CM and 'I'M = the compliance and mech
the large cone. 1M = the force-generated in the voice coil. The graph shows the pressu
C. Cross-sectional view of a single-cone, double-coil loudspeaker with the voice coil networ
mechanical system. In the voice coil electrical network, e = the internal voltage of the ge
resistance of the generator. rEI and LI = the electrical resistance and the inductance of the l
resistance and inductance of the small coil. CE = the electrical capacitance shunting the la
mi = the mass of the large coil.
m2 = the mass of the small coil.
ZMI = the mechanical i
CM and 'I'M = the compliance and mechanical resistance of the corrugation separating the l
force generated in the large coil. 1M2 = the force generated in the small coil. The graph s
characteristic. D. Cross-sectional view of a double-cone, double-coil loudspeaker with t
mechanical network of the mechanical system. In the voice coil electrical network, e = t
rEG = the internal electrical resistance of the generator.
rEI and LI = the electrical resist
coil. r/i:2 and L2 = the electrical resistance and the inductance of the small coil. C E = th
In the mechanical network, mi = the mass of the large coil. m2 = the mass of the small co
of the large cone. ZM2 = the mechanical impedance of the small cone. CM and 'I'M = the
of the corrugation separating the large cone and coil and the small cone and coil. 1MI =
1M2 = force generated in the small coil. The graph shows the pressure response frequency c
'0
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FIG. 6.19.
A. Cross-sectional view of a conventional single-coil direct radiator, dynamic lo
of the mechanical system. In the mechanical circuit, m = the mass of the voice coil. ZM
cone and suspension system. The graph shows the pressure response frequency characteri
40
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DIRECT RADIATOR LOUDSPEAKERS
147
handling capacity as a direct radiator loud­
speaker may be obtained at the low frequencies.
6.7. Mechanical Networks for Control­
ling the High-Frequency Response of a
Loudspeaker.-In general, in radio and other
forms of sound reproduction it is desirable to
attenuate the response above a certain high­
frequency limit. In some cases, it may be
desirable to attenuate a certain band as, for
example, 10,000 cycles in radio reproduction to
eliminate the adjacent channel beat note.
Electrical networks and filters are usually quite
costly compared to mechanical filters for cer­
tain applications in sound reproduction. It is
the purpose of this section to describe the con­
struction and performance of several mechani­
cal networks and filters for suppressing certain
frequency bands or for attenuating the high­
frequency response of a loudspeaker.
A relatively light weight, 8-inch loudspeaker
was chosen for these tests. This type of loud­
speaker is used in small radio receivers. Due
to the small mass of the cone and coil the
response is well maintained at the high fre­
quencies. The principles involved are applic­
able to all loudspeakers. The loudspeaker was
mounted in a 3-foot irregular baffle. The
response was obtained employing a velocity
microphone located on the axis of the speaker
at a distance of two feet.
A. Conventional Single-Coil Loudspeaker.­
The response frequency characteristic of the
conventional loudspeaker, referred to above,
is shown in Fig. 6.19A. The mechanical cir­
cuit of the mechanical system is also shown in
Fig.6.19A. The constants have been indicated
as the mass of the voice coil, mI, and the com­
pliance of the centering suspensions, the cone
mechanical impedance including the cone out­
side suspension and the radiation mechanical
resistance, etc., lumped as ZMC. The response
is well maintained to 12,000 cycles. For this
reason, this loudspeaker is well adapted to
illustrate the performance of mechanical net­
works for controlling the response at the higher
frequencies.
148
ACOUSTICAL ENGINEERING
B. Loudspeaker with a Compliance Shunting the Cone Mechanical Imped­
ance.-One of the simplest means for attenuating the high-frequency response
of a loudspeaker is a compliance inserted between the voice coil and the
cone. This compliance, CM , may take the simple form of a bead or corruga­
tion pressed into the voice coil form. The response characteristic of a
conventional loudspeaker with a compliance between the voice coil and
cone is shown in Fig. 6.19B. In the mechanical network the compliance,
CM, shuts the cone mechanical impedance, ZMC. Comparing with Fig.
6.l9A it will be seen that there is some attenuation at the higher frequencies.
However, the attenuation is not large. This is due to the fact that the
mechanical impedance, ZMC, does not increase appreciably with frequency.
At the higher frequencies a light cone, in particular, does not vibrate as a
piston. In a large diameter light cone the action changes gradually from
piston action to wave propagation at the higher frequencies. As a conse­
quence, the mechanical impedance does not increase directly with the
frequency. In some loudspeakers the mechanical impedance, ZMC, actually
decreases with frequency at the higher frequencies.
C. Loudspeaker with a Compliance Shltnting; a Compliance and Mass in
Parallel, Connected in Series with the Cone Mechanical Impedance.-In a
radio receiver it is desirable to attenuate the response at 10,000 cycles so
that the 1O,000-cycle adjacent channel beat note willFlot be reproduced.
A parallel circuit inserted in series with a line causes high attenuation at
the resonance frequency. By inserting a parallel circuit in series with the
voice coil and cone the response will be reduced at the resonant frequency.
The amount of attenuation will depend upon the magnitude of the mechani­
cal resistance in the compliance. An example of this system is shown in
Fig. 6.19C. The mass and compliance are designated as m2 and CM2.
Comparing with Fig. 6.19A the attenuation at 10,000 cycles is about 25 db.
This system is also easy to fabricate. Two suitable corrugations are pressed
into a single voice coil form.
D. Loudspeaker with a "T" Type Filter Connecting the Voice Coil Mass
and the Cone Mechanical Impedance.-This system, Fig. 6.19D, consists
of two parallel resonant mechanical circuits, or a parallel resonant mechani­
cal circuit, m2 and CM2, connected to the bottom of the voice coil of the
system of Fig. 6.19C. The mechanical network is also shown in Fig. 6.19D.
The system then is a "T" type low-pass mechanical filter connecting the
coil and cone. Very high attenuation is obtained at the resonant frequency
of the arms. The response frequency characteristic of this system is shown
in Fig. 6.19D. Comparing with Fig. 1.19A the attenuation at 10,000 cycles
is 35 db. The attenuation is also quite high above 10,000 cycles. As in
the other systems it is made by simply pressing three corrugations into a
single voice coil form.
Several mechanical networks for controlling and suppressing the response
of a loudspeaker at the high frequencies have been described. Some of
these systems are in use in practically all loudspeakers. The cost of the
system is very small compared to an electrical network for accomplishing
DIRECT RADIATOR LOUDSPEAKERS
149
the same result because the mechanical networks are made by simply
placing corrugations in the voice coil form. These examples also illustrate
the value of analogies of electrical circuits in designing and in predicting
the action of mechanical systems.
6.8. Loudspeaker Baffles.-A baffle is a partition which may be used
with an acoustical radiator to increase the effective length of the acoustical
transmission path between the front and back of the radiator. The term
baffle is commonly applied to a plane surface. When a direct radiator
loudspeaker is mounted in a baffle, there exists at 180 0 phase difference
between the front and back of the cone. When the baffle is small compared
to the wavelength the system is an acoustic doublet (see Secs. 2.3 and 5.14).
In this frequency range the power output for constant velocity is proportional
to the fourth power of the frequency (see Sec. 5.14). When the baffle is
large compared to the wavelength, the two sides of the cone act independently
and the sound power output is proportional to the square of the frequency
(see Secs. 2.2 and 5.8). In the case of a mass controlled system the velocity
is inversely proportional to the frequency. A mass controlled system is a
system in which a positive mechanical reactance is the controlling mechani­
cal impedance. Therefore, in the case of the large baffle the sound power
output will be independent of frequency (see Sec. 6.2). However, when
the dimensions of the baffle are small compared to the wavelength, the power
output in the case of a mass controlled system is proportional to the square
of the frequency. In this frequency range the low-frequency response falls
off rapidly. The transition between doublet operation and independent
operation is quite marked. This transition point occurs when the dimen­
sions of the baffle are slightly less than one-half wavelength.
It is the purpose of this section to consider the action of various types of
baffles and loud-speaker systems.
A. Irregular Bajjle.-In the case of a cone in a square baffle the path
from the front to the back is practically the same for all possible paths.
Therefore, some peculiarities in the response would be expected when the
acoustical path from the front to back is equal to a wavelength. At this
frequency the sound that is diffracted around the baffle and transmitted
forward will interfere destructively with the radiation from the front.
The pressure response characteristics of Fig. 6.20A show "dips" in the
response when the acoustical path from front to back is a wavelength.
Using an irregular baffle, Fig. 6.20B, it is possible to reduce this interference
and obtain a uniform response characteristic. In this baffle the various
paths from front to back differ and the destructive interference is spread
over a wide frequency range. The pressure response frequency charac­
teristics of an irregular baffle, Fig. 6.20B, show that the dip in the response
frequency characteristic of the square baffle is eliminated by the use of an
irregular baffle.
B. Large Bajjle, Different Resonant Frequencies.-The radiation me­
chanical resistance of a vibrating piston in an infinite baffle is proportional
to the square of the frequency in the range below the frequency where the
150
ACOUSTICAL ENGINEERING
radiation resistance attains its ultimate value. Referring to equation 6.S
it will be seen that the power output of a direct radiator loudspeaker will
be independent of the frequency in the frequency range above the resonant
frequency up to the frequency of ultimate mechanical resistance, and will
CONEEr
CONEfJI
r- FT.1
A
B
4
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ell
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:r'"
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V
1000
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20
V
z
v
6
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30
30
12
6
""'
,/
o /'
20
1000
100
FREQUENCY
FIG. 6.20.
Pressure response frequency characteristics of mass-controlled,
direct radiator, dynamic loudspeaker mechanisms, with to-inch diameter
cones, mounted in square baffles. In A, the loudspeaker mechanism is
mounted in the center of the baffle. In B, the loudspeaker mechanism is
mounted unsymmetrically to eliminate interference.
be proportional to the fourth power of the frequency below the resonant
frequency. The measured pressure response frequency characteristic of
a direct radiator loudspeaker having a fundamental resonance of 50, 100,
and 200 cycles is shown III Fig. 6.21. It will be seen that the pressure
~o
CYCLES
100
200
CYCLES
",24
",24
o
... 16
ell
Z
~ 12
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0
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CYCLES
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o
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0:
V
100
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500
0
100
FREQUENCY
5 00
6
go
'j
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V
100
500
FREQUENCY
FIG. 6.21. Pressure response frequency characteristics of direct radiator, dynamic
loudspeaker mechanisms, with 10-inch diameter cones, with resonant frequencies
of S0, 100, and 200 cycles, mounted in very large baffles.
response is independent of the frequency in the frequency range above the
resonant frequency. Below the resonant frequency the pressure response falls
off 12 db per octave. These results agree with that predicted by theory.
C. Low Resonant Frequency, Different Baffle Sizes.-The radiation mechan­
ical resistance of a vibrating piston in a finite baffle is proportional to the
fourth power of the frequency when the dimensions of the baffle are small
151
DIRECT RADIATOR LOUDSPEAKERS
compared to the wavelength, doublet operation, and proportional to the
square of the frequency when the dimensions are comparable to or greater
than the wavelength in the range below the ultimate mechanical resistance
(see Sees. 2.2, 5.8, and 5.14). If the considerations are confined to the
frequency range above the resonant frequency of the mechanism, the velocity
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50 a
100
I'REQUENCY
500
100
FREQUENCY
FIG. 6.22. Pressure response frequency characteristics of mass-controlled, direct
radiator, dynamic loudspeaker mechanisms, with to-inch diameter cones, mounted
in square baffles of 8, 4, and 2 feet on a side.
of the cone will be inversely proportional to the frequency. Under these
conditions, the pressure response will be proportional to the frequency in
the range where the system behaves as a doublet and independent of the
frequency where the system behaves as a simple radiator. The experimental
results of Fig. 6.22 substantiates these predictions for 2-, 4-, and 8-£00t
baffies. Above the range where the system changes from doublet to singlet
t-- 8
FT.--l
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B
SO CYCLES
3
30
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",18
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II' 6 11
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100
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500
200
3a
f--
T
CYCLES
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o
/'-
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-
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<II
12
-f-
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o
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a
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2fT.
no-:
1
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-j
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6
100
FREQUENCY
500
6
a
/
/
/
100
FREQUENCY
50 a
FIG. 6.23. Pressure response frequency characteristics of direct radiator, dynamic
loudspeaker mechanisms, with 10-inch diameter cones, operating under the follow­
ing conditions: A. Square baffle 8 feet on a side and a loudspeaker resonant fre­
quency of 50 cycles. B. Square baffle 4 feet on a side and a loudspeaker resonant
frequency of 100 cycles. C. Square baffle 2 feet on a side and a loudspeaker with a
resonant frequency of 200 cycles.
152
ACOUSTICAL ENGINEERING
operation the pressure response is independent of the frequency. Below
this transition point the pressure response falls off 6 db per octave.
D. Different Resonant Frequencies and Different Baffle Sizes.-If the
resonant frequency of the loudspeaker is placed near the doublet-singlet
transition frequency the pressure response will be independent of the fre­
quency above this frequency and will be proportional to the cube of the
frequency below this frequency. The experimental results of Fig. 6.23
combines the loudspeaker mechanism of Fig. 6.21 with the baffies of Fig.
6.22. The resonant frequency is placed slightly lower than the doublet­
singlet transition frequency so that the output is quite uniform above the
resonant frequency. Below the resonant frequency and the doublet-singlet
transition frequency the pressure response falls off 18 db per octave. Again
the experimental results are in agreement with theory.
6.9. Cabinet Loudspeakers. 13 ,14,15,16-The most common housing for
a direct radiator loudspeaker is the conventional open-back cabinet which
VOICE
COIL
ELECTRICAL
CROSS ­ SECTIONAL VIEW
CIRCUIT
MECHANICAL
CIRCUIT
Of
THE
MECHANICAL
SYSTEM
FIG. 6.24.
Cross-sectional view of a single-coil, single-cone, direct radiator,
dynamic loudspeaker mechanism mounted in open-back cabinet. In the voice
coil circuit, e = the internal voltage of the generator. rEG = the internal
electrical resistance of the generator. rEO and L = the electrical resistance and
inductance of the voice coil. ZEM = the motional electrical impedance. In the
mechanical circuit, me = the mass of the cone and voice coil. eMS = the
compliance of the suspension system. rMS = the mechanical resistance of
the suspension system. mA = the mass of the 'air load. rMA = the mechanical
resistance of the air load. 1M = the mechanomotive force in the voice coil.
ZMl = the mechanical impedance due to the cabinet load on the cone.
ZAI = the
acoustical impedance at the closed end of the cabinet. ZA2 = the acoustical
impedance at the open end of the cabinet. S. = the area of the cone.
also houses the radio chassis or phonograph mechanism. These range in
size from the largest console type to the smallest midget. From the stand­
point of sound reproduction the principle is the same in all, namely, to
provide a baffie for the loudspeaker. In the case of the midget cabinets the
sound path from the front to the back is very small and the low-frequency
sounds are not reproduced. In the case of the large console cabinets the
Olson and Preston, Radio and Tel. News., Vol. 51, No.2, p. 69, 1954.
Olson. H. F., Audio Eng.. Vol. 35, No. 11, p. 34, 1951.
15 Olson, H. F., Radio and Tel. News, Vol. 45, No.5, p. 53, 1951.
16 Meeker, Slaymaker, and Merrill, Jour. Acous. Soc. Amer., Vol. 22, No.2, p. 206,
1950.
13
14
153
DIRECT RADIATOR LOUDSPEAKERS
acoustic path length is sufficiently large to insure good reproduction of low
frequencies. One of the most troublesome acoustical factors in conventional
cabinets is the resonance in the enclosure back of the cone. This resonance
is termed cabinet resonance. The system may be considered from the
standpoint of lumped or distributed constants. In the case of most systems,
the latter viewpoint seems to yield better agreement with experiment. The
cabinet enclosing the back of the cone may be considered to be a pipe with
distributed constants.
A cross-sectional view of a direct radiator loudspeaker mounted in an
open-back cabinet and the mechanical circuit of the mechanical system is
shown in Fig. 6.24. The input acoustical impedance of a finite cylindrical
pipe has been considered in Sec. 5.25. In this chapter it has been more
convenient to use mechanical impedance instead of acoustical impedance.
The mechanical impedance due to the cabinet in terms of the acoustical
impedance is
ZMl = ZAlSc 2
6.14
where ZMl = mechanical input impedance of the cabinet, in mechanical
ohms.
ZAI = acoustical impedance of the cabinet, in acoustical ohms, and
Sc = area of the cone in square centimeters.
The power output of the system may be determined from the mechanical
and electrical circuits of Fig. 6.24 and the constants of the system.
It is the purpose of the sections which follow to consider the performance
3f various types of cabinets and loudspeaker systems.
A. Low Resonant Frequency, Different Cabinet Sizes.-The pressure
response frequency characteristics of a direct radiator loudspeaker mechan­
[sm, having a resonant frequency of 20 cycles mounted in cabinets of
various sizes, is shown in Fig. 6.25. The resonant frequencies at 80, 150,
0
30
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V
100
FR~QUENCY
500
~o
100
,REQUENCY
500
FIG. 6.25. Pressure response frequency characteristics of mass-controlled, direct
radiator, dynamic loudspeaker mechanisms with lO-inch cones mounted in square
open-back cabinets. A. Cabinet, 4 feet X 4 feet X 12 inches in depth. B.
Cabinet, 2 feet X 2 feet X 8 inches in depth. C. Cabinet, 1 foot X 1 foot X
6 inches in depth.
154
ACOUSTICAL ENGINEERING
and 250 cycles for the 4-, 2-, and I-foot cabinets, respectively, is quite
evident. In this region the output is somewhat exaggerated in spite of
the fact that the cabinets are relatively shallow. Below the resonant fre­
quency the system behaves as a doublet. Therefore, with a mass-controlled
mechanism the response falls off 6 db per octave.
r-2FT--1..1...
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500
100
~o
500
r-...
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a
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.........
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tt
3
30
0
1fT
W'.,
FREQUENCY
I'r­
100
fREQUENCY
500
FIG. 6.26.
Pressure response frequency characteristics of direct radiator,
dynamic loudspeaker mechanisms, with 10-inch diameter cones, operating in
open-back cabinets under the following conditions: A. Cabinet, 4 feet X 4 feet X
12 inches in depth and a loudspeaker resonant frequency of SO cycles. B. Cabinet,
2 feet X 2 feet X 8 inches in depth and a loudspeaker resonant frequency of
100 cycles. C. Cabinet, 1 foot X 1 foot X 6 inches in depth and a loudspeaker
resonant frequency of 200 cycles.
B. Different Resonant Frequencies and Different Cabinet Sizes.-In most
of the cabinets and mechanisms in use today the resonant frequencies of
the two systems are quite close together. This situation comes about in
a perfectly natural way due to manufacture procedures and design limitations
involved in low-cost, direct radiator mechanisms. The pressure response
B
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30
30
.,24
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o
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a
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en V
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Z
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(\
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o
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c
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'"cr
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w
100
FREQUENCY
500
6
030
100
fREQUENCY
500
FIG. 6.27.
Pressure response frequency characteristics of mass-controlled, direct
radiator, dynamic loudspeaker mechanisms with lO-inch diameter cones operating
in open-back cabinets 2 feet x 2 feet and the following depths: A. 8 inches.
B. 16 inches.. C. 24 inches.
155
DIRECT RADIATOR LOUDSPEAKERS
frequency characteristics of combinations of various cabinets and mechan­
isms having different resonant frequencies are shown in Fig. 6.26. These
characteristics show a marked increase in output of about 6 db at the region
of cabinet and mechanism resonance. Below this frequency range the
pressure response falls off 18 db per octave.
C. Effect of the Depth of the Cabinet.-A consideration of the open-back
cabinet system of Fig. 6.24 shows that the depth of the cabinet will influence
the response, particularly at the resonant frequency. The pressure re­
sponse frequency characteristics of a mass-controlled loudspeaker mechan­
ism mounted in 2-foot cabinets with depths of 8, 16, and 24 inches are
shown in Fig. 6.27. It will be seen that the accentuated response in the
region of cabinet resonance becomes more pronounced as the depth of the
cabinet is increased.
6.10. Back-Enclosed Cabinet Loudspeaker.-A loudspeaker mechan­
ism with the back of the cone completely enclosed by the cabinet is shown
in Fig. 6.28. At the low frequencies the system is a simple source (see
VOICE
COIL
MECHANICAL
OF
ELECTRICAL CIRCUIT
MECHANICAL
CIRCUIT
THE
$YSTEM
eRO$$ -$ECTIONAL VIEW
FIG. 6.28. Cross-sectional view of a single-coil, single cone, direct radiator,
dynamic loudspeaker mechanism mounted in closed-back cabinet. In the voice
coil circuit, e = the internal voltage of the generator. 'YEG = the internal electri­
cal resistance of the generator. rEC and L = the electrical resistance and induc­
tance of the voice coil. ZEM = the motional electrical impedance. In the
mechanical circuit, mc = the mass of the cone and voice coil. CMS = the com­
pliance of the suspension system. rMS = the mechanical resistance of the
suspension system. mA = the mass of the air load. rMA = the mechanical
resistance of the air load. C MB = the compliance of the cabinet. 1M = the
mechanomotive force in the voice coil.
Sec. 2.2). Under these conditions the radiation mechanical resistance is
proportional to the square of the frequency up to the frequency of ultimate
mechanical resistance. The mechanical circuit of Fig. 6.28 shows that,
under these conditions, the output will be independent of the frequency
above the resonant frequency of the system.
A consideration of the mechanical circuit shows that the fundamental
resonance is influenced by the compliance of the cone suspension, and the
compliance of the enclosure. The compliance of the enclosure in terms of
the acoustical capacitance is given by
CA
6.15
CMB = Sc 2
156
where
ACOUSTICAL ENGINEERING
compliance of the cabinet, in centimeters per dyne,
CA = acoustical capacitance of the cabinet, in (centimeters)5 per
dyne, and
Sc = area of the cone, in square centimeters.
CMB =
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lRJl
A
150
30
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CD 24
c
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z
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If)
w
'­
12
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II:
6
6
I
o
30
100
FREQUENCY
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100
FREQUENCY
1--24'~
W1
c
30
30
CYCLES
30
30
CYCLES
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c
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z
i;'
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II:
30
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CRJl
D
,,-...
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150
30
c
II:
[Rli
B
CYCLES
100
FREQUENCY
500
6
0
30
100
FREQUENCY
500
FIG. 6.29. Pressure response frequency characteristics of
direct radiator, dynamic loudspeaker mechanisms with
lO-inch diameter cones operating in open- and closed-back
cabinets. A. Open-back cabinet, 2 feet X 2 feet x 8 inches
in depth and a loudspeaker resonant frequency of 150 cycles.
B. Closed-back cabinet, 2 feet x 2 feet X 8 inches in depth
and a loudspeaker resonant frequency of 150 cycles. C. Open­
back cabinet, 2 feet X 2 feet X 8 inches in depth and a loud­
speaker resonant frequency of 30 cycles. D. Closed-back
cabinet. 2 feet X 2 feet X 8 inches in depth and a loudspeaker
resonant frequency of 30 cycles.
From the expression for the acoustical capacitance of an enclosure,
equation 5.7 and equation 6.15, the compliance of the cabinet is given by
V
CMB = ~S2
6.16
pc c
DIRECT RADIATOR LOUDSPEAKERS
where V
157
volume, in cubic centimeters,
p = density of air, in grams per cubic centimeter, and
c = velocity of sound, in centimeters per second.
The pressure response frequency characteristic of a loudspeaker mechan­
ism having a resonant frequency of 150 cycles mounted in a 24-inch open­
back cabinet is shown in Fig. 6.29A. The response falls off 18 db per
octave below 150 cycles.
The pressure response frequency characteristic of a loudspeaker mechan­
ism, having a resonant frequency of 150 cycles, mounted in a completely
enclosed 24-inch cabinet is shown in Fig. 6.29B. The fundamental resonant
frequency of the system is 200 cycles. The increase in the resonant frequency
is due to the addition of the compliance of the enclosure. The response falls
off 12 db per octave below the resonance frequency.
The pressure response frequency characteristic of a loudspeaker mechan­
ism having a resonance frequency of 30 cycles mounted in a 24-inch open­
back cabinet is shown in Fig. 6.29C. In this case the response falls off 6 db
per octave below the doublet-singlet transition frequency.
The pressure response frequency characteristic of a loudspeaker mechan­
ism having a resonant frequency of 30 cycles mounted in a completely
enclosed cabinet is shown in Fig. 6.29D. The compliance of the cabinet
raises the fundamental resonant frequency of the entire system to 70 cycles.
The response is maintained down to 40 cycles. The response of this system
is superior to that of the 8-foot baffle with a low-frequency resonant
mechanism as shown in Fig. 6.16 or to the 4-foot open cabinet with a low
resonance mechanism, Fig. 6.19.
6.11. Compound Direct Radiator Loudspeaker,l7-A consideration
of the back-enclosed cabinet loudspeaker has been given in Sec. 6.10. The
analysis shows that cabinet volume influences the response in the low­
frequency range when the compliance of the cabinet is comparable to the
compliance of the suspension system. In order to obtain adequate response
in the low-frequency range, some means must be provided for increasing the
effective volume of the cabinet. The effective volume of a cabinet may be
increased by decreasing the stiffness presented to the radiating loudspeaker
mechanism by the cabinet in the low-frequency range. It is possible to
reduce the effective stiffness of the cabinet by the use of an auxiliary loud­
<;peaker mechanism which drives the radiating loudspeaker mechanism.
fhis system has been termed a compound direct radiator loudspeaker.
A front and sectional view of a compound direct radiator loudspeaker is
shown in Fig. 6.30. A schematic sectional view and the electrical and the
mechanical network of the system is shown in Fig. 6.31. It will be seen
that the two voice coils are connected in series. Thus, in the low-frequency
range the same current flows in both voice coils. Referring to equation 6.16,
it will be seen that the compliance of the cabinet is inversely proportional
to the square of the cone area. Thus, it will be seen that the compliance
17
=
Olson, Preston, and May, Unpublished Report.
158
ACOUSTICAL ENGINEERING
FIG. 6.30. Front and sectional views of a compound direct radiator loudspeaker.
(After Olson, Preston, and May.)
~::r~4~~
f '"
f"2
MECHANICAL NETWORK
ELECTRICAL NETWORK
CROSS-SECTIONAL VIEW
FIG. 6.31. Cross-sectional view and the electrical and acoustical networks of a
compound direct radiator loudspeaker. In the electrical network, e = internal
voltage of the generator. rEG = internal electrical resistance of the generator.
YEI and Ll = electrical resistance and inductance of the voice coil of the radiating
loudspeaker. ZEMI = electrical motional impedance of the radiating loudspeaker.
YE2 and L2 = electrical resistance and inductance of the secondary loudspeaker.
ZE'M2 = electrical motional impedance of secondary loudspeaker.
CE"2 = elec­
trical capacitance. In the acoustical network, Y.>fA and m.ll = mechanical
resistance and mass of the air load upon the cone of the radiating loudspeaker.
11'11 = mass of the cone and coil of the radiating loudspeaker.
YMI and C Ml =
mechanical resistance and acoustical resistance and compliance of the suspensions
of the radiating loudspeaker. fill = mechanomotive force in the voice coil of the
radiating loudspeaker. C,IIY = compliance of the cabinet volume. rM2 and
C M 2 = mechanical resistance and compliance of the suspensions of the secondary
loudspeaker. 11'12 and mA2 = masses of the cone and air load of the secondary
londspeaker. C MVl = compliance of the volume between the two loudspeakers.
fW2 ~ mechanomotive force in the voice coil of the secondary loudspeaker.
DIRECT RADIATOR LOUDSPEAKERS
159
of the cabinet can be increased by making the diameter of the driving loud­
speaker small. By this expedient the resonant frequency of the driving
loudspeaker and cabinet can be made lower than the radiating loudspeaker
and the cabinet. By this two-stage system the response in the low-fre­
quency range can be extended. A IS-inch loudspeaker is the radiating
loudspeaker in the system of Fig. 6.30. The response-frequency charac­
teristic of the duo-cone loudspeaker mechanism in a back-enclosed cabinet
of two cubic feet is shown in Fig. 6.32. The response frequency charac­
teristic of the compound direct radiator loudspeaker housed in the same
30
25
cn20
o
!!:
ljll5
z
oQ.
lJV', ,
, ---...
l'
....:
/
,/A
...
<II
0::10
5
60
80
100
200
300
400
FREQUENCY IN CYCLES PER SECOND
6.32. A. Response frequency characteristic of a
1S-inch duo-cone loudspeaker mechanism in a cabinet
of 2 cubic feet. B. Response frequency characteristic
of a compound direct radiator loudspeaker housed in
the same cabinet and employing the same radiating
loudspeaker mechanism.
FIG.
cabinet is also shown in Fig. 6.32. It will be seen that a substantial gain in
low-frequency response is obtained by the use of the compound direct
radiator loudspeaker system.
6.12. Acoustical Phase Inverter Loudspeaker.-The acoustical phase
inverter loudspeaker18 system consists of a direct radiator loudspeaker
mechanism mounted in a completely closed cabinet save for a port coupling
the cabinet volume to the air, Fig. 6.33. The phase of the velocities on
the two sides of the cone differs by 180°. Referring to the mechanical
network of Fig. 6.33, it will be seen that the velocities in the branches 1
and 2 may differ by as much as 180° for positive mechanical reactances and
no mechanical resistances in branches 1 and 2 and a pure compliance in
18
Dickey, Caulton, and Perry, Radio Engineering, Vol. 8, No.2, p. 104, 1936.
160
ACOUSTICAL ENGINEERING
branch 3. The phase angle will be reduced as mechanical resistance is intro­
duced. However, the mechanical resistance in direct radiator loudspeaker
systems is small compared to the mechanical reactance and the constants
may be chosen so that the phase angle between the velocity of the cone and
the port is very small. This system increases the radiation mechanical
resistance of a direct radiator loudspeaker at the low frequencies. The
pressure response frequency characteristic of a loudspeaker mechanism
mounted in an open-back cabinet is shown in Fig. 6.34A. The pressure
'··'·'b
VOICE
COIL
ELE CTRICAL
CROSS -SECTIONAL
CIRCUIT
MECHANICAL NETWORK
OF
THE
MECHANICAL
SYSTEM
VIEW
FIG. 6.33. Cross-sectional view of a single-coil, single-cone, direct radiator, dynamic
londspeaker mechanism mounted in closed-back cabinet with a port. In the voice coil
circuit, e = the internal voltage of the generator. 'YEG = the internal electrical resistance
of the generator. 'YEG and L = the electrical resistance and inductance of the voice coil.
ZEM = the motional electrical impedance.
In the mechanical circuit, me = the mass of
the cone and voice coil. CMS = the compliance of the suspension system. 'YMS = the
mechanical resistance of the suspension system. mA = the mass of the air load. 'YMA =
the mechanical resistance of the air load. CMF' = the compliance of the cabinet.
mp = the mass of the air in the port.
'YMP = mechanical resistance of the air load on
the port. 1M = the mechanomotive force in the voice coil.
A
o
]
C
f+-18',-!
30
., 24
o
",24
V\
o
~ 18
'" 18
z
~ 12
'"
z
o
/
'"0:'" 6
"''"
~o
V r-
'" 24
0
.......
w
<II
100
FREQUENCY
500
18
iYO
12
2J1
z
0
"<II
"- 12
0:
f+- 18'-+1
30
0
w
cr
6
o
30
100
FREQUENCY
6
3
V
o
500
~
30
100
500
FREQUENCY
FIG. 6.34. Pressure response frequency characteristics of a direct radiator, dynamic
loudspeaker mechanism with a 10-inch diameter cone and a resonant frequency of
30 cycles operating under the following conditions: A. Open cabinet, 2 feet x 2 feet
X 18 inches in depth. B. Closed cabinet, 2 feet x 2 feet X 18 inches in depth.
C. Phase inverter cabinet, 2 feet x 2 feet x 18 inches in depth and various port
openings, 1. Small port. 2. Medium port. 3. Large port.
DIRECT RADIATOR LOUDSPEAKERS
161
response frequency characteristic for the same mechanism mounted m a
closed cabinet of the same dimensions is shown in Fig. 6.34B. The pressure
response frequency characteristic of the same mechanism and cabinet used
as an acoustical phase inverter for various port openings is shown in Fig.
6.34C. The low-frequency range is extended, the output is increased and
cabinet resonance is eliminated by the phase inverter system.
6.13. Drone Cone Phase Inverter,19-The acoustical phase inverter
loudspeaker system has been described in the preceding section. A study
of this system has shown that the particle velocity over the area of the port
VOICE COIL
ELECTRICAL CIRCUIT
MECHANICAL NETWORK
OF THE
MECHANICAL SYSTEM
CROSS-SECTiONAL VIEW
6.35. Cross-sectional view, electrical circuit and mechanical network of a phase
inverter cabinet equipped with a drone cone. In the electrical circuit, e = internal
voltage of the electrical generator. rEO = internal electrical resistance of the generator.
Land rEe = inductance and electrical resistance of the voice coil. ZBM = motional
electrical impedance. In the mechanical circuit, me and mA = mass of the cone and voice
coil and air load. eMS and rMS = compliance and mechanical resistance of the suspen­
sion system. rMA = mechanical resistance of the air load. CMF = compliance of the
cabinet. m'A = mass of the air load upon the drone cone. r'M A = mechanical resistance
load of the air load upon the drone cone. r'M8 = mechanical resistance of the suspension
of the drone cone. m'e = mass of drone cone. C'MS = compliance of the suspension of
the drone cone. (After Olson, Preston, and May.)
FIG.
is not uniform, either with respect to phase or amplitude. The result is a
loss in energy due to phase shift and friction incurred by viscosity. Another
problem in the ported cabinet is the difficulty of providing a port of large
cross-sectional area so that the particle velocity in the port will be relatively
low. The appropriate inertance in the port can be obtained with a large
cross-sectional area if the length of the port is increased by the required
amount. When this is done, the port becomes very long; and, as a result,
the losses due to viscosity are very large. Thus, it will be seen that simple
port in the phase inverter or bass reflex cabinet is not a satisfactory system
from the standpoint of maximum performance. These objectionable
features can be overcome by the use of an undriven cone, termed a drone
cone, instead of the port, as shown in Fig. 6.35. In this system the port
19
Olson, Preston, and May, Jour. Aud. Eng. Soc., Vol. 2, No.4, p. 219, 1954.
162
ACOUSTICAL ENGINEERING
area of the drone cone can be made the same as the active cone. The phase
and amplitude of the particle velocity are the same over the entire area of
the drone cone. Furthermore, the particle velocity is relatively low because
the area of the drone cone is large compared to a port. As a result, the losses
are low in the drone cone phase inverter.
A typical response-frequency characteristic of the drone cone phase
inverter is shown in Fig. 6.36. Two response-frequency characteristics of
the same loudspeaker in the same cabinet but with two different ports are
also shown in Fig. 6.36. In one case the frequency range of the port is the
same as the drone cone, but the output which is obtained with the port is
30
25
D\:E CONE
~--
m20
o
~
::lz
15
:;<
.--<
32 SQ. IN. PORT
,-::--..:.­
10 SQ. IN. FORT
~
!/l
'" 10
a:
5
60
80
100
200
300
400
FREQUENCY IN CYCLES PER SECOND
FIG. 6.36. Response frequency characteristics of a
is-inch duocone loudspeaker housed in a cabinet of
seven cubic feet comparing a drone cone-type phase
inverter with a port-type phase inverter.
lower. In the other case the output of the port is the same as the drone
cone, but the frequency range which is obtained is less. To summarize,
these characteristics show that a wider frequency range with greater output
can be obtained with the drone cone type of phase inverter, as compared
to the port type, the reason being that the losses in the drone cone are less
than in the port.
6.14. Acoustical Labyrinth Loudspeaker.20-The acoustical labyrinth
loudspeaker consists of an absorbent walled conduit with one end tightly
coupled to the back of the cone of a direct loudspeaker mechanism and the
other end opening in front or at the bottom of the cabinet within which it
is folded (Fig. 6.37). The labyrinth is a piston driven tube with absorbing
walls. At the first half wavelength resonance, the velocity at the open
end is in phase with that of the front of the cone. The radiation, then,
20
Olney, Benj., Jour. Acous. Soc. Amer., Vol. 8, No.2, p. 104, 1936.
163
DIRECT RADIATOR LOUDSPEAKERS
from both sources is additive and the response is increased. An increase
in response can be obtained over about an octave. The rising absorption
of the tube lining with increase in frequency damps out the higher reso­
nances. The transmission through the tube is very low above 150 cycles.
An anti-resonance occurs when the tube is one-quarter wavelength long.
The deleterious effect of the fundamental resonance of the cone with its
suspension system upon the response may be eliminated by choosing the
constants so that fundamental resonance of the loudspeaker coincides
with the quarter wavelength anti-resonance of the tube. The pressure
response characteristic of a direct radiator loudspeaker with and without a
labyrinth is shown in Fig. 6.37. These characteristics show that the
accentuated response due to cabinet resonance has been eliminated and that
the low-frequency range has been extended.
A
5
:" \A
.,20
J
C
)
..,1 5
\. ,''v
~
'"
Z
"
~
"­
210
..,'"
cr 5
'MOUTH
SECTION THROUGH A-A'
'10
,,
100
1000
F"REQUENCV
FIG. 6.37. Acoustical labyrinth loudspeaker.
The pressure response frequency
characteristic of an acoustical labyrinth loudspeaker is labeled B on the graph. The
pressure response frequency characteristic of the corresponding open-back cabinet
loudspeaker is labeled A on the graph. (After Olney.)
6.15. Combination Horn and Direct Radiator Loudspeaker. 21 -One
form of the combination horn and direct radiator loudspeaker consists of a
horn coupled to the back side of a direct radiator loudspeaker mechanism
and an acoustical capacitance for changing the output from the horn to
the open side of the cone for reproduction of the mid- and high-frequency
ranges (Fig. 6.38).
At low frequencies the mechanical reactance of the compliance, CMI,
is large compared to the mechanical impedance, ZMI, at the throat of the
horn. Therefore, the cone is coupled directly to the horn in this frequency
range. In the system shown in Fig. 6.38 the mechanical reactance of the
compliance, CMI, becomes equal to the throat mechanical impedance, ZMI,
at 150 cycles. Therefore, above 150 cycles, the response from the horn is
attenuated and the major portion of the output issues from the front of
the cone and the system behaves as a direct radiator loudspeaker. The
use of a horn as a coupling means makes it possible to obtain large power
21
Olson and Hackley, Proc. Inst. Rad. Eng., Vol. 24, No. 12, p. 1557, 1936.
164
ACOUSTICAL ENGINEERING
outputs from a small diameter cone. In addition, the combination of a
horn and a direct radiator loudspeaker mechanism yields high efficiency
and smooth response at the low frequencies. A cone with a single coil may
be used for reproduction to 7000 cycles. For reproduction to 12,000 cycles
a double voice coil is used. The pressure response frequency characteristic
of the combination horn and direct radiator loudspeaker with double voice
coil driving system is shown in Fig. 6.38.
..
A'
o
B
mtWJ
SECTION 8-B'
S' SECTION A-A
SECTIONAL VIEWS
~20
~
~rHn---~~-H~+-~~-+++~~H
Q.
~ IO~~H---~r1-H~+-~~-+++ITH--H
a:
OLLLU~IO~O----~~~IOtO~O~~-L~ll,O~O~O~O~
FREQUENC'(
FIG. 6.38. Sectional views of the combination horn and double-voice coil, direct
radiator loudspeaker. In the voice coil circuit, e = the internal voltage of the generator.
rEG = the internal electrical resistance of the generator.
rEI and Ll = the electrical
resistance and inductance of the large coil. rE2 and L2 = the electrical resistance and
inductance of the small coil. C E = electrical capacitance. In the mechanical network,
ml = the mass of the large coil. m2 = the mass of the small coil. C M2 and rMa = the
compliance and mechanical resistance of the corrugation separating the large and small
coils. ma = the mass of the cone. rMS and CMS = the mechanical resistance and com­
pliance of the suspension system. m4 and YM2 = the mass and mechanical resistance of
the air load on the front of the cone. CMl = the compliance of the chamber behind the
cone. ZMl = the mechanical impedance at the throat of the horn. hill and JM2 = the
mechanomotive forces generated in the large and small voice coil sections. The graph
shows the pressure response frequency characteristic of the combination horn and direct
radiatorloudspeaker. The overlap between the horn and direct radiator action is shown
by the dotted and dashed characteristics. (After Olson and Hackley.)
A combination horn and direct radiator loudspeaker for operation in the
corner of a room is shown in Fig. 6.39. A sectional view depicting the horn
of the combination horn and direct radiator loudspeaker is shown in Fig.
6.40. The theory and action is the same as the system shown in Fig. 6.38.
There is some advantage in the operation of a relatively small horn loud­
speaker in the corner of the room in that the radiation resistance presented
to the mouth is increased. See Secs. 2.2D and 5.24. There is one unique
feature of the horn system 22 shown in Figs. 6.39 and 6.40, namely, that it
provides true corner operation in that the horn mouth feeds into the
boundaries of the loudspeaker and the two walls and the floor as well. If
22
Olson, H. F., Unpublished Report.
r
DIRECT RADIATOR LOUDSPEAKERS
165
the floor is not employed as the case of most corner systems, full advantage
of corner operation cannot be realized.
In a modification of the systems of Figs. 6.38,6.39, and 6.40 direct radiator
operation is not used, that is, one side of the cone is coupled to the horn
and the other side is coupled to an enclosed cavity. See Sec. 7.4B.
A phase inverter loudspeaker may also be operated in the corner of
the room to obtain increased output in the very low audio-frequency
range.
Another form of the combination horn and direct radiator 10udspeaker23
consists of a direct radiator loudspeaker with a large cone for the repro­
duction of the low-frequency range and a small horn loudspeaker for the
FIG. 6.39. A perspective view of a corner-type com­
bination horn and direct radiator loudspeaker.
reproduction of the high-frequency range. Two different designs of this
type of loudspeaker are shown in Fig. 6.41. In Fig. 6.41A the center pole
for the low-frequency loudspeaker constitutes the small portion of the horn
and a flared type cone in the direct radiator loudspeaker provides a con­
tinuation of the horn. In Fig. 6.41B the center pole also constitutes the
small portion of the horn. A small cellular horn, coupled to the small
portion in the pole, completes the horn. An electrical dividing network is
used to allocate the input to the low- and high-frequency units in the
appropriate frequency ranges.
23
Lansing,
J.
B., Jour. S oc. Mot . Pic. Eng. , Vol. 46, No.3, p. 212, 1946.
166
ACOUSTICAL ENGINEERING
LOUDSPEAKER
SECTION B- a'
LOUDSPEAKER
SOUND
PATH
FIG. 6.40. Cross-sectional views of a corner-type
combination horn and direct radiator loud­
speaker.
INPUT
rRONT
VIEW
A
SECTION A-fI<
INPUT
A'
rRONT
VIEW
B
SECTION A-fI<
FIG. 6.41. Combination horn and direct radiator loudspeakers. A. A direct radiator
loudspeaker is used for the reproduction of the low-frequency range; and a horn loud­
speaker, in which the pole and low-frequency cone form the horn, is used for the repro­
duction of the high-frequency range. B. A direct radiator loudspeaker is used for the
reproduction .of the low-frequency range and a cellular horn loudspeaker is used for the
reproduction of the high-frequency range.
DIRECT RADIATOR LOUDSPEAKERS
167
A modification of the system of Fig. 6.41A, shown in Fig. 6.42, includes
a diverging acoustic lens 24 ,25 at the mouth of the high-frequency horn unit.
Referring to Sec. 2.19, it will be seen
that the high-frequency radiation from
a simple horn is emitted in a relatively
narrow beam. It has been shown in
Sec. 1.11 that a diverging acoustic lens
will spread out the impinging wave­
front. An acoustic lens placed in the
mouth of the high-frequency horn will
increase the spread of the high-frequency
radiation and thus provide improved
directivity.
6.16. Loudspeaker Mechanisms
for Small Space Requirements. 26­
There are many applications in the field
of sound reproduction where space is
limited and the volume occupied by
the loudspeaker must be kept to a
mmlmum. This is particularly the
case in pocket type personal radio
receivers where the loudspeaker is the
largest single component. For these
FIG. 6.42. Combination horn and
applications, an inverted magnetic­ direct radiator loudspeaker equipped
field structure has developed as shown with a lens in the mouth of the high­
in Fig. 6.43. The field structure con­ frequency horn.
sists of three parts, namely, the top and
bottom plates and the magnet. The cone is located between the top and
bottom plates. The top and bottom plates are perforated for efficient
transmission of sound. The performance of the loudspeaker is the same
OPENINGS
BACK VIEW
OPENINGS
SECTIONAL VIEW
FRONT VIEW
6.43. Loudspeaker mechanism with an inverted field structure.
(After Bleazey and Preston.)
FIG.
24 Koch and Harvey, Jour. Acous. Soc. Amer., Vol. '21, No.5, p. 471, 1949.
2. Frayne and Locanthi, Jour. Soc. Mot. Pic. Tel. Eng., Vol. 63, No.3, p. 82, 1954.
26 Bleazey and Preston, RCA Review, Vol. 17, No.2, p. 211, 1956.
168
ACOUSTICAL ENGINEERING
as that of the conventional design with a cone of the same diameter and
weight and a magnet of the same weight. An examination of Fig. 6.43
reveals that the space occupied by the inverted-magnet field structure is
considerably less than in the case of the conventional design of Fig. 6.1.
6.17. Feedback Applied to a Loudspeaker.-Feedback in a transmis­
sion system or a section thereof is the returning of a fraction of the output
to the input. Negative feedback is feedback which results in decreasing the
amplification. Among the sources of nonlinear distortion and nonuniform
response in a reproducing system may be the power amplifier and loud­
speaker. It is possible to reduce distortion and improve the res.ponse as a
function of the frequency of an amplifier by making the amplification
deliberately higher than necessary and then feeding the output back in
such a way as to throwaway excess gain. In the same way this system
may be made to include the loudspeaker. It is not an easy proposition
..
0
I
5
V
30
V\ 11
~I\
m
Oz5
3
'"
"'ZO
z
o
"­
:::1 5
,
- --- -~
0::
0
A
B
5
0
'10
y'
V
100
FREQUENCY
IN
1000
CYCLES
PER
SECOND
10000
FIG. 6.44. Loudspeaker and amplifier feedback systems. A. The output of the pickup
coil is fed into the input side of the amplifier. B. The output of the microphone is fed
into the input side of the amplifier. The graph shows the pressure response frequency
characteristic of System A: 1. Without feedback. 2. With feedback. 3. With feed­
back and compensation.
to employ feedback in this way because of the very special control required
of phase shifts in the amplifier and loudspeaker system. Unless certain
phase relations 27 ,28 are maintain~d, oscillation will occur. Fig. 6.44 shows
feedback systems applied to an amplifier and loudspeaker. In Fig. 6.44A
a pickup coil is attached to the cone. The output from the pickup coil is
fed into the input of the amplifier out of phase with the signal input. The
response characteristic of the amplifier loudspeaker without feedback is
shown in Fig. 6.44. The same loudspeaker and amplifier with 15 db nega­
tive feedback from the pickup coil are also shown in Fig. 6.44. It will be
seen that the response at the high frequencies is improved. This system
tends to drive the cone at constant velocity for constant signal voltage
27
28
Nyquist. R .• Bell Syst. Tech. Jour .• Vol. 11. No. 1, p. 126. 1932.
Black, R. C.• Bell Syst. Tech. Jour., Vol. 13. No. 1, p. 1, 1934.
DIRECT RADIATOR LOUDSPEAKERS
169
input. Therefore, the response will fall off below the point of ultimate
resistance, because the radiation resistance falls off 6 db per octave in this
range (see Fig. 6.2). The response may be made uniform with respect to
frequency by compensation of the input to the system.
A feedback system employing an amplifier, loudspeaker and microphone
is shown in Fig. 6.44B. If a pressure operated microphone having uniform
sensitivity with respect to frequency is used the response characteristic of
the loudspeaker will become more uniform as the amount of feedback is
increased.
6.18. Cabinet Configuration. 29 ,30,31-The outside configuration of the
cabinet influences the response due to diffraction effects introduced by the
edges of the cabinet. The effects of
diffraction for various shapes are given
in Sec. 1.11. The response frequency
characteristics of Fig. 1.7 shows that
cabinet diffraction effects may introduce
variations of several decibels in the res­
ponse of a loudspeaker mechanism with
an otherwise smooth response frequency
characteristic. As a result of a study of
cabinet shapes, a cabinet has been
developed in which the deleterious effects
of diffraction have been reduced to practi­
cal limits. The cabinet which has been
evolved is shown in Fig. 6.45. It will be
seen that the sharp front edges of a
standard rectangular parallelopiped type
cabinet, which should set up diffracted
FIG . 6.45.
A cabinet designed to
waves, have been eliminated. As a result, eliminate the deleterious effects of
the variations in the response due to diffraction in the response of the
diffraction effects have been reduced to a loudspeaker.
negligible amount.
6.19. Loudspeaker Mounting Arrangement in the Cabinet Wall. 32 ,33
-The mounting arrangement of the loudspeaker mechanism in the front
wall of the cabinet influences the response due to the resonances of the
cavity in front of the mechanism. In addition, variations in the response
are produced by reflections and diffractions from the circular boundary of
this cavity. The standard mounting arrangement for loudspeaker mechan­
isms which has been used for years is shown in Fig. 6.46A. Referring to
Fig. 6.46A, it will be seen that the cabinet wall forms a cavity in front of the
loudspeaker. The resonances and anti-resonances of this cavity, as well
Olson,
Olson.
31 Olson.
32 Olson.
33 Olson.
29
30
H. F., Radio and Television News, Vol. 45, No.5. p. 53, 1951.
H. F .• Audio Engineering, Vol. 35, No. 11. p . 34. 1951.
Preston, and May, Jour. Aud. Eng. Soc ., Vol. 2. No.4. p. 219. 1954.
H. F .• Radio and Television News. Vol. 45. No. 5. p. 53. 1951.
Preston, and May. Jour . Aud. Eng. Soc .• Vol. 2, No.4, p. 219. 1954.
170
ACOUSTICAL ENGINEERING
as reflections and diffractions of this
wall edge, introduce variations in the
response frequency characteristic as
shown by curve A in Fig. 6.47. These
variations in response can be reduced
by the improved loudspeaker mechan­
ism mounting arrangement, as shown
in Fig. 6.46B. It will be seen that the
cavity in front of the loudspeaker
mechanism has been materially re­
duced. The reflecting edge of the cut­
out in the cabinet wall has been
completely eliminated. The sharpness
of the edge has also been reduced
A
B
which mitigates the diffraction effects
FIG. 6.46. A. Direct radiator louddue to this edge. The response fre­
speaker mechanism mounted on the
quency characteristic of a loudspeaker
back of the cabinet wall. B. Direct
mechanism mounted as shown in
radiator
loudspeaker
mechanism
mounted flush with the front of the
Fig. 6.41B is shown by curve B in
cabinet wall.
Fig. 6.47. Comparing the response fre­
quency characteristics of curves A and
B of Fig. 6.47, it will be seen that a considerable improvement in response
can be obtained with the mounting arrangement shown in Fig. 6.46B.
30
25
CD
o
20
r-...
~
llll5
z
oQ.
'"'"
a:
..,......r,::
r-..,,t:. ~
,
I
\ IA
\I
V
~ ~r­-
\\ /
/
~
I
II
I
I
'~
"
10
5
o
1000
2000
3000 4000
6000 8000 10000 15000
FREQUENCY IN CYCLES PER SECOND
FIG. 6.47. A. Response frequency characteristic of a direct
radiator loudspeaker mechanism mounted as shown in Fig. 6.46A.
B. Response frequency characteristic of a direct radiator loud­
speaker mechanism mounted as shown in Fig. 6.46B.
DIRECT RADIATOR LOUDSPEAKERS
171
6.20. Loudspeaker Locations in Television Receivers.-The proper
placement of the loudspeaker mechanism in television receivers is difficult
to achieve because of the large area of the surface of the kinescope on the
front of the receiver. This is a particularly difficult problem in table model
television receivers where there is no space on the front of the receiver for
the loudspeaker mechanism. As a consequence the loudspeaker mecha­
nism must be placed in one or more of the four locations LSI, LS2, LS3,
and LS4, shown in Fig. 6.48. In all of these locations the listener-viewer
is located at approximately 90° from the axis of the loudspeaker. The
I
1 I
~'~1
1)4
~tS3
I
I
:
/~.\
~
'\:~I
/
1
L~I
./
LS4'~
\
I
I
\
/
~
\
I~~__
\
1
TOP VIEW
r;
C"1
I
f
I'"
I )J
t LS3
1
I
/;:1:
I,
~
LSI
'1'1I
l~
LS4
LS2
~o~_
I
o
I
1
-'
FRONT VIEW
SIDE VIEW
TABLE MODEL
FIG. 6.48. Loudspeaker mechanism locations in a table model
television receiver.
loudspeaker mechanisms employed are usually small ones with 4- or 5-inch
diameter cones. Referring to the directional patterns of a four-inch cone
of Fig. 6.11, it will be seen that it is practically nondirectional in the low­
frequency range. However, in the high-frequency range it becomes quite
directional. Under these conditions with the mounting arrangements of
Fig. 6.48 there will be considerable frequency discrimination in the high­
frequency range. Some compensation in high-frequency areas is provided
by accentuating the high-frequency response.
The placement of the loudspeaker mechanism in consoles is a somewhat
simpler problem, as shown in Fig. 6.49, in that the loudspeaker mechanism
may be located on the front of the cabinet. The most common arrange­
ment is a single loudspeaker LS 1. Other arrangements are as follows:
Two loudspeaker mechanisms LS2 and LS3 are placed in the corner and
angled to increase the coverage. Three loudspeaker mechanisms LSI, LS2,
and LS3 placed in the front and corners of the cabinet. Three loudspeakers
with the loudspeaker mechanism LSI in the front of the cabinet and loud­
speaker mechanisms LS4 and LS5 in the sides of the cabinet. If separate
172
ACOUSTICAL ENGINEERING
loudspeaker mechanisms are used to cover the low- and high-frequency
ranges, the low-frequency loudspeaker mechanisms may be placed in one
or more of the locations LSI, LS2, LS3, LS4, and LS5 and the high-frequency
,
I
:
I
l
LS4
LssA
\
r'",
t).! /
: /
I
\
r( I
\ . . '~
\ I
IL~Ls6 ~~{LS7 F~{~J
r.t~.c""_":>':\f.~~
TOP VIEW
SIDE VIEW
FIG. 6.49.
Loudspeaker mechanism locations in a console model
television receiver.
I
I
)
I
/
1
I
/
:
:I
I
I
I
\
I
\
I
I
\
\
/
\
I
\
I
I
I
\
/
\
1
1
LS3
\
ItI--------
LSI
L~2YA
\
~\.. L2:/
I
I
'~
TOP VIEW
I
t~I,~-~~
I
O~-\
I
I
I
I
I
I
I
I
;{/
0
'\_-_/
===1
o, - '
LSI
~~,
\==:1
-I
'...,..,."
I
I
L53 J
L _______________
FRONT VIEW
SIDE VIEW
FIG. 6.50.
Loudspeaker mechanism locations in a console model
television receiver.
DIRECT RADIATOR LOUDSPEAKERS
173
loudspeaker mechanisms may be placed in either or both locations LS6 and
LS7.
Telecasts are usually viewed within an angle of ±4S o with respect to the
axis of the kinescope. For that reason there appears to be no object in the
use of loudspeaker locations LS2, LS3, LS4, and LSS because a single
loudspeaker mechanism will give adequate coverage over a total angle of
90°. Furthermore, multiple loudspeaker mechanisms decrease the intel­
ligibility of speech reproduction. There is one objectionable feature in the
cabinet of Fig. 6.49, namely, that the television chassis and kinescope are so
large that the loudspeaker mechanism must be located very close to the
floor. It has been established that the most natural sound reproduction is
obtained when the loudspeaker mechanism is located at ear level of the
listener.
A cabinet design in which the loudspeaker may be located at a greater
distance from the floor is shown in Fig. 6.50. In the most common arrange­
ment, a simple loudspeaker LSI is used to cover the entire frequency range
and placed near the top of the cabinet. In another arrangement an addi­
tional loudspeaker mechanism LS2 may be added. In a further modifica­
tion one loudspeaker mechanism may be used to cover the high-frequency
range and another loudspeaker mechanism may be used to cover the low­
frequency range. Another loudspeaker location is LS3. This loudspeaker
may be used in various combinations with LSI and LS2. For example, in
one arrangement, each of the three loudspeakers covers a section of the
frequency range. In another combination LS2 may be omitted and LSI
and LS3 may be operated in parallel.
In another arrangement of Fig. 6.50, loudspeakers may be symmetrically
located on both sides of the kinescope.
6.21. Loudspeaker Locations in Phonographs.-There are two general
types of phonograph cabinets, namely, table and console models. There
are many possible locations for loudspeaker mechanisms in phonograph
cabinets as depicted in Figs. 6.51, 6.52, and 6.53.
The seven most common loudspeaker mechanism locations in table
model phonographs are shown in Fig. 6.51. Obviously, not all of these
loudspeaker locations are employed in a single instrument. For example,
the most common arrangement is that of a single loudspeaker mechanism
in location LSI of Fig. 6.51. In a multiple arrangement, two loudspeaker
mechanisms LS2 and LS3 of Fig. 6.51 are placed in the corners of the cabinet
and angled to increase the coverage. In another multiple arrangement, two
loudspeaker mechanisms LS4 and LSS are placed in the two sides of the
cabinet. In a multiple arrangement of three loudspeaker mechanisms,
LSI, LS4, and LSS are placed in the front and the two sides of the cabinet,
respectively, to obtain wide angle coverage. In another multiple arrange­
ment the three loudspeakers LSI, LS2, and LS3 are placed in the front of
the cabinet. If separate loudspeaker mechanisms are used to cover the
low- and high-frequency ranges, the low-frequency loudspeaker mechan­
isms may be placed in one or more of the locations LSI, LS2, LS3, LS4,
174
ACOUSTICAL ENGINEERING
and LS5 and the high frequency loudspeaker mechanisms may be placed
in either one or both of the locations LS6 and LS7.
The two most common console type phonograph cabinets and the loud­
speaker locations in these cabinets are shown in Figs. 6.52 and 6.53. Seven
loudspeaker locations are shown in Fig. 6.52. The locations LSI, LS2,
LS3, LS4, and LS5 are employed for mechanisms with full frequency range.
A single loudspeaker at location LSI is the most common arrangement.
Other arrangements are as follows: Two loudspeaker mechanisms LS2 and
LS3 are placed in the corner and angled to increase the coverage. The two
loudspeaker mechanisms LS4 and LS5 are placed in the two sides of the
TOP VIEW
-l
r
i
"'----------:::l,
~=~-~.;~:-.:~=-~~----4
t,',
/'"
I ' l ' ,-~
';J - /
"
\
)lLS!5
I
I
I
I
~~--~'::-.::::-----~J
L 0
0
CHASSIS
SIDE VIEW
FIG. 6.51. Loudspeaker mechanism locations in a table model
phonograph.
cabinet. The latter arrangement is seldom used. Three loudspeaker
mechanisms LSI, LS4, and LS5 are placed in the front and the two sides
of the cabinet, respectively, to obtain wide angle coverage. Three loud­
speaker mechanisms LSI, LS2, and LS3 are placed in the front of the
cabinet. If separate loudspeaker mechanisms are used to cover the low­
and high-frequency ranges, the low-frequency loudspeakers may be placed
in one or more of the locations LSI, LS2, LS3, LS4, and LS5 and the high­
frequency loudspeakers placed in locations LS6 and LS7. In the simplest
arrangement a single high-frequency loudspeaker LS6 and a single low­
frequency loudspeaker LSI are used. Other arrangements include various
arrangements of LSI to LS7. In all arrangements the loudspeakers should
be located at as large a distance from the floor as possible. The advantage
of the console phonograph shown in Fig. 6.53 is that the loudspeaker
DIRECT RADIATOR LOUDSPEAKERS
I:
._--....
~
1
i
iI
'"
'II
/TURNTABLE\II
1,
,
i/
I
n
).
LS4
I
N
1/1
\
1/
,
(/; '
" /
ILS2 ~...
1.,./-'"-",
0
LS6
L ---~Y-
I
n
I
I'
1
I
L\ I
LS5~ 1
'i
LSI ......... __ -r-*S3 I
"';::\..... LS7 / , - .... I
~~------~
TOP VIEW
FRONT VIEW
SIDE VIEW
6.52. Loudspeaker mechanism locations in a console model
phonograph.
FIG.
r
I
I
I
I
I
I
I
1
I
I
TOP VIEW
1-;;;:;'- -
III '1
1==;=:0=·:::----~1=-..::I1 \~Y
----1
I
1
L.3~~<?..._J
IILS2
I
I
I
TURNTABLE
_C...::.:-:::rr==-:l__
I
r
=
=. -=--...::.::.::: =:..""
I
I
I
I
I
I
1_ _ ­
FRONT VIEW
FIG. 6.53.
Loudspeaker mechanism locations in a console model
phonograph.
175
176
ACOUSTICAL ENGINEERING
mechanism can be placed at a greater distance from the floor. It has been
established that the most natural sound reproduction is obtained when the
loudspeaker is located at a distance from the floor corresponding to the ear
level of the listener. In the cabinet of Fig. 6.53 the most common
arrangement is that of a single loudspeaker mechanism covering the entire
frequency range and placed near the top of the cabinet. In another
arrangement a high-frequency loudspeaker mechanism LSI is used to cover
the low-frequency range. Another loudspeaker location is LS3. This loud­
speaker may be used in various combinations with LSI and LS2. For
example, in one arrangement, each of the three loudspeakers covers a section
of the frequency range. In another combination LS2 may be omitted and
LSI and LS3 may be operated in parallel.
6.22. Loudspeaker Locations in Radio Receivers.-Radio receivers
may be classed as follows: console, table, portable, personal, and automobile.
Each of these types requires a different arrangement and location for the
loudspeaker mechanism.
In general, the loudspeaker mechanism locations in the console type
radio receiver are the same as the console type phonograph. See Sec. 6.21
and Figs. 6.52 and 6.53.
The six most common locations for loudspeaker mechanisms in table
model radio receivers are shown in Fig. 6.54. The most common arrange­
ment of the loudspeaker mechanism in a table model radio receiver is the
use of a single loudspeaker in any of the locations LSI, LS2, LS3, LS4, LS5,
or LS6. In a multiple arrangement, two loudspeaker mechanisms LS2 and
LS3 are placed and angled to increase the coverage. In another multiple
arrangement two loudspeaker mechanisms LS4 and LS5 are placed in the
two sides of the cabinet. In a multiple arrangement of three loudspeaker
mechanisms, the loudspeaker mechanisms LSI, LS4, and LS5 are placed
in the front and the two sides of the cabinet, respectively, to obtain wide
angle coverage. In another multiple arrangement of three loudspeakers,
the loudspeaker mechanisms LSI, LS2, and LS3 are used.
Since the cabinet size of a table model receiver is relatively small, the
response falls off rapidly in the low-frequency region. See Sec. 6.9. Suit­
able compensation may be employed in the audio amplifier to maintain the
low-frequency response. Employing this expedient, the limitation is the
maximum allowable excursion of the loudspeaker cone. The amplitude
of the cone for a certain sound power output is inversely proportional to the
area of the cone. Thus, it will be seen that the low-frequency response that
can be obtained is a function of the total radiating surface.
Personal radio receivers employ very small enclosures in the form of a
rectangular parallelopiped. The cubical content of the smallest receivers
is of the order of 25 cubic inches. The loudspeaker mechanism is usually
placed in the front panel of the radio receiver. The order of power available
for feeding the loudspeaker is of the order of 5 to 50 milliwatts. Since the
power is limited to a very low value and the loudspeaker mechanism and
cabinet are both small, it is impossible to obtain adequate low-frequency
DIRECT RADIATOR LOUDSPEAKERS
177
response for natural sound reproduction. In the personal radio the principal
use is for the reproduction of speech. Therefore, the acoustical problem is
to obtain good intelligibility on speech together with adequate sound level
output. This can be accomplished in the personal radio receiver by proper
and suitable design of the loudspeaker and case. See Sec. 6.9.
The acoustic problem in the larger portable radio receiver is similar to
the personal radio receiver. Of course, the larger the case and loudspeaker
the greater the low-frequency response.
Automobile radio receivers are housed in an enclosure of about 1000
cubic inches or less. In general, a loudspeaker mechanism is also placed in
the same housing. In some cases the radio chassis and loudspeaker are
I~LS5
,'''
I"
I )J
;~~
LS4)
;.:-'
../ 1
,-\
ill
LS3\~;LS2"\ 1
{..-c--~
"~
'--~/
B
S""~~
'\..i,,;--..J
_...I
I
\8
~""=-~l5
I
I(
I
I,
0
1,_"
l -.::......,..
TOP VIEW
r~
I
.!
FRONT VIEW
A
I
1
I
SIDE
VIEW
~
r~\)d:':-,
I : ~!
I tli.... . _ j
I
1
L
0
0
I
I
FRONT VIEW
1
LS4 _
Lr' _____
,z
CHASSIS
SIDE VIEW
FIG. 6.54. A. Loudspeaker mechanism locations in a
table model radio receiver. B. Loudspeaker mechanism
location in a personal radio receiver.
mounted in separate enclosures. An additional loudspeaker is sometimes
placed in the deck behind the rear seat. See Sec. 11.25. Since the loud­
speaker in the rear deck operates in the trunk compartment, there is no
enclosure problem. However, the space allocated in the loudspeaker
mechanism is limited when it is mounted with the chassis or separately for
reproduction in the forward portion of the motor car. Employing a low
resonant eight-inch loudspeaker or the equivalent in an elliptical loudspeaker,
very good reproduction of low frequencies can be obtained in an enclosure
of 1000 cubic inches.
6.23. Loudspeaker Locations in Combination Instruments.-The
location of the loudspeaker mechanism in combination radiophonograph
cabinets is similar to that of the phonograph of Sec. 6.21. In both the
table model and console phonographs it is customary to add a radio tuner
so that both radio and phonograph reproduction can be obtained.
178
ACOUSTICAL ENGINEERING
6.24. Concentrated Source Loudspeaker. 34-There are applications
where it is necessary to produce a high sound signal level over a limited zone
without producing a high sound signal level surrounding the zone. These
conditions may be obtained by means of the system shown in Fig. 6.55.
In the particular example of Fig. 6.55, 67 small direct radiator loudspeaker
mechanisms are mounted on a section of a spherical surface. The output
of the loudspeakers are all in phase at the center of the spherical surface or
the focus of the system. By means of this system a gain of more than 20
COUDSPEAKERS
LOUDSPEAKERS
® ®
® ® ®
®
®
®
®
® ®
® ® ®
® ®
®
®
®
®
®®®
®
®
®
® ® ®
®
®
®®®
®
®
®
®
®
® ®
® ® ®
®
®
®
® ® ®
®
®
® ® ®
®
FRONT VIEW
FIG. 6.55.
SECTIONAL VIEW
Front and sectional views of a concentrated source loudspeaker.
db in signal level at focus compared to other locations removed from the
focus may be obtained.
6.25. Transient Response. 3s-The subject of transient response em­
braces a wide variety of physical phenomena. Electrical transients concern
electrical circuits and the components of electrical systems. Acoustical
transients concern acoustical and mechanical systems. In view of the fact
that the sound reproducing and collecting systems are mechanical, the
general tendency is to assume that these systems exhibit very poor transient
response characteristics. In properly designed acoustical elements the
performance is very often far superior to the other components used in
sound reproducing systems.
The behavior of a loudspeaker may be analyzed by solving the differen­
tial equations of the dynamical system. In other words, find the velocities
of the elements of the system which, when substituted in the differential
equations, will satisfy the initial and final conditions. The solution of a
differential equation may be divided into the steady state term and the
Olson, H. F., Unpublished Report.
Olson, .. Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J.,
1943.
34
35
DIRECT RADIATOR LOUDSPEAKERS
179
transient term. The operational calculus is of great value in obtaining
the transient response of a mechanical or acoustical system to a suddenly
impressed force or pressure.
The general analysis used by Heaviside is applicable to any type of
vibrating system whether electrical, mechanical, or acoustical. It is the
purpose of this section to show the response of the conventional direct
radiator loudspeaker to a suddenly applied unit force.
The mechanical circuit of the dynamic loudspeaker at the low frequencies
is shown in Fig. 6.1. The differential equation for the system of 6.1 is
mx + YMTX + ;M =/MO
where
x
=
/MO =
m
=
eM =
YMT =
6.17
displacement,
mechanical driving force, in dynes,
total mass, in grams,
compliance of the suspension system, m centimeters per
dyne, and
total mechanical resistance, in mechanical ohms.
The total mechanical resistance is
YMT = YMS
where
YMS =
YMR =
YME =
+ YMR + YME
6.18
mechanical resistance due to losses in the suspension system,
etc., in mechanical ohms,
mechanical radiation resistance, in mechanical ohms, and
mechanical resistance due to the electrical system, in mechan­
icalohms.
The mechanical resistance, YMS, is the sum of all the losses in the suspension,
the viscosity of the grill and cloth coverings and the viscosity loss due to the
air forced through the slit formed by the air gap and voice coil.
From equation 6.10 the mechanical impedance, ZME, due to the electrical
circuit is
(Bl) 2
ZME = YME = - ­
YET
where
B
=
l
=
6.19
flux density, in gausses,
length of the voice coil conductor, in centimeters,
+ YEG,
rET =
YEC
rEa =
damped electrical resistance of the voice coil, in abohms, and
internal electrical resistance of the generator (the vacuum
tube), in abohms.
YEG =
The inductance of the voice coil is negligible. The mechanical radiation
resistance, rMR, is given by the equation 5.10. It may be obtained directly
from the graph of Fig. 5.2.
180
ACOUSTICAL ENGINEERING
The mass, m, is the sum of the cone mass and the mass of the air load
upon the cone. The mechanical mass reactance of the air load upon a
cone may be obtained from equation 5.10. It may be obtained directly
from the graph of Fig. 5.2.
Heaviside's un extended problem 36 ,37,38 is as follows: Given a linear
network of n meshes in a state of equilibrium, find its response when a
unit force is applied to any mesh. The unit function is defined to be a
force which is zero for t < 0 and unity for t > O.
The indicial mechanical admittance of the mechanical circuit of Fig. 6.1 is
1
A(t) =
YMT
1
6.20
+ mp + GMp
where p is employed as a symbol for the differentiation with respect to the
independent variable, time.
Let
YMT
2m
a =
w
=
Jm~M
-
a2
The indicial mechanical admittance may be written
pw
A(t) _ _
1
-
mw
(P
+ a)2 + w 2
6.21
From tables of operational formulas, the solution is
A (t)
=
_1_
mw
sin wt
€-at
6.22
Fig. 6.56 shows the effect of the electrical impedance of the vacuum tube
upon the transient response of a loudspeaker. In this case the loudspeaker
is connected to the following generators: a very high electrical impedance
corresponding to pentode or Class" B" operation; a generator of one-half
the electrical resistance of the loudspeaker corresponding to class "A"
operation; and to a generator of :very low electrical impedance corresponding
to inverse feedback operation. The electrical impedance characteristic of
the loudspeaker is shown by the uppermost left-hand graph of Fig. 6.57.
This example shows that the damping exerted by the electrical system is
of consequence. However, there is very little difference between Class
" A" and feedback operation. When a loudspeaker operates from a high
36 Carson, " Electric Circuit Theory and Operational Calculus," McGraw-Hill Book
Company, New York, N.Y., 1926.
37 Bush, "Operational Circuit Analysis," John Wiley and Sons, New York, N.Y.,
1937.
38 Berg, "Heaviside's Operational Calculus," McGraw-Hill Book Company, New
York, N.Y., 1936.
181
DIRECT RADIATOR LOUDSPEAKERS
electrical impedance vacuum tube amplifier, the internal mechanical resist­
ance of the loudspeaker is the major factor influencing the transient response.
Fig. 6.57 shows response of a 12-inch (to-inch diameter cone) loudspeaker
to a unit force for various values of mechanical resistance. In order to
correlate the response with actual systems, the electrical impedance frequency
characteristic for each system is also shown. These characteristics are for
a loudspeaker coupled to a generator with very high internal electrical
A
!~?MfHlH
~ 1.00
B
0.1
!ff~!-tlllill
d
c
0.05
1.00
0.05
0.1
iJtH j-IIIIII
o
0.05
TIME
IN
0.1
SECONDS
FIG. 6.56. The transient response of a direct radiator, dynamic
loudspeaker, with a 12-inch diameter cone, to a unit force for various
types of electrical generators. A. Generator of very high electrical
resistance. B. Generator having an electrical resistance of one half
of the loudspeaker electrical impedance. C. Generator of zero electri­
cal impedance.
impedance. For this type of operation it is customary to provide a large
mechanical resistance, rMS, the second and third conditions of Fig. 6.57.
Figs. 6.56 and 6.57 show that the "hangover" in properly designed and
operated loudspeakers is very small. Of course, the systems are improved
as the fundamental resonant frequency is lowered. In some of the small
receivers employing relatively high electrical impedance power amplifiers
driving loudspeakers having the fundamental resonance above tOo cycles,
the response to transients is usually very poor because the internal mechanical
resistance is not sufficiently large. Of course, the steady state response
with respect to frequency is usually not very smooth and the nonlinear
distortion is quite large in these receivers. As a consequence, the poor
transient response is masked by these distortions.
182
ACOUSTICAL ENGINEERING
Another means for depicting the transient response of a loudspeaker is
the application of a tone burst signal. 39 ,40,41 A tone burst signal consists of
a sine wave with a rectangular envelope containing ten or more cycles. The
deviation in the sound output from the rapid growth and decay charac­
teristics and steady state characteristic of an applied tone burst signal
depicts the transient response of a loudspeaker. Referring to the response
frequency characteristic of a loudspeaker with an undamped suspension
60
40
20
V
r-­
f-r-"'"
100
300
100
300
z
20
1---­
20
I - -I--"
~o
100
rREQUENCY
300
!twi4+HffH
t
1.00
0.05
0.1
!~N1d+H J \ \ I
cl
1.00
0.05
0.1
iJtH IIIII11
0
0.05
TIME
IN
0.1
SECONDS
FIG. 6.57.
The transient response of a direct radiator, dynamic loud­
speaker, with a 12-inch diameter cone, for different values of the suspension
mechanical resistance. The electrical impedance frequency characteristic
indicates the degree of internal damping.
system shown in Fig. 6.74A, it will be seen that there is a peak in the response
at 800 cycles and a dip at 1100 cycles. The sound outputs from this loud­
speaker with applied tone burst signals of 800 cycles and 1100 cycles are
shown in Fig. 6.58. In the case of the peak in the response frequency
characteristic there is a slow growth and a slow decay in the response of
the loudspeaker to an applied tone burst signal. In the case of a dip in the
response frequency characteristic there is a rapid growth followed by a
decrease in output then followed again by an increase in output in the
response to an applied tone burst signal. It has been suggested that a
39
40
41
Olson, H. F., Audio Engineering, Vol. 34, No. 10, p. 5, 1940.
Olson, Preston, and May, Jour. Aud. Eng. Soc., Vol. 2, No.4, p. 219, 1954.
Corrington, M. S., Jour. Aud. Eng. Soc., Vol. 3, No. 1, p. 35, 1955.
DIRECT RADIATOR LOUDSPEAKERS
183
measure of the transient distortion be designated as the remaining response
after the applied tone burst signal has been cut off, as shown in Fig. 6.58.
Apparatus for measuring the transient response by means of a tone burst
signal is described in Sec. 1O.3G.
800 CYCLES
1100 CYCLES
ELECTRICAL INPUT
ELECTRICAL IN PUT
LOUDSPEAKER OUTPUT
LOUDSPEAKER OUTPUT
-+ ---1­
TRANSIENT DISTORTION
TRANSIENT DISTORTION
FIG. 6.58. The transient response of a loudspeaker having the
response frequency characteristic of Fig. 6.74A at 800 cycles
and 1100 cycles. The waves show the tone burst input to the
loudspeaker, the sound output from the loudspeaker, and the
output from the loudspeaker after the input to the loud­
speaker has been stopped.
i
!
6.26. Distortion.-The general trend in all types of radio receivers and
phonographs is more output without a corresponding increase in the size
of the loudspeaker. As a result, the maximum amplitude of the loud­
speaker is also increased. Many apparently peculiar activities are mani­
fested by the loudspeaker when the amplitude or excursion of the cone is
large. Under these conditions, the operation exceeds the linear portion
of the stress-strain characteristic of the cone material.
Most of the unusual phenomena are due to the nonlinear characteristics
of the cone suspension system. One of the effects of a nonlinear cone
suspension system is a jump phenomena in the response characteristic.
Another effect is the production of harmonics and subharmonics due to the
nonlinear cone suspension system. Frequency modulation of a high­
frequency signal by a large low-frequency amplitude of the cone is another
form of distortion. The nonlinear characteristics of the air also introduce
distortion. It is the purpose of this section to consider the various types
of distortion produced in a direct radiator type loudspeaker system.
A. Distortion Due to Nonlinear Cone System. 42 ,43-Nonlinear distortion is
42
43
Olson, H. F., Audio Engineering, Vol. 34, No. 10, p. 5, 1950.
Olson, Preston, and May, Jour. Aud. Eng. Soc., Vol. 2, No.4, p. 219,1954.
184
ACOUSTICAL ENGINEERING
generated in the cone when the operation exceeds the linear portion of the
stress-strain characteristic of the cone. The strength of the cone depends
upon the thickness of the paper.
12
The weight in turn is proportional
to the thickness of the cone. The
....... sound
9
output of a typical direct
.,.;'
III
radiator loudspeaker for a certain
o
value of nonlinear distortion as a
... 6
:>
function of the weight of the cone
11.
...:::.
is shown in Fig. 6.59. The rela­
o
/
3
tive output of a direct radiator as a
function of the weight of the cone
is shown in Fig. 6.4. A review of
o
1.0
1.5
2.0
2.5
3.0 Figs. 6.4 and 6.59 shows that high
RELATIVE CONE MASS
sensitivity by the use of a light
FIG. 6.59. The sound output of a typical
cone is not compatible with low
direct radiator loudspeaker, for a certain fixed
nonlinear distortion. In order to
value of nonlinear distortion, as a function of
obtain low nonlinear distortion, a
the mass of the cone.
relatively heavy cone must be used.
B. Nonlinear Suspension System. 49-The force displacement charac­
teristic of a typical, direct radiator loudspeaker cone suspension system
V
V
/
X
DISPLACEMENT
FIG. 6.60. Force displacement characteristic of the
suspension system of a direct radiator loudspeaker.
is shown in Fig. 6.60. It will be seen that for small amplitudes the suspen­
sion system is linear. However, for large amplitudes the suspension system
is nonlinear.
44
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 16, No. 1, p. 1, 1944.
DIRECT RADIATOR LOUDSPEAKERS
185
The force defiection characteristic of the loudspeaker cone suspension
system of Fig. 6.60 may be approximately represented by the expression
/M
=
/(x)
=
ax
+ f3x 3,
6.23
where a = constant> 0, f3 = constant> 0, and /M = applied force which
produces the displacement x.
The compliance of the suspension system of Fig. 6.60 may be obtained
from equation 6.23 as follows:
x
1
eM = JM
T =
f3 2
6.24
a
x
The differential equation of the vibrating system in Fig. 6.1 is
+
mx + YMX +
where
J
M =
F cos wt,
6.25
=
displacement,
velocity,
acceleration,
m = mass of the cone, coil, and air load,
YM = mechanical resistance due to dissipation in the air load and
suspension system,
eM = compliance of the suspension system,
F = Eli,
B = magnetic fiux density in the air gap,
l = length of the voice coil conductor,
i = amplitude of the current in the voice coil,
w = 21r/,
/ = frequency, and
t = time.
Substituting the expression for eM of equation 6.24 in equation 6.25,
the differential equation becomes
x
x=
x=
mx + YMX + aX + f3x 3 =
F cos wt
6.26
Since the mechanical resistance, YM, is quite small compared to the
mechanical reactance, save over a very narrow frequency range near the
resonant frequency, equation 6.26 can be written as follows:
mx + aX + f3x 3 =
F cos wt.
6.27
A number of investigators have obtained an approximate solution of this
differential equation.
If f3 is considered to be small, the relation
+ !f3A 2 _ £
m
m
Am
between the arbitrary amplitude A and w may be obtained,
w2
=~
6.28
186
ACOUSTICAL ENGINEERING
An approximate solution of the differential equation, for unit mass, is
f3A 3
1
x
=
+ 32 ex + !f3A2 _
A cos wt
6.29
(F/A) cos 3wt.
The sections which follow will show that these equations predict the
perfonnance of a loudspeaker with a nonlinear cone suspension system.
C. Distortion Characteristics of Nonlinear Suspension System.-The well­
known experimental result of a nonlinear cone suspension system is the
production of odd order harmonics when a sinusoidal input is applied to the
loudspeaker. The wave shape under these conditions is shown in Fig. 6.61.
ELECTRICAL
INPUT
ACOUSTICAL
OUT PUT
6.61. The wave shapes of the electrical input and the sound pressure
output of a loudspeaker with a nonlinear suspension system.
FIG.
The third harmonic is the preponderant distortion component. Equation
6.29 shows that a third harmonic tenn is introduced due to the suspension
system. In the case of a direct radiator loudspeaker, the amplitude is
inversely proportional to the square of the frequency for constant sound
power output in the frequency region below the frequency of ultimate
30
\
I­
:z 25
w
5Il. zo
~ 15
z
2
\
10
I-
ao
~ 5
ell
Q
~o
\
\
1\
II
1\
\ \
U
~
"- I"-
olWIAITTS
~WIATTS
'
~~ATT
...I'
~
100
fREQUENCY
400
IN
CYCLES
PER
1000
SECOND
FIG. 6.62. Distortion frequency characteristics of a direct
radiator, dynamic loudspeaker mechanism with a lO-inch
diameter cone and a nonlinear suspension system for electrical
inputs of 2, 5, and 10 watts.
resistance. Consequently, the greatest distortion will occur at the low­
frequency end of the frequency range as shown by typical, experimental,
nonlinear distortion frequency characteristics of Fig. 6.62. The manifesta­
tion and effect of this type of distortion upon the reproduction of sound are
well known. Distortion occurs in all amplifiers as well as loudspeakers.
DIRECT RADIATOR LOUDSPEAKERS
187
As a matter of fact, it is more troublesome in amplifiers because the distortion
occurs over the entire audio frequency range, whereas the distortion is
confined to the low-frequency range in loudspeakers.
In the above considerations, the distortion produced by the nonlinear
element comprises harmonics of the fundamental. Distortion components
with frequencies of t, t, t ... lin of the frequency of the applied force
also occur in nonlinear systems. Those familiar with the performance of
loudspeakers have noticed the production of subharmonics. In general,
these are very pronounced in the mid-frequency range. In the mid-fre­
quency range the subharmonics are due to the nonlinear properties of the
cone. Particular solutions of equation 6.26 have been obtained which
show that subharmonics are possible in a loudspeaker due to a nonlinear
cone suspension system. As pointed out above, the amplitude of the cone
of a direct radiator loudspeaker is inversely proportional to the square of
the frequency for constant sound output. The large amplitudes are con­
fined to the low-frequency range. Therefore, these subharmonics will be
of a very low frequency and difficult to detect. Careful experimental
investigations have shown the existence of subharmonics due to a nonlinear
cone suspension system as predicted from theoretical considerations.
m,
m2
CROSS-SECTIONAL VIEW
f
FIG. 6.63.
A system consisting of a mass ml driven by a crank at a
frequency f and a second mass m2 supported by a spring coupled to
ml vibrating with a frequency tf.
The cross-sectional view of the
cone shows a similar system and illustrates how subharmonics may be
produced by a loudspeaker.
Fig. 6.63 illustrates the mechanism of one type of subharmonic. The
driven mass ml at the end of the bar vibrates at a frequency j while the mass
m2 vibrates at a frequency tf.
In the same way a cone, Fig. 6.63, will
vibrate at a sub harmonic frequency. The existence of subharmonics in
direct radiator loudspeakers is well known. However, in horn loudspeakers
the diaphragms are relatively small and quite rigid. Consequently the
conditions for the production of subharmonics are not particularly favourable.
Circular corrugations in the diaphragm or cone may be used to increase
the stiffness and thereby reduce the tendency to break into subharmonic
vibrations.
188
ACOUSTICAL ENGINEERING
D. Response Frequency Characteristics of a Direct Radiator Loudspeaker
With a Nonlinear Suspension System.-The velocity frequency charac­
teristic of a loudspeaker with a nonlinear suspension system may be obtained
from the equation 6.29. A theoretical response frequency characteristic
is shown in Fig. 6.64.
Suppose that a constant current is applied to the voice coil of the loud­
speaker and at a low-frequency point A of Fig. 6.64. Then as the frequency
is increased, the velocity increases steadily to the point C. At this point
the velocity drops suddenly, in a jump, to point E. From point E on, the
velocity steadily decreases. Now start at
8
F and decrease the frequency. The
velocity steadily increases to the point D.
7
At point D the velocity suddenly jumps
6
to the point B. From point B on, the
velocity steadily decreases.
5
Irc
Typical experimental velocity frequency
>
tI
characteristics are shown in Fig. 6.65.
U4
e'l
9w
The velocity frequency characteristic for
>3
an increase in frequency is shown in
Fig. 6.65. The velocity frequency charac­
2
teristic for a decrease in frequency, is also
l~
/
shown in Fig. 6.65. These characteristics
I
~--.!.
are quite similar to the theoretical charac­
~
200
teristic of Fig. 6.64.
60
100
30
fREQUENCY
E. Distortion Due to Inhomogeneity of
the Air-Gap Flux.-Inhomogeneity of the
FIG. 6.64. Theoretical response
frequency characteristic of a direct
flux density through which the voice coil
radiator loudspeaker mechanism
moves is another source of distortion.
with a nonlinear suspension system.
The
result is that the driving force does
The unstable portion of the response
not correspond to the voltage developed
frequency characteristic is indicated
by a dashed line.
by the generator in the electrical driving
system. Furthermore, the motional elec­
trical impedance is a function of the amplitude. This type of distortion is
similar to that due to a nonlinear suspension system. The wave-shape dis­
tortion is similar to that of Fig. 6.61.
The force, in dynes, developed by the interaction of the current in the
voice coil and the magnetic field is
jfr
""
f=
where B
=
Bli
6.30
flux density, in gausses,
1 = length of the voice coil conductor, in centimeters, and
i
=
current, in abamperes.
Equation 6.30 shows that the force is directly proportional to the current
if Bl is a constant. If the Bl product varies with the position of the voice
189
DIRECT RADIATOR LOUDSPEAKERS
8
8
7
7
6
6
5
1/
~
t­
U4
g
>­
U4
o
-'
w
> 3
II
w
>3
/
j
2
V
""
30
I
j
2
/
I
5
t­
I
'--­
60
V
100
200
fREQUENCY
~
~o
V
V
60
\
r----_
100
200
fREQUENCY
00
~
6.65. Experimental response frequency characteristic of a direct radiator
loudspeaker mechanism with a nonlinear suspension system for an increasing and
a decreasing applied frequency.
FIG.
coil, the force will not be proportional to the current and distortion will
be produced.
A typical flux distribution in an air gap is shown in Fig. 6.66. A con­
sideration of the flux distribution shows that the Bl product will be practi­
cally a constant if the voice coil is made longer than the gap, as shown in
Fig. 6.66B, because, as the coil moves into the weaker tufting field on one
end, it moves into a stronger field on the other end. From the standpoint of
efficiency at the higher frequencies, this method is not particularly desirable
11)20
'"
'""
3
16
If
'" 12
>- 8
...
~
4
L
V
o
x 0
w
3 -;4
"­
~
~
::J
/
\
\
AIR GAP
AXIAL
LENGTH
I
,2
AXIAL
0
LENGTH
\
\
.2
IN
,.
INCHES
FIG. 6.66.
Graph of the flux distribution in an air gap. A. Typical field map
of the flux lines in an air gap. B. A voice coil longer than the air gap. C. A
voice coil shorter than the air gap.
190
ACOUSTICAL ENGINEERING
because part of the voice coil is in a weak field. This type of distortion
can also be eliminated by making the air gap of sufficient axial length so that
the voice coil remains at all times in a uniform field as shown in Fig. 6.66C.
The latter method is used for high-frequency loudspeakers of high efficiency.
F. Frequency-Modulation Distortion. 45 -The amplitude of the cone in a
direct radiator loudspeaker for constant sound output, in the frequency
range below the ultimate radiation resistance, is inversely proportional to
the square of the frequency. If the cone is radiating both at high- and low­
frequency, the source of high-frequency energy may be considered to be
moving back and forth at the low frequency. The high-frequency energy
will be modulated. The resulting frequency-modulated wave may be
represented by a carrier and a double infinity of sidebands.
The square root of the ratio of the power in the sidebands to the total
power in the sound wave, in per cent, is
D = 2900 kVtJ;.
6.31
/1 2d2
where h
=
II =
PI =
d=
A
modulated frequency, in cycles,
modulating frequency, in cycles,
acoustical output at /1, in watts, and
cone diameter, in inches.
IS"CONE
B
100 CYCLES
.... 10
....
..,z
..,..
u 8
Q.
z
o
6
IWAT~V
4
j:
.. 2
o
>­
-
/
4"CONE
1000 CYCLES
10r---~--,----r,---~
w
. 8r----+----+----++----I
u
w
Q.
6r----+----+----++---:Il
~
Y ~~
z 4f---t-----t----t--h'<---j
o
j:
~ 2r----+-----r-~~:
I--"V~ ~I--" ....
VI
5 ~02
/
z
4
a 10'
~ o~-==:F-~~"'*::t:;:::::::=-!
103
2
• 104 0
FREQUENCY
FIG. 6.67. A. Frequency modulation distortion characteristics of a 15-inch
diameter cone with outputs of 1. 0.1. and 0.01 acoustical watts at 100 cycles for
a second applied frequency over the range 100 to 10.000 cycles. B. Fre­
quency modulation distortion characteristics of a 4-iuch diameter cone with
outputs of 1, 0.1, and 0.01 acoustical watts at 1000 cycles for a second applied
frequency over the range of 1000 to 10.000 cycles.
Frequency modulation distortion characteristics for a cone 15 inches
in diameter and a cone 4 inches in diameter for acoustical outputs of 1,
0.1, and 0.01 watts are shown in Fig. 6.67.
G. Air Nonlinear Distortion.-In general, the distortion generated in
the air between the cone of a direct radiator loudspeaker and the listener
is considered to be negligible. I t is very much less than this type of dis­
tortion in a horn type loudspeaker. However, if small distortions are of
45
Beers and Belar, Proc. IRE, Vol. 31. No.4, p. 132. 1943.
DIRECT RADIATOR LOUDSPEAKERS
191
consequence, then some consideration must be given to the distortion
generated in the air between the cone and the air of the listener.
The ratio of the second harmonic pressure to the fundamental pressure,
at a distance r centimeters from a spherical radiator of radius rl centimeters,
is
P2r = {y + l)wPlrr loge':'
6.32
Plr
2V2ypoc
rl
where y
Plr
=
P2r
=
=
Po =
w =
ratio of specific beats (1.4 for air),
fundamental sound pressure at a distance r, in dynes per
square centimeter.
second harmonic sound pressure at a distance r, in dynes per
square centimeter.
atmospheric pressure, in dynes per square centimeter,
27Tj,
j = frequency, in cycles per second,
c = velocity of sound, in centimeters per second.
Equation 6.32 applies to any diverging wave system in which the sound
pressure varies inversely as the distance. It may be mentioned in passing
that the pressure in a sound wave in free space varies inversely as the
distance.
The second harmonic pressure,46 in dynes per square centimeter, gen­
erated in a distance x, in centimeters, in a plane wave is
{y + l)w Pl2x
6.33
2 V2ypoc
fundamental pressure, in dynes per square centimeter, and
the other quantities are the same as equation 6.32.
P2
where PI
=
=
Equation 6.33 applies to"a plane wave, as, for example, a sound wave in
a pipe.
In the case of a direct radiator loudspeaker the wave is diverging. At
a distance equal to the radius of the cone the system can, from the stand­
point of distortion, be replaced by a spherical radiator equal to the radius
of the cone. The distortion generated in the volume between the cone
and the spherical surface may be determined by approximations by em­
ploying equation 6.33. The complete expression for the second harmonic
distortion, in per cent, generated between a cone of radius rl, in centimeters,
and an observation point at a distance r, in centimeters, from the front of
the baffle is
D
where Plr
46
=
P2r 100
+
+
{y
l)wPl [.85r
r loge ':.-] 100
6.34
2V2ypoc
rl
pressure at the observation point, in dynes per square centi­
meter.
=
Plr
=
Thuras, Jenkins, and O'Neil, Jour. Acous. Soc., Amer. Vol. 6, No.3, p. 173, 1935.
192
ACOUSTICAL ENGINEERING
The distortion frequency characteristics, for a distance of 3 meters (about
10 feet) and various pressures at the observation point, for direct radiator
loudspeakers with cones having diameters of 2 inches and 8 inches, are
shown in Fig. 6.68.
A
0
B
8
4
...z
2
U
I
.8
w
.
..,It
"
/
/ /
/
/ V /.1
,O~VV V.I
/ /
~///
B .2
i=
a:
o .I
:;; .06
a
.04
~~
~~~q,/
'//7,,~
1/ /
L/
,L// / /I
.02
.0
,0bQ~/ /
,0 QQ;
4
8 103
2
I/~~/ V
V V V ~~ ~Q; V
~Q"
V / /~V P
V/ V V
8 104 10 2
4
,REQUENCY
r-­
IN
CYCLES
2
PER
8 103
SECOND
FIG. 6.68.
Distortion frequency characteristics depicting the distortion
generated in the air between the cone and the observer in a direct radiator
loudspeaker. A. Loudspeaker with a cone 8 inches in diameter. B.
Loudspeaker with a cone 2 inches in diameter. Labels on characteristics
indicate sound levels at observation point. 0 db = .0002 dyne per square
centimeter. Distance 3 meters.
6.27. Diaphragms, Suspensions, and Voice Coils.-The diaphragm or
cone of practically all direct radiator loudspeakers is made of paper. Typical
cones shown in Fig. 6.69, are made by a felting process employing a master
screen having the shape of the diaphragm. The mixture of pulp and water
is drawn through the screen leaving a thin deposit of compressed pulp.
When this deposit is dried it can be removed from the screen and the result
is the finished diaphragm. The outside suspension system can also be
felted as part of the cone.
There are two types of felted diaphragms in general use-namely, the
circular and the elliptical cone shown in Fig. 6.69. In certain cabinets it
is possible to obtain a larger diaphragm area by employing the elliptical
cone. The directional pattern of the elliptical cone is sharper in the plane
containing the major axis of the ellipse and axis of the cone and is broader
in the plane containing the minor axis of the ellipse and axis of the cone
than the circular cone with the same area.
There are three types of cross sections in general use in felted cones­
namely: the conical shape, Fig. 6.70A; the flared shape, Fig. 6.70B; the
193
DIRECT RADIATOR LOUDSPEAKERS
corrugated conical shape, Fig. 6.70C. The shapes of Fig. 6.70 may be
employed in either circular or elliptical cones.
The flared shape is somewhat more rigid than the conical shape. For
this reason, the directional pattern in the high-frequency range is very much
sharper. The use of corrugations increases the radial rigidity and slows
o
A'
FRONT VIEW
FRONT VIEW
~ ~ ~......-------~
SECTIONAL VIEW
SECTION
A-A'
A
SECTION
B -
s'
B
FIG. 6.69. Felted cones for direct radiator, dynamic loudspeakers.
B. Elliptical cone.
A
B
A. Circular cone.
c
FIG. 6.70. Sectional views of felted cones for direct radiator, dynamic loud­
speakers. A. Conical shape. B. Flared Shape. C. Conical shape with corruga­
tions.
propagation of the wave in the cone and thereby broadens the directional
pattern.
The three types of suspension systems, shown in Fig. 6.71, are in general
use. The leather or kidskin suspension system shown in Fig. 6.71A has
gradually gone out of use. It has been displaced by the one-piece felted
cone and suspension system shown in Fig. 6.71B. The latter system is
much simpler and less costly in manufacture. One of the principal dis­
advantages of the felted suspension system is the nonlinear characteristics
which introduce distortion (see Sec. 6.16). The stiffness of the suspension
system may be decreased and the distortion reduced by means of a folded
or double suspension system as shown in Fig. 7.61C. The reduction in
194
ACOUSTICAL ENGINEERING
A
B
c
FIG. 6.71. Sectional views of cone suspension systems for direct radiator loudspeakers.
A. Leather suspension. B. Felted corrugated suspension. C. Folded or double-felted
corrugated suspension system.
FIG. 6.72. A. Outer suspension system for a cone of a
direct radiator loudspeaker. B. Normal vibration of the
suspension. C. Resonant vibration of the suspension in
phase with the cone. D. Resonant vibration of the sus­
pension out of phase with the cone.
stiffness makes it possible to obtain a low fundamental resonant frequency
in small light cones and thereby extend the low-frequency range (see Sec.
6.2).
An enlarged view of the corrugated outside suspension system of the type
of Fig. 6.71B is shown in Fig. 6.72A.
The normal mode of vibration of the cone and suspension is shown in
Fig. 6.72B. The maximum excursions of each part of the suspension and
DIRECT RADIATOR LOUDSPEAKERS
195
cone are shown by the dotted lines. It will be seen that the amplitude falls
off gradually in the suspension from the edge of the cone to the fixed outside
edge. Unfortunately, a limp suspension employed to obtain a low resonant
frequency does not behave in this manner throughout the frequency range,
but breaks into resonance in the mid-frequency range. The amplitude of the
suspension may be greater than that of the cone, as shown in Fig. 6.72C.
The vibration of the suspension is in phase with the cone, as shown in
Fig. 6.72C, and out-of-phase, as shown in Fig. 6.72D.
In the past, the procedure has been to coat the suspension with some
highly viscous material, thereby providing damping which reduces the
amplitude at resonance. In this way, the response is smooth and free from
the peak and dip. The objection to the use of viscous materials is that
these materials tend to dry out, with the result that the damping efficiency
is reduced. The vibration 47 ,48 can be controlled and the effects of any
resonance reduced to a negligible amount by means of a special rubber
damping ring, as shown in Fig. 6.73. The curve of Fig. 6.74A taken without
FIG. 6.73.
The output suspension system of the cone of a
direct radiator loudspeaker equipped with a foam rubber
damper. (After Olson, Preston, and May.)
the damping ring can be compared with that of Fig. 6.74B taken with the
damping ring. It will be seen that the response frequency characteristic
without the damping ring exhibits a peak and a dip at 800 and 1100 cycles,
respectively.
In the direct radiator loudspeaker the sound vibrations start at the voice
coil, flow out in the cone, and then into the suspension system. In the low­
frequency range the phase shift, in degrees, along the cone is relatively small
and the cone behaves essentially as a piston. However, in the high-fre­
quency range the phase difference between the voice and suspension may
be several radians. In this frequency range it is important that the wave
that travels into the suspension system is absorbed and not reflected back.
The latter condition would lead to standing waves which would produce a
ragged response frequency characteristic. The sponge-rubber damping ring
serves as a suitable acoustical termination, thereby absorbing the vibrations
which flow into the suspension system. The response frequency charac­
teristics without and with the damping ring are shown in Figs. 6.7SA and
6.7SB, respectively. It will be seen that response is smoother with the
damping ring.
47
48
Olson, H. F., Radio and Television News, Vol. 51. No.2, p. 69,1954.
Olson, Preston, and May, Jour. Aud. Eng. Soc., Vol. 2, No.4, p. 219, 1954.
196
ACOUSTICAL ENGINEERING
30
25
,
-
AI'
I
rJl20
0
..
~
\
B
/
~
!
III
III 15
2
0
I
-\ /1 ~
~
-
j
I
11
\j
D-
...
11:10
c/)
y"
5
o
400
600
800 1000
2000
3000
FREQUENCY IN CYCLES PER SECONO
FIG. 6.74.
Mid-frequency response frequency
characteristics of direct radiator loudspeakers.
A. A conventional suspension system. B. A
suspension equipped with a foam rubber damper.
30
25
L.
~
~'rr<7
'A
-.
'_F~
~~
"
5
o
2000
3000 4000
6000 800010000 15000
FREQUENCY IN CYCLES PER SE.COND
FIG. 6.75. High-frequency response frequency
characteristics of direct radiator loudspeakers.
A. A conventional suspension system. B. A
suspension equipped with a foam rubber damper.
Centering suspensions for keeping the voice coil aligned in the air gap
are shown in Fig. 6.76. An inside slotted centering suspension, usually
made of fiber, is shown in Fig. 6.76A. An inside felted paper centering
197
DIRECT RADIATOR LOUDSPEAKERS
suspension with corrugations is shown in Fig. 6.76B. This type of suspen­
sion is usually employed where the amplitude is small. An outside felted
paper suspension with corrugations is shown in Fig. 6.76C. The outside
INSIDE
SLOTTED
INSIDE
CORRUGATED
CENTERING
01 SK
OUTSIDE
CORRUGATED
CENTERING
DISK
DUST
CAP
::rr~e~~~~~'~
COIL
COIL
FIG. 6.76. Sectional views of voice coil centering systems for direct radiator dynamic
loudspeakers. A. Inside slotted fiber disk. B. Inside felted corrugated centering disk.
C. Outside felted corrugated centering disk.
centering suspension can be made large in diameter and thereby obtain a
very low value of stiffness.
Voice coil construction used in direct radiator loudspeakers are shown
in Fig. 6.77. A voice coil wound on a cylindrical paper form with round wire
is shown in Fig. 6.77A. Cement is used to bind the voice coil to the form.
The cement also serves to bind the adjacent turns of wire together. Three
types of self-supporting voice coils are shown in Fig. 6.77A, C, and D.
B
c
D
6.77. Sectional views of voice coil constructions. A. Round enameled wire
wound on a paper form. B, C, and D. Self-supporting voice coils held together with
thermosetting cement. B. Round wire. C. Square wire. D. Edgewise wound ribbon.
A
FIG.
Thermosetting cement is used to bind the entire assembly. The use of a
self-supporting coil eliminates the cylindrical paper form and thereby re­
duces the space required in the air gap. The use of square wire or ribbon
effects a further reduction in the space required in the air gap.
6.28. High-Frequency Sound Distributor.-The diameter of the
vibrating surface of multiple cones decreases with increase in frequency and
as a result the directional pattern is essentially independent of the frequency.
When a single uncorrugated cone is used to cover the high-frequency range
the directional pattern becomes quite narrow at the higher frequencies.
198
ACOUSTICAL ENGINEERING
By means of a distributor consisting of vanes it is possible to spread the
high-frequency radiation and thereby maintain uniform directional charac­
teristics with respect to the frequency. The high-frequency contours of
equal phase for a cone with and without a distributor are shown in Figs.
6.78A and 6.78B. The radius of curvature of the wavefront with the dis­
tribut~;;;;,:han
SECTIONAL
A
VIEW
that of the Plan.:,',:ne
HORIZONTAL
~~t
the
SECTION
B
VERTICAL
SECTION
FIG. 6.78. High-frequency sound distributors for direct radiator loudspeakers.
A. The contour of equal phase for a plain cone. B. The horizontal cross­
sectional view of a cone with a vane distributor and the contour of equal phase.
c. The vertical cross-sectional view of a cone with a vane distributor.
distributor broadens the radiation pattern. The vertical section, Fig.
6.78C, shows that the distributor will not broaden the pattern in this direc­
tion. In general, in radio or phonograph reproduction, the required vertical
plane of spread is quite small. If a broader pattern is required in this plane
crossed vanes may be used.
6.29. Field Structures.-Six typical electromagnetic and permanent
magnet field structures used in direct radiator dynamic loudspeakers are
shown in Fig. 6.79. For many years the loudspeakers employing the
field structures shown in A and B of Fig. 6.79 were universally used in a-c
powered radio receivers and phonographs. The field coil was used as a
choke in the filter system of the high voltage power supply. The advent
of the new permanent magnet alloys consisting of combinations of aluminum,
nickel, cobalt, and iron and termed "Alnico" made it possible to design
efficient magnetic structures of reasonable size. The air-gap flux densities
obtained with field coil excitation could be duplicated with permanent
magnets. Another important factor is the lower cost of the permanent
magnet loudspeaker as compared to the field coil excitation even when the
power supply is considered as part of the problem. As a consequence,
the magnetic structures shown in C, D, E, and F of Fig. 6.79 are universally
used in dynamic direct radiator loudspeakers today.
The design of field structures involves both empirical and theoretical
considerations because the leakage flux cannot be predicted without some
experimental data. It is beyond the scope of this book to give a compre­
hensive discussion on the design of field structures. However, since field
199
DIRECT RADIATOR LOUDSPEAKERS
--~
A
'.~..""
SECTION A-~
t1.
·"w
SECTION A-A'
SECTION A-A'
$
1]]
A
I
-
1
I
Ii
SECTION A- A'
FIG. 6.79. Field structures. A and B, electromagnetic types.
C, D, E, and F, permanent magnet types.
structures are used in loudspeakers, microphones, and other transducers it
seems worthwhile to outline the fundamentals of magnetic circuits.
A few of the terms used in magnet systems will be defined.
Magnetic Flux-Magnetic flux is the physical manifestation of a condi­
tion existing in a medium or material subjected to a magnetizing influence.
The quantity is characterized by the fact that an electromotive force is
induced in a conductor surrounding the flux during any time that the flux
changes in magnitude. In the cgs system the unit of magnetic flux is the
maxwell.
Magnetomotive Force-Magnetomotive force in a magnetic circuit is
the work required to carry a unit magnetic pole around the circuit against
the magnetic field. In the cgs system, the unit of magnetomotive force
is the gilbert.
Reluctance-Reluctance is the property of the magnetic circuit to resist
magnetization. Thus the amount of magnetic flux resulting from a given
magneto motive force acting on a magnetic circuit is determined by the
magnetic reluctance of the circuit.
Maxwell-The maxwell is the cgs unit of magnetic flux. It is the flux
produced by a magnetomotive force of 1 gilbert in a magnetic circuit of
unit reluctance.
200
ACOUSTICAL ENGINEERING
Line-Line is a term commonly used interchangeably for a maxwell.
Gilbert-The gilbert is the cgs unit of magnetomotive force. It is the
magnetomotive force required to produce 1 maxwell of magnetic flux in a
magnetic circuit of unit reluctance.
Oersted-The oersted is the unit of field strength in the cgs system. It
is the magnetomotive force equivalent to 1 gilbert per centimeter of length.
Gauss-The gauss is the unit of flux density. One gauss equals 1 maxwell
per square centimeter.
Flux-Flux is the term applied to the physical manifestation of the
presence of magnetic induction.
Flux Density-Flux density is the number of lines or maxwells per
unit area in a section normal to the direction of the flux. In the cgs system
the unit is the gauss.
Ampere-Turn-Ampere-turn is the unit of magnetomotive force. It is
a product of the number of turns on a coil and the amperes passing through
the turns.
Magnetizing Force-Magnetizing force is the magnetomotive force per
unit length at any given point in a magnetic circuit. In the cgs system
the unit of magnetizing force is the oersted.
Leakage-Leakage is that portion of the magnetic field that is not useful.
Leakage Coefficient-Leakage coefficient is the ratio of the total flux
produced to the useful flux.
Induction, Intrinsic-Also known as ferric induction. Intrinsic induc­
tion is that portion of the induction in excess of the induction in a vacuum
for the same magnetizing force.
Induction, Magnetic-Magnetic induction is the magnetic flux per unit
area of a section normal to the direction of flux, resulting when a substance
is subjected to a magnetic field. This is also known as magnetic flux density.
In the cgs system the unit of magnetic flux density is the gauss.
Coercive Force-Coercive force is the magnetomotive force which must
be applied to a magnetic material in a direction opposite to the residual
induction to reduce the latter to zero.
Demagnetization-Demagnetization is the reduction of magnetization.
It may be either partial or complete.
Demagnetization Curve-The demagnetization curve is that portion of
the normal hysteresis loop in the second quadrant showing the induction
in a magnetic material as related to the magnetizing force applied in a
direction opposite to the residual induction.
Permeability-Permeability is the ratio of the magnetic induction in
a given medium to the induction which would be produced in a vacuum by
the action of the same magnetizing force.
The subject of analogies and the importance in the solution of problems
in vibrating systems have been discussed in Chapter IV. Analogies between
electrical and magnetic quantities may be used to solve problems in magnetic
systems by reducing the system to an electrical network. The performance
of the network may be determined by electrical circuit theory.
201
DIRECT RADIATOR LOUDSPEAKERS
The fundamental equation of magnetic circuits is given by
c/>=M
6.35
R
where c/>
M
R
=
=
=
total lines of flux, in maxwells,
magnetomotive force, in gilberts, and
reluctance, no unit.
Equation 6.35 is analogous to Ohm's Law in electrical circuits expressed
as follows:
.
2
e
6.36
=­
YE
where i = current, in abamperes,
e = electromotive force, in abvolts, and
YE = electrical resistance, in abohms.
The quantities, units, and symbols in electrical and magnetic systems are
shown in Table 6.2.
TABLE
6.2
Electrical
Magnetic
Symbol
Quantity
108
e
Magnetomotive
Force
Gilberts
M
109
YE
Reluctance
No Unit
R
Flux
Maxwells
Unit
Quantity
Electromotive
Force
Volts
Electrical
Resistance
Ohms
Current
Resistance
Amperes
X
X
X
10- 1
i
Unit
Symbol
'"
The analogies between electrical and magnetic systems will be used to
solve problems in magnetic systems in this book. The first illustration will
be the solution of two simple magnetic systems, shown in Fig. 6.80.
The magnetomotive force, in gilberts, in an electromagnetic system, as
shown in Fig. 6.80A is given by
MT = 4rrni
6.37
where n = number turns, and
i = current, in abamperes.
The magnetizing force, in oersteds, developed by the permanent magnet
in the permanent magnet system of Fig. 6.80B may be obtained from the
demagnetization curves of Fig. 6.81. In order to use the minimum amount
of material it is necessary to operate at the maximum value of B X H.
For Alnico V this is 470 oersteds per centimeter of length.
202
ACOUSTICAL ENGINEERING
The total lines in the system at the coil or magnet, in maxwells, is
1>T = 1>1 + 1>2
where
6.38
1>1 = lines in the air gap, in maxwells, and
1>2 = lines in the leakage field, in maxwells.
The total magnetomotive force, in gilberts, developed by the energized
coil or permanent magnet is
6.39
ffi
R:
RL
0.
T
T
L-=-.
~
CPe
<p.
MT
22
T..--L
MAGNETIC NETWORK
COIL
l\ TURNS
PERMANENT
MAGNET
SCHEMATIC VIEW
SCHEMATIC VIEW
6.80. Schematic views and magnetic networks of electromagnetic and
permanent field structures. M 1 = magnetomotive force drop in the air gap.
Ma = magnetomotive force drop in the air gap. M!J' = magnetomotive force
developed by the current i in the coil or by the permanent magnet. 4>1 =
total flux lines in the air gap. 4>2 = total leakage flux lines. 4>!J' = total lines
delivered by the coil or permanent magnet. R1 = reluctance of the air gap.
RL = reluctance of the leakage paths. R2 = reluctance of the iron path.
FIG.
The number of lines in the air gap, in maxwells, is
1>1
where Rl
=
=
M1
Rl
reluctance of the air gap, and
M1 = magnetomotive force across the air gap in gilberts.
6.40
203
DIRECT RADIATOR LOUDSPEAKERS
The reluctance of the air gap is
1
RI = ­
6.41
Al
where 1 = length of the air gap in the direction of the flux in centimeters,
and
Al = cross-sectional area of the air gap, in square centimeters.
14
12
~~
III
III
~IO
( /Y
"9
>:: 6
z
I
6
Q
r
u
J
04
Z
/'
2
o
600
--....::
~
V
/
1/
I
/; ~
Iii
0/ ~\
~
l);;?
700
600
500
400
DEMAGNETIZING fORCE
-~
;....­
~ t-­
-Irt- / ~~n J
I
300
200
IN
OERSTEDS
~~
100
~
~
~
'" D
1)/
V
V
V
V
o
2
3
EXTERNAL ENERGY
4
B x H xlO-"
FIG.6.81. Demagnetization and energy characteristics of permanent magnet materials.
I. Alnico I; II. Alnico II; III. Alnico III; IV. Alnico IV; V. Alnico V ; VB. Alnico VB;
VI. 36 per cent cobalt.
The reluctance in the iron structure
1
R2 =­
6.42
fA-A2
where 1 = length of the path in the iron. in centimeters,
A2 = cross-sectional area, in square centimeters, and
fA- = permeability.
The curves 49 of Fig. 6.82 show the relation between the intrinsic flux
density and the magnetizing force for different magnetic materials. The
permeability may be obtained from the curves of Fig. 6.82 and the following
relation
B
fA-= -
H
where B
=
H
=
49
flux density, in gausses, and
magnetizing force in oersteds.
Kentner, A. E., Gen. Etec. Rev. XL V, No. 11, p. 633,1942.
6.43
204
ACOUSTICAL ENGINEERING
The magnetomotive force drop in the iron is given by
6.44
M2 = R2 CP2
where R2 = reluctance in the iron structure, and
CP2 = lines in the iron, in maxwells.
The magnetomotive force drop in the iron should be made small com­
pared to the magneto motive force drop in the air gap.
en
w
z
V
!3=B-H
18
j
.J
9 16
)0
.,,- V
10
tZ 8
/
x 6
::J
.J
... 4
Vi
z
2
it
t~ 0
.01
~
r-
~ Yy
~ V.. . . /
4
8.1
~f
/ V1 r/
/
iii
w
a
II
k:::
/ J0
<ll.. 12
~
/
sf
/ ~
%~
7
/
'0/
V
• j /
..? I-­ V
V
810
FORCE
H
I II
I
"j
.-
­
V
•
V
V
8,
.,,­
V
IL
iI' r-v
'Ji
I'
MAGNETIZING
cY
~
///
q: rI11"
2
/'/ ~
f-- ~ ::::::
VV
"zl4
Y
.-!-­
V
';I
V
4
81022
IN
OERSTEDS
I
I
4
81032
6.82. D-c magnetization characteristics for various magnetic materials.
1. Permandur, 50 per cent cobalt and 50 per cent iron. 2. Electrolytic iron.
3. Armco iron. 4. Annealed cold drawn steel. 5. Medium hard silicon steel.
6. Nickel iron alloy, 47 per cent nickel. 7. Permalloy, 79 per cent nickel.
8. Allegheny, mumetal. 9. Pure nickel. 10. Cast iron. 11. Cobalt perman­
ent magnet steel. 12. Alnico permanent magnet alloy. (After Kentner.)
FIG.
The air-gap flux density, in gausses, is given by
B = <Pl = Ml
Al
l
6.45
By means of the above equations and the leakage flux it is possible to
design the field structure. The starting point is usually the desired air­
gap flux density. The air-gap flux, <Pl, and the magnetomotive force, M l ,
required to produce this flux may be obtained from equation 6.45 and the
flux density, B. If the leakage lines, <p2, are known then the total lines,
<PT, are given by equation 6.38. The magnetomotivc drop, M 2, in the iron
can be obtained from the number of lines, CP2, in the iron and the reluctance,
R2, as shown in equation 6.44. Then the total magnetomotive, M T, can
DIRECT RADIATOR LOUDSPEAKERS
205
be determined from equation 6.39. The number of ampere turns required
to produce this magneto motive is obtained from equation 6.36. In the
case of a permanent magnet, the length of magnet which delivers the re­
quired magnetomotive force is
I=M T
H
where
6.46
I = length, in centimeters,
M T = total magnetomotive force required, in gilberts, and
H = demagnetizing force, in oersteds (see Fig. 6.81).
For Alnico V, the demagnetizing force is 470 oersteds per centimeter of
length, when the flux density magnet is 9500 gausses. The required cross­
sectional area of the permanent magnet is the total flux cPT divided by 9500.
For relatively long magnets or large air gaps, the leakage flux of the magnet
must also be considered in obtaining the cross-sectional area. This con­
sideration results in a larger cross section at the center of the magnet.
Measurements of the air-gap flux density, the leakage flux and the flux
density in various parts of the magnetic circuit can be made by means of
a calibrated flux meter and a loop or coil. The air-gap flux density can be
obtained by means of a calibrated search coil and fluxmeter. The flux
in any part of the magnetic circuit can be obtained by placing a loop of
one or more turns around the section to be tested. The loop is connected
to the fluxmeter. This coil is then pulled out to a point where there is no
flux. The flux can be obtained from the deflection and calibration of the
fluxmeter and the number of turns in the coil. From these measurements,
data can be obtained which will give the total flux, the leakage flux, the
air-gap flux, and the flux density in the iron and permanent magnet struc­
ture. This data together with the equations and data in the preceding
considerations will indicate the direction of improvement from the stand­
point of air-gap flux density, leakage flux, and optimum cross section of the
iron and permanent magnet.
6.30. Electrostatic Loudspeaker.50,51,52~All of the considerations in
this chapter have been concerned with the dynamic direct radiator loud­
speaker. During the early stages of sound reproduction, magnetic and
electrostatic loudspeaker mechanisms were also employed. However,
during the past 25 years the dynamic direct radiator loudspeaker has been
universally employed for the reproduction of sound. Recently, some atten­
tion has again been given to electrostatic loudspeakers. The advent of
thin plastic sheets with excellent electrical properties has revived interest
in electrostatic loudspeakers. In view of these developments it seems
logical to analyze the action and describe some of the new developments in
50 Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J.,
1943.
51 Hanna, C. R., Jour. Acous. Soc. Amer., Vol. 2, No.2, p. 143, 1930.
,2 Hunt, "Electroacoustics," John Wiley and Sons, New York, N.Y., 1954.
206
ACOUSTICAL ENGINEERING
electrostatic loudspeakers. Consider the system of Fig. 6.83, consisting of
a vibrating surface moving normal to its plane and separated from a fixed
conductor. The force, in dynes, between the plates is
1M =
e2A
81Ta2
6.47
where e = electromotive force between plates, in statvolts,
a = normal distance between the plates, in centimeters, and
A = area of the plate, in square centimeters.
ELECTRICAL NETWORK
Zltl2
STATIONARY
PLATE
DIAPHRAGM
SCHEMATIC VIEW
MECHANICAL CIRCUIT
FIG. 6.83. Schematic view, electrical network, and
mechanical circuit of an electrostatic loudspeaker. In the
electrical network, ZEN = the normal electrical capacitance
of the loudspeaker. ZEM = the motional electrical im­
pedance of the loudspeaker. ZEl = the damped electrical
impedance of the loudspeaker. ZEl = l/jwC E l' CEl =
the damped electrical capacitance of the loudspeaker. In
the mechanical circuit, JM = driving force. ZMl = the
mechanical impedance of the movable plate. ZM2 = the
mechanical impedance of the load. (After Olson, "Dyna­
mical Analogies", D. Van Nostrand Company, Princeton,
N.J., 1942.)
Assume that the polarizing voltage is eo and that the alternating voltage
is e = emaxsin wt. The force, in dynes, between the plates is
jM
=
(eo2
j M = eo 2
+ emax sin wt).2 A
81Ta 2
+ 2eoemax sin wt + 1/2
e2 max 81Ta2
6.48
1/2 e2max cos 2wt A
6.49
The first and third terms in the numerator of equation 6.49 represent
steady forces. The fourth term is an alternating force of twice the frequency
of the impressed voltage. The second term is an alternating force of the
frequency of the impressed voltage. If the polarizing electromotive eo is
large compared to the alternating electromotive force emaxsin wt, the fourth
term will be negligible. The useful force, in dynes, then is the second term
DIRECT RADIATOR LOUDSPEAKERS
207
which causes the moving surface to vibrate with a velocity which cor­
responds to the impressed electromotive force.
fM
J,
= eoemax sin wt A = eoe
4n-a2
4n-a2
The electrical motion impedance of this system will now be considered.
charge, in statcoulombs, on the condenser is
q = CEeo
where eo
CE
=
=
6.50
The
6.51
potential difference between the plates, in statvolts, and
capacity per unit area, in statfarads.
The current, in statamperes, generated due to motion is
.
1
dq
6.52
dt
=
From equations 6.51 and 6.52 the generated current is
.
1
dCE dx
dx di
6.53
= eo
The electrical capacitance of the condenser, in stratfarads, is
A
4n-a
6.54
CE1=-
Let the movable plate be deflected a distance Llx away from the fixed plate.
The electrical capacitance is
A
6.55
CEl - LlCEl
4n-(a
Llx)
+
Now let the movable plate be deflected a distance Llx toward the fixed plate.
The electrical capacitance is
CEl
+ LlCEl =
A
4n-(a _ Llx)
6.56
The difference between the two conditions is
LlC El
=
ALlx
4n-a2 _ (LlX)2
ALlx
47Ta2
6.57
The change in electrical capacitance with respect to x is
dCEl
dt
A
4n-a 2
6.58
Substituting equation 6.58 in 6.53, the generated current, in statamperes, is
.
1 =
eoA.
4n-a2x
6.59
208
ACOUSTICAL ENGINEERING
From the mechanical circuit of Fig. 6.83, the mechanical rectilineal im­
pedance of the vibrating system is
+
where
ZM =
ZMl
=
ZM2 =
ZM = ZMl
ZM2
6.60
total mechanical rectilineal impedance of the vibrating system,
in mechanical ohms,
mechanical rectilineal impedance of the vibrating plate, in
mechanical ohms, and
mechanical rectilineal impedance of the load, in mechanical
ohms.
The mechanical rectilineal impedance at the plate is
1M
-;­
6.61
1617 2«41M
eo 2A2 i
6.62
ZM =
X
From equations 6.50 and 6.59
e
2
From equations 6.60, 6.61, and 6.62
ZEM =
where
ZEM
ZM
=
=
16172«4
eo 2A2
ZEN,
ZEN =
in statohms, of the condenseris
ZEIZEM
---,-­
ZEI
ZEI =
ZEM =
6.63
motional electrical impedance, in statohms, and
total mechanical rectilineal impedance presented to the
vibrating surface including the vibrating surface.
The normal electrical impedance
where
ZM
+ ZEM
6.64
damped electrical impedance of the condenser, in statohms, and
motional electrical impedance of the condenser, in statohms.
The motional electrical impedance as given by equation 6.64 may be
represented as in parallel with the blocked or damped electrical impedance
of the condenser as depicted by the electrical network in Fig. 6.83.
The preceding considerations show that the efficiency is proportional to
th~ SWIare of the area of the plates, proportional to the square of the polariz­
ing voltage, :'Dyprsely proportional to the fourth power of the spacing between
the plates, and inversely proportional to the mechanical impedance of the
vibrating system. Thus, it will be seen that in order to obtain high effi­
ciency the spacing must be small because the relationship between spacing
and efficiency is a fourth-power function. This, of course, limits the ampli­
tude. For example, the maximum amplitude of a IS-inch dynamic loud­
speaker is about ±i inch. To provide for this amplitude in a bilateral
electrostatic loudspeaker would require a spacing between the plates of at
least one inch, which is, of course, impractical. About 25 years ago a large
DIRECT RADIATOR LOUDSPEAKERS
209
bilateral condenser loudspeaker was built in which the diaphragm area was
100 square feet. The spacing between the fixed plates was i- inch. The
vibrating membrane was .010 rubber with gold leaf as the conductor. The
polarizing voltage was 4000 volts. In the low-frequency range below 100
cycles the performance of the electrostatic loudspeaker was not comparable
to a l2-inch dynamic loudspeaker from the standpoints of efficiency and
power handling capacity. Employing a thinner membrane would not have
CONDENSER
LOUDSPEAKER
.---------lFiK
PLATE
ELECTRICAL CIRCUIT
DIAGRAM
PERFORATED
BACK PLATE
METALLIC COATED
PLASTIC DIAPHRAGM
6
DIAPHRAGM
W /A
:y..--- BACK PLATE
SECTION A-A'
FIG. 6.84.
Perspective and sectional views and electrical circuit
diagram of a high-frequency unilateral electrostatic loudspeaker.
increased the efficiency because the mass of the air load was the major
factor in the mechanical impedance below 100 cycles. Using more than
4000 volts polarizing voltage leads to corona and other leakage effects.
These tests substantiate the theoretical consideration, namely, that the
operation of a practical electrostatic loudspeaker must be confined to the
upper portion of the audio frequency range, say above 1000 cycles.
A commercial electrostatic loudspeaker 53 for the frequency range above
7500 cycles is shown in Fig. 6.84. The diaphragm is a .001-inch plastic
with a thin metallic coating. The diaphragm rests directly upon the
perforated metal back plate. The electrostatic loudspeaker is coupled
53
Bobb, Goldman, and Roop, Jour. Acous. Soc. Amer., Vol. 27, No.6, p. 1128, 1955.
210
ACOUSTICAL ENGINEERING
directly to the vacuum tube; the plate supplies the polarizing voltage.
Since the effective spacing is about .001 inch, the sensitivity is comparable
to conventional dynamic loudspeakers. This small spacing limits the
maximum amplitude and confines the operation to the frequency range above
7500 cycles.
Front and sectional views and the electrical circuit diagram of a bilateral
or push-pull electrostatic loudspeaker54 is shown in Fig. 6.85. In commer­
cial versions of this loudspeaker the area of the diaphragm is one square
foot. The spacing between the movable and fixed plates is about /6 inch.
INSULATOR
A
roO-To
00-0-0-0-0-0-0-.0-01
DIAPHRAGM
10
0
STATIONARV
PLATES
10
0
10
0
0
0
0
0
0
0
0
0
Ie.
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
POLARIZING
VOLTAGE
AUDIO
INPUT
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0I
01
0
0
0
01
0
0
0
0
01
0
0 01
0
10
10
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
:0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
01
10
0
0
0
0
0
0
0
0
0
0
0
0
01
10
0
0
0
0
0
0
0
0
0
0
011
lo
0
0
0
0
0
0
0
0
0
0
0
0
,0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
0
0
10 0
~
0
0
0
1
01
01
01
01
_o_~!....~..2---..5'_o_~2....-!_oJ
A'
ELECTRICAL DIAGRAM
SECTION A- A'
FRONT VIEW
FIG. 6.85. Front and sectional views and electrical diagram of a
bilateral electrostatic loudspeaker.
A polarizing voltage of about 3000 volts is used. Under these conditions
the sensitivity must be confined to the frequency range above 1000 cycles
because the power handling capacity is inadequate below 1000 cycles due
to the limited amplitude. Since the directivity pattern of a diaphragm
of these dimensions is quite sharp in the high-frequency region, two or more
units are used and directed so as to obtain the required coverage.
6.31. Sound Power Emitted by a Loudspeaker.-The sound power,
PAN, in ergs per second, emitted by a point source or by a nondirectional
loudspeaker is given by
47Tr 2p2
PAN = - - ­
6.65
pc
where p = sound pressure, in dynes per square centimeter at the distance
r, in centimeters.
If the sound source or loudspeaker is directional, that is, if the sound
emitted varies with the direction with respect to some axis of the system,
then the total sound power from the geometry of Fig. 6.86, is given by
PAD = r-
2J2"J"p2(e, ,p, r) sin e de d,p
pc 0
64
0
Janszen. A. A.• Jour. Aud. Eng. Soc .• Vol. 3, No.2. p. 87. 1955.
6.66
DIRECT RADIATOR LOUDSPEAKERS
where
pro, 0/, r) =
oand 0/ =
211
sound pressure, in dynes per square centimeter, at a
distance r, in centimeters, and the angle (J and angle 0/,
and
angular polar coordinates of the system. The loud­
speaker axis coincides with the x axis.
z
.)<I!!13t-t------T-+-y
x
6.86. The geometry for obtaining the total sound
output of a loudspeaker.
FIG.
6.32. Loudspeaker Directivity Index.-The directivity factor of a
loudspeaker is the ratio of the sound power which would be radiated if the
free space axial sound pressure were constant over 4?T solid angles to the
actual sound power radiated. The directivity factor, Q, is given by
Q =P AN
6.67
PAD
. where PAN and PAD are given by equations 6.65 and 6.66.
The directivity index can be computed from the directional characteristics
of the loudspeaker.
The directivity index of a loudspeaker can be obtained from the directivity
factor as follows:
6.68
DI DB = 10glO Q
7
HORN LOUDSPEAKERS
7.1. Introduction.-Large-scale reproduction of sound, involving several
acoustical watts, is quite commonplace. Since high power audio-frequency
amplifiers are costly, it is logical to reduce the amplifier output to a mini­
mum by the use of high-efficiency loudspeakers. At the present time,
horn loudspeakers seem to be the only satisfactory high-efficiency system
for large-scale sound reproduction. A horn 10udspeaker1 ,2,3,4,5 consists of
an electrically or mechanically driven diaphragm coupled to a horn. The
principal virtue of a horn resides in the possibility of presenting practically
any value of acoustical resistance to the generator. This feature is ex;tremely
valuable for obtaining maximum over-all efficiency in the design of the
acoustical system. Employing a suitable combination of horns, directional
characteristics which are independent of the frequency, as well as practically
any type of directional pattern, may be obtained. The combination of
high efficiency and the possibility of any directional pattern makes the horn
loudspeaker particularly suitable for large-scale reproduction. For applica­
tions requiring high quality reproduction of intense sound, some considera­
tion should be given to the introduction of frequencies not present in the
output due to nonlinearity of the operating characteristics of the elements
which constitute the vibrating system of the loudspeaker. It is the purpose
of this chapter to consider the principal factors which influence and govern
the efficiency, distortion, and power handling characteristics of a horn
loudspeaker and to describe several horn loudspeaker systems.
7.2. Efficiency.6,7,8-The efficiency of a loudspeaker is the ratio of the
useful acoustical power output to the electrical power input. For all large­
scale reproduction of sound, efficiency is an important consideration.
Specifically, the efficiency depends primarily upon the flux density, the
Hanna and Slepian, jour. A.I.E.E., Vol. 43, No.3, p. 251, 1924.
Wente and Thuras, Bell Syst. Tech. jour., Vol. 7, No. 1, p. 1940, 1928.
3 Olson, H. F., jour. Acous. Soc. Amer., Vol. 2, No.4, p. 242, 1931.
4 Wente and Thuras, jour. A.I.E.E., Vol. 53, No. 1, p.17, 1934.
5 Olson, H. F., RCA Review, Vol. 2, No.2, p. 265, 1937.
6 Wente and Thuras, jour. A .I.E.E., Vol. 43, No.3, p. 251, 1924.
7 Olson, H. F., RCA Review, Vol. 2, No.2, p. 265, 1937.
B Massa, F., Electronics, Vol. 10, No.4, p. 30, 1937.
212
1
2
213
HORN LOUDSPEAKERS
mass and the density-resistivity product of the voice coil, the mass of the
diaphragm, the ratio of the diaphragm to the throat area, the dimensions of
the air chamber, the area of the diaphragm, and the voice coil temperature.
Some of the factors are interrelated and others are independent; as a con­
sequence, it is impossible to depict in one set of characteristics the effect of
the various parameters. Therefore, the design of a horn loudspeaker is
usuany a long and tedious task. The labor is further increased when
economic considerations are involved. It is believed that a general con­
sideration of the problem, together with a series of characteristics, is valuable
for initiating the design of a loudspeaker and for facilitating the determina­
tion of the ultimate constants. The throat acoustical impedance and direc­
tional characteristics of a large number of representative horns were given
in Secs. 5.28 and 2.19. From these characteristics it is possible to inter­
polate the characteristic of practically any horn and thus eliminate con­
siderable initial work in the design of a horn loudspeaker. It is the purpose
of this section to consider the effect of the various parameters, referred to
above, upon the efficiency of a horn loudspeaker and to include charac­
teristics depicting the influence of these parameters upon the performance.
A. The Relation Between the Voice Coil Mass, the Load 1tlechanical Re­
sistance, and the Initial Efficiency.-Initial efficiency is the ratio of sound
power output to electrical power input in the system in which the mechani­
cal reactance is negligible and in which all the mechanical resistance may
be attributed to radiation. In most loudspeakers the mechanical reactance
of the vibrating system is negligible in the upper low-frequency range.
Near the cutoff of the horn the mechanical reactive component at the throat
of the horn is relatively large. Furthermore, the mechanical reactance due
to the stiffness of the diaphragm may be comparable to the other mechanical
impedances in the system. Therefore, the starting point in most horn
loudspeaker designs is a determination of the initial efficiency. This is
logical because the mechanical reactances referred to above are usually
chosen so their effect upon the efficiency characteristic in the upper low­
frequency range is very small. It is the purpose of this section to discuss
briefly the factors which influence the initial efficiency and to include a
family of curves showing the effect of the flux density, the voice coil mass,
the throat area, and the diaphragm diameter upon the initial efficiency.
The motional electrical impedance, 9 ZEM, in ohms, is given by
ZEM =
(Bl) 2
-- X
ZM
10-9
7.1
where B = flux density, in gausses,
I = length of wire in the voice coil, in centimeters, and
ZM = mechanical impedance of the vibrating system, in mechanical
ohms, at the pointjM in Fig. 7.1.
•
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.].,
1943.
9
214
ACOUSTICAL ENGINEERING
The efficiency, fL, in per cent, is
fL =
rED
where
rEM =
rED =
VOICE
+
100
7.2
electrical resistance component of the motional electrical
resistance, in ohms, and
damped electrical resistance of the voice coil, in ohms.
COIL
ELECTRICAL
rEM
X
rEM
MECHANICAL NETWORK
OF THE
MECHANICAL SYSTEM
CIRCUIT
CROSS-SECTIONAL VIEW
FIG. 7.1 Cross-sectional view of a horn loudspeaker, the electrical circuit and mechanical
network of the vibrating system. In the voice coil ci.rcuit; e = the internal voltage of
thc generator. YEG = the internal electrical resistance of the generator. L = the
inductance of the voice coil. YED = the damped electrical resistance of the voice coil.
ZEM = the motional electrical impedance.
In the mechanical network; mA and YMA =
the mass and mechanical radiation resistance due to the air load on the back of the
diaphragm. mo and mD = the masses of the voice coil and diaphragm. CMS and CMt
= the compliances of the suspension and air chamber. ZMH = the mechanical impe­
dance at the throat of the horn. ZME = the mechanical impedance due to the electrical
circuit. 1M = the force generated in the voice coil. /MO = the force of the mechanical
generator.
In the mechanical network, Fig. 7.1, the mechanical impedance,
mechanical ohms, at 1M is given by
.
ZM = JwmA
where
mA =
me =
mD =
rMA =
CMS =
CMl =
ZMH =
rMH =
XMH =
ZM,
..
1
ZMH
+ rMA + Jwme
+ JwmD + JW
""'---c
+.JWCMlZMH
+1
MS
in
73
.
mass of the air load on the back of the diaphragm, in grams,
mass of the voice coil, in grams,
mass of the diaphragm, in grams,
mechanical resistance load on the back of the diaphragm, in
mechanical ohms,
compliance of the suspension, in centimeters per dyne,
compliance of the air chamber, in centimeters per dyne,
rMH
jXMH = mechanical impedance of the throat of the
horn, in mechanical ohms,
mechanical resistance of the throat of the horn, in mechanical
ohms, and
mechanical reactance of the throat of the horn, in mechanical
ohms.
+
•
215
HORN LOUDSPEAKERS
For initial efficiency considerations, the mechanical reactance of the
mechanical system is assumed to be negligible compared to the radiation
mechanical resistance, that is, mA, me, mD, CMI , IjCMs, and XMH are zero.
rMA is also negligible.
Then
AD2
7.4
ZM = rMH = 42 - ­
AT
where AD
AT
=
=
area of the diaphragm, in square centimeters, and
area of the throat, in square centimeters.
Substituting equations 7.1 and 7.4 in equation 7.2
B2
fL =
( 42AD2rED) 109
l2AT
+ B2
X
100
7.5
The electrical resistance,IO rED, in ohms, is given by
rED
where Kr
=
l
S
=
=
=
Krl
5
X
7.6
10-6
resistivity of the voice coil material, in microhms, per centi­
meter cube (see Table 6.1),
length of the conductor, in centimeters, and
area of the conductor, in square centimeters.
Then equation 7.5 becomes
fL =
B2
42A 2K
( lSADr) 103
T
X
+ B2
100
7.7
The mass of the coil, me, is
me = lSp grams
7.8
where p = density, in grams per cubic centimeter (see Table 6.1).
efficiency may be written, employing equation 7.8, as
The
B2
fL =
(42A 2K
rp) 103
meAT
D
+ B2
X 100
7.9
For a particular material, KrP is a constant. Equation 7.9 gives the effi­
ciency in terms of B2, me, and A D2jA T . The efficiency as a function of
I
10 The voice coil electrical circuit is shown in Fig. 7.1.
rED is the total damped
electrical resistance of the voice coil and includes skin effect and hysteresis losses in the
iron. L is the inductance of the voice coil. As shown in Fig. 6.6, the electrical
impedance of the voice coil increases at the high frequencies due to the electrical react­
ance of L and an increase in electrical resistance due to skin effect and hysteresis losses
in the iron circuit. In order to simplify these considerations the damped electrical
resistance will be assumed to be the same as the ohmic (d-c) electrical resistance.
216
ACOUSTICAL ENGINEERING
AD2/AT for aluminum voice coils of 0.1,0.25,0.5, 1, 2, 4, and 8 grams and
flux densities of 22,000 and 14,000 gausses is shown in Fig. 7.2. The
characteristics of Fig. 7.2 also apply to a copper voice coil if the abscissa are
multiplied by 0.5. Equation 7.9 and Fig. 7.2 show the factors which
influence the initial efficiency of a horn loudspeaker.
VOICE
COIL.
MASS
22000 GAUSSES
100
~
z
!:! 10
8
4
2
"~
8
~
i;:
0.5
w
0.25
...
100
i;:
...
"'
0.1
I
FIG.
7.2.
2
I
0.5
["
I
10000
10
4
"
Z
~IO
U
I
U
VOICE
COIL.
MASS
14000 GAUSSES
~
10
0.25
10000
The initial efficiency, in per cent, of a horn loudspeaker as a function of
AI)2/A1' for aluminum voice coils having masses of 0.1, 0.25, 0.5, 1.2, 4, and
8 grams and flux densities of 22,000 and 14,000 gausses. AI) and A l' are the areas
of the diaphragm and throat, respectively. in square centimeters. The above
graphs may be applied to a copper voice coil by multiplying the ratio AI)2/A l' by
one-half.
B. The Effect of the Mass of the Vibrating System upon the Efficiency.-In
the preceding section the mechanical reactance of the vibrating system
was assumed to be negligible compared to the mechanical resistance. The
mechanical mass reactance of the diaphragm and voice coil influences
the efficiency when this mechanical reactance becomes comparable to the
mechanical resistance. It is the purpose of this section to consider the
effect of the mechanical reactance of the vibrating system upon the effi­
ciency.
The real part of the motional electrical impedance, equation 7.1, is
rEM =
where
rM =
XM =
(
(Bl)2rM )
rM 2
XM 2
+
10-9 ohms
7.10
mechanical resistance of the vibrating system, in mechanical
ohms, and
mechanical reactance of the vibrating system, III mechanical
ohms.
In this discussion, let
XM =
w (mD
+ me)
7.11
At the high frequencies the mechanical reactance due to CMS and XMA is
negligible compared to the mechanical reactance due to the mass of the dia­
phragm. In order to divorce the effect of the air chamber from the effect
of the mass of the diaphragm, the compliance, CMt, will be assumed to be
HORN LOUDSPEAKERS
217
zero. For the same reason rMA will be assumed to be zero. These effects
will be considered in following sections. The mechanical resistance, rM,
then becomes the horn throat resistance rMH. The throat mechanical
resistance is given by
AD2
rMH =
42 AT
7.12
where AT = area of the throat, in square centimeters.
equation 7.10 in 7.2, the efficiency, in per cent, becomes
p. =
(
rED rMH
2
+(Bl):;~~
XM
+ (Bl)2rMH
X
Substituting
100
7.13
This expression shows that the efficiency is a function of the flux density,
the coil mass and material, the diaphragm mass, the throat resistance, and
I
22000 GAUSSES
COIL·DIAPHRAGM
= I' I
10022000 GAUSSES
-.
COIL'DIAPHRAGM-I' 2
~
10
\
\
I
20
100
10 14000 GAUSSES
0
1000
FREQUENCY
10000
-.
COIL· DIAPHRAGM - 1'1
I
20
100
100
14000 GAUSSES
1000
FREQUENCY
\
10000
COIL • DIAPHRAGM
I' 2
~
r;
\~
"
z
!!!I 0
Il
...
b
\
10
100
1000
FREQUENCY
\
10000
\
I
20
100
1000
FREQUENCY
\
10000
FIG. 7.3. The efficiency. in per cent. as a function of the frequency of horn loud­
speaker systems having ratios of voice coil mass to diaphragm mass of 1:2 and
1: 1. flux densities of 22.000 and 14.000 gausses. and initial efficiencies of 20. 40.
60. and 80 per cent for an aluminum coil. The above graphs may be applied to a
copper voice coil by multiplying the frequency by one-half.
the frequency. The efficiency characteristics for ratios of voice coil mass
to diaphragm mass of 1 : 1 and 1 : 2, and flux densities of 22,000 and 14,000
gausses for an aluminum voice coil are shown in Fig. 7.3. The character­
istics of Fig. 7.3 are applicable to a copper voice coil by multiplying the
abscissa by 0.5. In order to connect with the characteristics of initial
efficiency of Fig. 7.2, these curves are depicted in terms of the initial
efficiency (20, 40, 60, and 80 per cent). These data show that it is a
218
ACOUSTICAL ENGINEERING
comparatively simple matter to obtain high efficiencies at the lower
frequencies. However, at the higher frequencies the efficiency is limited by
the mass of the diaphragm and voice coil.
C. The Effect of the Air Chamber upon the Efficiency.ll,l2,l3,l4,l5,l6-The
results of the preceding sections were obtained by assuming the compliance
of the air chamber to be zero. In general, it is impractical to design a high
efficiency loudspeaker to cover a wide frequency range without an air
chamber, because the diaphragm area is usually larger than the throat
area. In order to eliminate interference, the dimensions of the elements of
the air chamber are usually made small compared to the wavelength.
When these conditions obtain, the volume of the air chamber appears as
a compliance. At the higher frequencies, the mechanical impedance at
the throat of the horn is resistive, the mechanical reactance of the suspen­
sion is very small, and the mechanical impedance of the diaphragm system
is a mechanical mass reactance. The mechanical network reduces to a
mechanical resistance and compliance in parallel connected in series with a
mass. It is the purpose of this section to show the effect of the air chamber
upon the efficiency from the standpoint of this mechanical network. The
mechanical impedance of a mechanical resistance and compliance in parallel,
which is the equivalent of the throat mechanical resistance and compliance
of the air chamber, is given by
7.14
where
rMH =
mechanical resistance at the horn throat, in mechanical ohms,
and
CMl =
compliance of the air chamber, in centimeters per dyne.
The throat mechanical resistance, YMH, is given by equation 7.12.
mechanical compliance, Sec. 5.7, of the air chamber is given by
CA
V
CMl = = -2-2
AD2
pc AD
where C A
=
V
=
p =
c
11
12
13
14
15
16
=
The
7.15
acoustical capacitance of the air chamber, in (centimeters)5
per dyne.
volume of the air chamber, in cubic centimeters,
density of air, in grams per cubic centimeter, and
velocity of sound, in centimeters per second.
Hanna and Slepian, Jour. A.I.E.E., Vol. 43, No.3, p. 251, 1924.
Wente and ThUfas, Bell Syst. Tech. Jour., Vol. 7, No. 1, p. 140, 1928.
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 2, No.4, p. 242, 1931.
Wente and Thuras, Jour. A.I.E.E., Vol. 53, No. 1, p. 17. 1934.
Olson, H. F., RCA Review, Vol. 2, No.2, p. 265, 1937.
Smith, Bob H., Jour. Acous. Soc. Amer., Vol. 25, No.2, p. 305, 1953.
219
HORN LOUDSPEAKERS
Referring to the system shown in Fig. 7.1, it is obvious that the effect of
the air chamber will be to reduce the mechanical reactance of the system
at the high frequencies and thereby increase the efficiency over a wide range.
Figure 7.4 shows the efficiency characteristics of a system consisting of a
voice coil and diaphragm having a mechanical reactance of 1 ohm at 1000
cycles coupled to the throat of a horn having a mechanical resistance of 1
ohm and an air chamber having a mechanical reactance of 1 ohm at 1000
cycles, 2000 cycles and an infinite mechanical reactance for an initial effi­
ciency of 20 per cent, 40 per cent, 60 per cent, and 80 per cent. These
characteristics are applicable to other frequencies by multiplying the
100
100
-"~
~
0
~
C B
I
20
100
A
1000
10000
rREQUENCY
\'
B
1000
10000
fREQUENCY
~
1\
B
20
\C
100
100
100
I
I
20
100
1000
FREQUENCY
f<'.
10000
C B
I
20
100
1000
fREQUENCY
~
10000
FIG. 7.4.
The efficiency, in per cent, as a function of the frequency of a voice coil
and diaphragm having a mechanical reactance of 1 mechanical ohm at 1000 cycles
coupled to a throat of a horn having a mechanical resistance of 1 mechanical ohm
and an air chamber having the following mechanical reactances: A. An infinite
mechanical reactance. B. A negative mechanical reactance of 1 mechanical ohm
at 2000 cycles. C. A negative mechanical reactance of 1 mechanical ohm at
1000 cycles for initial efficiencies of 20, 40, 60, and 80 per cent. These charac­
teristics are applicable to other frequencies by multiplying the frequency by any
number and multiplying the mass and the compliance by the reciprocal of the
number.
abscissa by any number and, of course, multiplying the mass and the com­
pliance by the reciprocal of the number. These characteristics are also
applicable for other values of mass and mechanical resistance by simply
multiplying these two factors by the same number and the compliance by
the reciprocal of that number. The characteristics shown in Fig. 7.4 have
included mass-compliance products which cover the useful range of values.
220
ACOUSTICAL ENGINEERING
-larger products result in a peaked characteristic, smaller values do not
show much deviation from zero value of compliance.
D. The Effect of the Generator Electrical Impedance and the Mechanical
Impedance at the Throat of the Horn upon the Efficiency.-Due to the im­
practicability of a horn mouth diameter comparable to the wavelength for
low-frequency loudspeakers, it is interesting to note that a relatively smooth
output response frequency characteristic can be obtained from a horn
having a mechanical impedance characteristic varying over wide limits.
Near the cutoff of both finite and infinite exponential horns, the radiation
mechanical resistance at the throat is small and the positive mechanical
reactance large. The compliance of the suspension system should be chosen
so that its negative mechanical reactance balances the positive mechanical
reactance due to the throat. For example, consider a moving coil mechan­
ism coupled to the throat of a horn and fed by a vacuum tube amplifier;
the sound power output is the real part of
Power
=
(IZ;TI)
where the motional electrical impedance,
B
where
=
2ZEM
ZEM,
7.16
in ohms, from equation 7.1, is
air gap flux, in gausses,
1 = length of wire in the voice coil, in centimeters,
AD = area of the diaphragm, in square centimeters,
r AH = acoustical resistance at the throat, in acoustical ohms,
XAH = acoustical reactance at the throat, in acoustical ohms, and
XM = mechanical reactance of the diaphragm, suspension and coil
system, in mechanical ohms.
From the voice coil electrical circuit, Fig. 7.1, the total electrical impedance,
in ohms, at e is
ZET,
ZET
where
rED =
rEG
=
L =
e=
=
rED
+ rEG + jwL + ZEM
7.17
voice coil resistance, in ohms,
amplifier output resistance, in ohms,
inductance of the voice coil, in henries, and
amplifier open circuit voltage, in volts.
Equation 7.16 shows that the throat acoustical resistance may vary over
wide limits without introducing large variations in the power output. As a
specific example, Fig. 7.5 shows the power output as a function of the
frequency lor a horn, having all dimensions two and one-half times
221
HORN LOUDSPEAKERS
that of Fig. S.l1B and driven
0
by a vacuum tube having the
a:J
o
constants indicated by the cap­
......
tion of Fig. 7.5. Although the
I­ 2 /
\ /
:>
variation in acoustical resistance 1--4
is 6 to 1, the variation power out­ o:>
put is only 2 db.
E. The Effect of the Voice Coil
Temperature upon the Ejjiciency.l7
100
200
500
-The effect of the temperature
fREQUENCY
of the voice coil upon the effi­ FIG. 7.5. Acoustical power output frequency
ciency is usually ignored in con­ characteristic of the horn (Fig. 5.11B with all
siderations of the performance dimensions multiplied by 2!) coupled to a
of a loudspeaker. In high power 10!-inch diameter, 10-gram diaphragm driven
loudspeakers, where the tem­ by a 5-gram aluminum voice coil in a field of
20,000 gausses. Damped electrical resistance
perature of the voice coil becomes of voice coil 20 ohms. Electrical impedance
quite high, considerable loss in effi­ of vacuum tube through a transformer,
ciency may result as will be shown 35 ohms.
in the discussion which follows.
The efficiency, in per cent, of a loudspeaker, when the temperature cor­
rection is added, may be expressed.
--
Q.
fL =
rEM
rEDO(l
where
rEDO =
a =
t
=
rEM =
+ at) + rEM
X
100
7.18
damped electrical resistance of the voice coil at 0° Centigrade,
temperature coefficient of resistance, 0.00423 for aluminum
and 0.00427 for copper,
temperature of the voice coil, in degrees Centigrade, and
motional electrical resistance of the voice coil.
The efficiency as a function of the temperature for various values of initial
efficiency at 0° Centigrade is shown in Fig. 7.6. These characteristics
show that the relative loss in efficiency with increase in temperature is
considerably greater for a loudspeaker with low efficiency.
F. The Effect of the Sound Radiation from the Unloaded Side of the Dia­
phragm upon the Ejjiciency.-In the consideration of the efficiency, usually
very little cognizance is taken of the sound radiation from the back of the
diaphragm of a horn loudspeaker. In view of the large amount of sound
that is radiated from the back of the diaphragm, some consideration should
be given to the effect of this radiation upon the efficiency. Since this
radiation cannot be used, it must be considered as a loss the same as the
electrical resistance loss in the voice coil. The loss due to the reactive
component of the mechanical impedance is usually small compared to the
mechanical reactance of the remainder of the system.
17
Olson, H. F., RCA Review, Vol. 1, No.4, p. 68, 1937.
222
ACOUSTICAL ENGINEERING
100
80
60
!Z 40
w
U 30
a:
w
0..
20
~
10
V
i:: 8
"­
uJ
----- ­­
-I--
r--
>­
u
3
--r-r-r-­
I-- I--­
I-- I-- r-­
-- -
r--
r-- r-- I-
--
6
50
100
TEMPERATURE
I-­
--
r-­
t--......
I'--­
-I-- i-­
ISO
RISE
-
- -­
200
IN DEGREES
250
-
300
CENTIGRADE
FIG. 7.6.
The efficiency, in per cent, as a function of
the temperature of a voice coil for various values of
initial efficiency at 0° Centigrade.
DIAPHRAGM DIA.2IN AREAs3 14 SQ IN
IZ
0
~
~ 20
'\:
a:
It 40
!
60
'"
gao
10 900
I­
''""
1000
FREQUENCY
DIAPHRAGM DIA'81N AREA'SO 2SQ IN
0
z
~2 0
a:
:;'4
THROAT
AREA
SQ IN
39
78
'0
I.57
DIAPHRAGM DIA.=4IN AREA: 12 56SQ IN
I­
Z
tj
o
t---:
20
~
~ 40
~
"-
3.14
:!:
6.28
60
6.28
I 2.6
25.1
:2 80
3
10900
10000
THROAT
AREA
SQIN
6.28
I 2.6
"-
1000
fREQUENCY
DIAPHRAGM DIA=16IN AREA=201 SQ IN
o
,""", ........ r--
'\ "".1'
25.1
0
10000
THROAT
AREA
so IN
25.1
50.2
I 00.
t'-­
50.2
201.
402.
I00.
'"
'"
(g80
..J
100100
THROAT
AREA
SQ.IN.
I .57
3.14
1000
FREQUENCY
10000
"'8 0
o
--'
10900
1000
fREQUENCY
10000
FIG. 7.7. Characteristics depicting the loss in per cent of the total sound radiation,
due to the radiation of sound from the back of the diaphragm of a horn loudspeaker
for diameters of 2, 4, 8, and 16 inches and throat areas of 2,1. !, t, and t times the
diaphragm area.
The radiation from the back of the diaphragm may be assumed to be
the same as that from a piston in an infinite baffle (see Sec. 5.8 and Fig.
5.2). The percentage of the total radiation which is lost due to the radia­
tion from the back is given by.
Efficiency loss
=
rMA
r~A
rMH
X
100
7.19
HORN LOUDSPEAKERS
where
rMH =
rMA =
223
radiation mechanical resistance at the throat of the horn, in
mechanical ohms, and
radiation mechanical resistance of the back of the diaphragm
from Sec. 5.8, in mechanical ohms.
The characteristics depicting the loss due to radiation from the back of
the diaphragm as a function of the frequency for diaphragm diameters of
2, 4, 8, and 16 inches and various ratios of throat area to diaphragm area
are shown in Fig. 7.7. These characteristics show that the loss is indeed
quite high.
7.3. Distortion.-In general, the electrical power input to (or the
acoustical power output of) a loudspeaker is limited by the generation of
spurious harmonics or subharmonics. The limiting factor may be due to
air overload, excessive amplitudes where Hooke's law no longer holds,
nonlinear elements, variable voice coil air-gap flux product, or nonfunda­
mental vibration modes of the diaphragm. It is the purpose of this section
to consider the most common forms of distortion in horn loudspeakers.
A. Distortion Due to Air Overload in the Horn. 18 ,19,20-A sound wave of
large amplitude cannot be propagated in air without a change in the wave
form and, as a result, the production of harmonics. If equal positive and
negative changes in pressure are impressed upon a mass of air the resultant
changes in volume will not be the same. The volume change for an increase
in pressure will be less than the volume change for an equal decrease in
pressure. From a physical viewpoint the distortion may be said to be due
to the nonlinearity of the air.
In the derivation of the fundamental wave equation the second order
terms were omitted. If these terms are included the magnitude of the
harmonic frequencies may be determined from the differential equation.
The subject has been investigated both theoretically and experimentally
by a number of investigators. In the case of an exponential horn for
constant sound power output, the distortion is proportional to the frequency.
Further, the nearer the observation frequency is to the cutoff frequency
the smaller the distortion.
The distortion due to nonlinearity of the air is at the present time one of
the most important as well as the most troublesome factors in the design
of high-efficiency loudspeakers for large outputs. In order to obtain high
efficiency, particularly at the higher frequencies, it is necessary to couple
the relatively heavy diaphragm to a throat small in area compared to the
diaphragm. For a certain allowable distortion the power output is directly
proportional to the area of the throat. Obviously, to deliver large sound
outputs with small distortion requires a very large throat which may be
suitably coupled to a correspondingly large diaphragm or a large number of
lightly driven small throat units.
Rocard. Comtes Rendus. Vol. 196. p. 161. 1933.
Thuras. Jenkins, and O·Neil. Jour. Acous. Soc. A mer .• Vol. 6. No.3. p. 173. 1935.
20 Goldstein and McLachlin. Jour. Acous. Soc. Amer.. Vol. 6. No.4. p. 275, 1935.
18
19
224
ACOUSTICAL ENGINEERING
The second harmonic distortion, at the mouth, in per cent of the funda­
mental, generated in an exponential horn is given by
+ l)PIt w (1 _ c
V2ypocm
D = p2 100 = (y
PI
mx/ 2 )100
7.20
where y = ratio of specific beats, y = 1.4 for air,
PIt = sound pressure at the throat, in dynes per square centimeter,
w = 21Tf,
J = frequency, in cycles per second,
m = flare constant of the exponential horn (see Sec. 5.22),
x = length of the horn, in centimeters,
po = atmospheric pressure, in dynes per square centimeter, and
c = velocity of sound, in centimeters per second.
The power21 which can be transmitted per square centimeter of throat
area of an infinite exponential horn as a function of the ratio of the fre­
quency under consideration to the cutoff frequency with the production
10
•
I'..
III
f-
.1
8
!(
~
4
Z
2
I
f-'
::J
Q.
•
!;
4
o
a:
'" "
"'"
""'" ...... i'..
.....
I......
,
""- 1010
'" '"
"I"-
I"
"-
z
'" '"
3 "(0
"
'"o~.Ol•
I'
~1o
Q.
.00 I
I
2
4
5
8
7 8 9 10
FREQUENCY f
.....1'..
"
....... r-,.
'"
CUTOFF
I"'­
f'..
'-.
I"­
2
FREQUENCY
7.8. The power output of infinite exponential horns, per
square centimeter of throat area, for 1, 3, and 10 per cent distor­
tion, as a function of the ratio of the frequency under considera­
tion to the cutoff frequency.
FIG.
of 1, 3, and 10 per cent distortion is shown in Fig. 7.8. For the sake of
generality the curves shown in Fig. 7.8 refer to an infinite horn. How­
ever, the increase in power which may be transmitted by a practical finite
21
Olson, H. F., RCA Review, Vol. 2, No.2, p. 265, 1937.
HORN LOUDSPEAKERS
225
horn is only a few per cent greater than that shown in Fig. 7.8, because
very little distortion is generated in the large cross-sectional area near the
mouth of the horn.
It may be mentioned in passing that the multiple flare horn (see Sec. 5.25)
provides a means of decreasing the distortion because the rate of flare is
very rapid near the diaphragm and, therefore, the pressures are rapidly
reduced with respect to the distance from the diaphragm.
B. Distortion Due to Variation in Volume of the Air Chamber.22-In
general, acoustical, mechanical, and electrical networks are assumed to be
invariable; that is, the constants and connections of the network do not
vary or change with time. A network which includes a circuit element that
varies continuously or discontinuously with time is called a variable net­
work. In some cases the variable elements are assumed to be certain
functions of the time; that is, the variations are controlled by outside forces
which do not appear in the equations or statement of the problem. In
another type of variable circuit element the variation is not an explicit
time function, but a function of the current and (its derivatives) which is
flowing through the circuit.
An example of the latter type of circuit element in an acoustical system
is the air chamber capacitance in a horn loudspeaker. The excurSIOns
of the diaphragm change the acoustical capacitance. The acoustical
capacitance of the air chamber of Fig. 7.9 is given by
CAl =
~
pc 2
where
A(d
+ x)
pc 2
7.21
density of air, in grams per cubic centimeter,
c = velocity of sound, in centimeters per second,
V = volume of the air chamber, in cubic centimeters,
A = projected area of the air chamber upon the diaphragm, in
square centimeters,
d = distance between the diaphragm and front boundary of the air
chamber in the absence of motion, in centimeters, and
x = displacement of the diaphragm, in centimeters.
p =
The acoustical network of the acoustical system shows the effect of the
nonlinear element upon the sound power output. In the case of a single
frequency the distortion which this element introduces is small, because,
for constant sound power output, the amplitude of the diaphragm is in­
versely proportional to the frequency. At low frequencies where the
amplitude of the diaphragm may be so large that the volume of the air
chamber becomes alternately zero and two times the normal volume, the
acoustical reactance of the acoustical capacitance is very large compared to
the acoustical resistance of the horn (see Fig. 7.9). At the high frequencies
22
Olson, H. F., RCA Review, Vol. 2, No.2, p. 265, 1937.
226
ACOUSTICAL ENGINEERING
where the acoustical reactance of the acoustical capacitance is comparable
to the acoustical resistance, the amplitude of the diaphragm for the same
output is so small that the variation in acoustical capacitance may be
neglected (see Fig. 7.9). However, the conditions are different when both
a high and a low frequency are impressed upon the same system. Under
these conditions considerable change in the acoustical capacitance occurs
due to the large amplitudes of the diaphragm for the impressed low fre­
quency. The resultant change in acoustical capacitance introduces a
variable element for the impressed high frequency which may have varia­
tions in acoustical impedance as large as the impedance of the other elements
of the system. The result is shown in Fig. 7.9. When this condition
ELECTRICAL
A~
ACOUSTICAL
NETWORK
INPUT
ACOUSTICAL
.~
~A
LOW
C
CROSS-SECTIONAL
a' '\PlM'V'I'J'v'V\PoV\/'IJ%
FREQUENCY
orfNVltv., DVViJV\j!JV
COMBINATION
~
FREQUENCY
B "rlVoWI'JVVtIVVVVVv
HIGH
OUTPUT
HIGH
C,LlJoA..lLAMM-"--=--''-'-\I0+n-_ _---,,-,
' VI.t\;\N.J..fJ"
AND LOW FREQUEIoiCY
VIEW
FIG. 7,9. A mechanism with an air chamber coupling the diaphragm to the
horn. In the acoustical network: M I = the inertance of the diaphragm and
voice coil. CAl = the acoustical capacitance of the air chamber. YAI = the
acoustical resistance at the throat of the horn. p = the driving pressure.
p = BliIA. B = the flux density in the air gap. I = the length of the voice
coil conductor. i = the current in the voice coil. A = the area of the dia­
phragm. The variation in volume of the air chamber introduces a nonlinear
element in the form of the acoustical capacitance CAl. The wave shapes of the
electrical input and acoustical output for a low, high, and a combination of a
high and a low frequency illustrates the effect of the nonlinear element upon
the acoustical output.
obtains, particularly with close spacing between the diaphragm and the
front boundary of the air cha!llber, the distortion may be tremendous.
Physically the low frequency modulates the high frequency.
In the above discussion the air chamber is assumed to be a pure acoustical
capacitance. This assumption is not correct at the higher frequencies
where the dimensions of the air chamber are comparable to the wavelength.
Regardless of the form of the acoustical impedance, it is, nevertheless, a
function of the spacing between the diaphragm and the air chamber and is
therefore a nonlinear element.
e. Distortion Due to the Diaphragm Suspension System. 23_The outside
suspension is another example of a variable circuit element in a mechanical
23
Olson, H. F., RCA Review. Vol. 2. No.2, p. 265. 1937.
227
HORN LOUDSPEAKERS
system. In certain types or, as a matter of fact, for unlimited amplitudes
in all types of suspension systems the stiffness is not a constant, but a
function of the amplitude and, in general, increase for the larger ampli­
tudes (see Sec. 6.26).
In the case of a horn loudspeaker the amplitude of the diaphragm for
constant sound power output is inversely proportional to the frequency.
Furthermore, the mechanical impedance of the suspension system becomes
comparable to the other mechanical impedances in the system at the lower
frequencies. Consequently, the greatest distortion due to the suspension
system will occur at the low-frequency end of the working range.
The mechanical circuit of the mechanical system, Fig. 7.10, shows the
effect of the nonlinear element. When the stiffness of the suspension
system increases with amplitude, the third harmonic is the preponderant
distortion (see Sec. 6.26C). The wave shape under these conditions is
shown in Fig. 7.10. A distortion frequency characteristic of a diaphragm
coupled to a large throat horn is shown in Fig. 7.10.
M,",
MECHANICAL
v
CIRCUIT
ELECTRICAL
INPUT
WAVE
SOUND
OUTPUT
SHAPES
0- 1
~
U
... B\
0:
Q.
~
6
z
Q 4
0­
0:
o
0­ 2
0/)
CROSS-SECTIONAL
VIEW
o ~o
\.
1\,
r­
100
rREQUENCY
1000
FIG. 7.10.
Mechanism having a diaphragm with a nonlinear suspension
system. In the mechanical circuit: ml = the mass of the diaphragm and
voice coil. CMl = the compliance of the diaphragm suspension system.
YMl = the mechanical resistance at the throat of the horn. 1M = the
driving force. The mechanical circuit of the vibrating system and the
wave shapes indicate the effect of the nonlinear element. The graph shows
a typical distortion characteristic obtained on an 8-inch diameter dia­
phragm coupled to a large throat horn and delivering an acoustical power
output of 3 watts.
D. Distortion Due to a Nonuniform Magnetic Field in the Air Gap.­
Inhomogeneity of the flux density through which the voice coil moves is
another source of distortion. The result is that the driving force does not
correspond to the voltage developed by the generator in the electrical
system.
228
ACOUSTICAL ENGINEERING
The force, in dynes, developed by the interaction of the current in the
voice coil and the magnetic field is
f
where B
I
=
i
=
=
=
Eli
7.22
flux density, in gausses,
length of the voice coil conductor, in centimeters, and
current, in abamperes.
Equation 7.22 shows that the force is directly proportional to the current
if BI is a constant. If the BI product varies with the position of the voice
coil the force will not be proportional to the current and distortion will
result. A typical flux distribution in an air gap is shown in Fig. 6.66. A
consideration of the flux distribution shows that the El product will be
practically a constant if the voice coil is made longer than the air gap
because, as the coil moves into the weaker tufting field on one side, it moves
into a stronger field on the other side. From the standpoint of efficiency
at the higher frequencies this method is not particularly desirable because
part of the voice coil is in a weak field. This type of distortion can also
be eliminated by making the air gap of sufficient axial length so that the
voice coil remains at all times in a uniform field as shown in Fig. 6.66C.
The latter method is usually used for high-frequency loudspeakers of high
efficiency (also see Sec. 6.26E).
E. Subharmonic Distortion.-The distortions referred to above have
been concerned with higher harmonics, that is, multiples of the funda­
mental. It has been shown in Sec. 6.26C that sub harmonics are generated
in vibrating systems with nonlinear elements. The existence of sub­
harmonics in direct radiator loudspeakers is well known. However, in
horn loudspeakers the diaphragms are relatively small and quite rigid.
Consequently, the conditions for the production of subharmonics are not
particularly favorable.
F. Power Handling Capacity and the Voice Coil Temperature. 24-The
maximum allowable distortion may determine the power rating for the
loudspeaker. However, in certain loudspeakers the maximum allowable
temperature of the voice coil determines the power rating. This is par­
ticularly true of high-frequency loudspeakers.
By making the efficiency a maximum, the dissipation in, and the resulting
temperature of, the voice coil for a certain acoustical output will be a
mlmmum. Practically all the heat energy developed in the voice coil is
transmitted across the thin air film between the voice coil and the pole
pieces and from the pole pieces to the field structure and thence into the
surrounding air. In this heat circuit practically all the drop in temperature
occurs in the thin air film. The temperature of the voice coil approaches
the temperature of the pole pieces as the thickness of the air film is de­
creased. The temperature rises as a function of the power dissipated in
the voice coil for various clearances between the voice coil and pole pieces.
24
Olson, H. F., RCA Review, Vol. 2, No.2, p. 265, 1937.
229
HORN LOUDSPEAKERS
This is shown in Fig. 7.11. These results are obtained for no motion of
the voice coil. When motion occurs, the thermal impedance of the air film
is reduced and the temperature of the voice coil is diminished.
...o
160
/
~140
/
"~
z
tjl20
If)
/'
~ 100
~
/'
~ 80
V
...
~
60
w
v:V
a:
./
w
../. ~. - 'V
j:!40
~
~ 20
w
I­
/
~Y
~
c
......
V
,/
./
,/
/
.-'
/
...... . - '
V
V
5
10
POWER
INPUT
15
IN
20
25
WATTS
7.11. The temperature rise as a function of the power
delivered to a voice coil for air-gap clearances as follows:
A. 0.021 inch, B. 0.015 inch, C. 0.009 inch. Coil 1! inches in
diameter, and 0.25 inch in length.
FIG.
G. Power Handling Capacity and the Amplitude of the Diaphragm. 25­
The maximum allowable amplitude of the diaphragm is another factor
which may determine the maximum allowable acoustical power output.
The acoustical power output, in watts, of a horn loudspeaker in which the
diaphragm is terminated in an acoustical resistance is
P
=
pc(27rJ)2d2An2 10- 7
2AH
where
7.23
density, in grams per cubic centimeter,
c = velocity of sound, in centimeters per second,
J = frequency, in cycles per second,
d = maximum amplitude from its mean position, in centimeters,
An = area of the diaphragm, in square centimeters, and
AH = area of the throat of the horn, in square centimeters.
p =
The amplitude of various diameter diaphragms coupled to a horn throat
of 1 square inch for 1 acoustical watt output is shown in Fig. 7.12.
25
Massa, F., RCA Review, Vol. 3, No.2, p. 196, 1938.
ACOUSTICAL ENGINEERING
230
I
••
VI
.""
z
2
is'"
~
.1
...'"
4
C
~
••
~
:J
"
......
"
"
""­
'"
'-.:
2
I'...
r...... i'..
8
'" •
II.
4
"r-.
.l.SQ.IN.
" ".......
""
-<
~.Ol
r-,
'" I'...
I'
2
i"
.
'"
I'..
~ r......
SQJn.
I'..
,
,
"- " ,
.......
..........
~SQ.I"il'.,....
4
5 ' 78V102
t'-r-,
I'..
~I.
2
.001
,
.l.SQJ~
'<:
.......
:I
FREQUENCY
•
4
5 ' 78V10'
IN
CYCLES
2
PER
r-,
.......
3 4
r-,
t-....
'"
'"
5 '7 8Vl04
SECOND
......
2
7.12. The peak amplitude frequency characteristics of vibrating
pistons of various areas in square inches, coupled to the throat of a horn
having an area of 1 square inch, for 1 watt output.
FIG.
7.4. Horn Loudspeaker Systems.-A. Single-Horn, Single-Channel
System.-The single-horn, single-channel system consists of a single horn
driven by a single diaphragm. A diaphragm coupled to an exponential
horn constitutes the simplest and most widely used system. The efficiency
frequency characteristic of a simple exponential horn coupled to a diaphragm
and coil having a mass ratio of 2 operating in a field of 22,000 gausses is
shown in Fig. 7.13. Two efficiency frequency characteristics are shown
with initial efficiencies of 80 per cent and 50 per cent. Although it is possible
to obtain reasonably high efficiency over a wide frequency range with a
single horn coupled to a diaphragm, the efficiency can be increased by
employing a multiple flare horn.
To obtain maximum efficiency in a horn loudspeaker at any frequency,
the effective mechanical reactance of the entire system should be equal to
the effective mechanical resistance. This, in general, means that to obtain
maximum efficiency the throat mechanical resistance of the horn should
be proportional to the frequency, since the mechanical reactance is primarily
mass reactance and, therefore, proportional to the frequency. The surge
mechanical resistance of the exponential horn is independent of the frequency.
However, the acoustical resistance 26 of a multiple flare horn increases with
frequency as shown in Sec. 5.26. Therefore, the efficiency is higher over a
wide range than in the case of a horn with a single rate of flare. The effi­
26
Olson, H. F., Jour. Soc. Mot. Pict. Eng., Vol. 30, No.5, p. 511, 1938.
231
HORN LOUDSPEAKERS
100
80
,,
~
z
... 40
IJ
-/
---­
/B
IC
I,
a:
..,20
a.
!:
,
,
B
-
C
"I"
I
i"­
1\
10
8
>­
IJ
'\
~ 4
U
i:
......
"\.
\.A
\
2
I
20
100
rREQUENCY
IN
CYCLES
1000
PER
\
10000
SECOND
FIG. 7.13. A. Efficiency frequency characteristic of a horn loudspeaker
employing the horn of Fig. 5.11D with the dimensions multiplied by three
and drive'll by 4 cones, 12 inches in diameter, with 5-gram copper voice
coils operating in a field of 14,000 gausses. B. Efficiency frequency
characteI:istic of a horn loudspeaker employing the horn of Fig. 5.10D
with the dimensions multiplied by one-half and driven by a diaphragm
and an aluminum voice coil having a mass ratio of two to one operating
in a field of 22,000 gausses. C. Same as B except that the horn dimen­
sions of Fig. 5.10D are multiplied by two.
ciency frequency characteristic of the multiple flare horn described in Sec.
5.30 coupled to a diaphragm and coil having a mass ratio of 2 operating in
a field of 22,000 gausses is shown in Fig. 7.14. This efficiency frequency
characteristic is only a few per cent below the ultimate efficiency frequency
characteristic obtained from the enevelope of the family of characteristics
shown in Fig. 7.3.
10
80
~
z
tj40
a:
:>,20
....
,
" 'A
to .... B
~
~I 0
8
t
...z
4
.....,
2
...Q
I
20
,
~
100
rREQUENCY
IN
1000
CYCLES PER
10000
\
SECOND
FIG. 7.14. Efficiency frequency characteristic of a diaphragm coupled
to the horn of Fig. 5.13 and driven by an aluminum voice coil of one­
half the diaphragm mass in a field of 22,000 gausses. A. Without air
chamber. B. With air chamber.
232
ACOUSTICAL ENGINEERING
The two preceding horn loudspeakers are suitable for high quality repro­
duction of speech and music. For certain types of announce installations
it is desirable to project intelligible speech over very great distances (1
~~,.,
I
M2
MECHANICAL
NETWORK
100
80
/"
/'
60
1,\
\
>-
~ 40
V
'" 30
U
V
lo.
lo.
W 20
CROSS-SECTIONAL VIEW
/
/
10
400
\
\
\
1000
FREQUENCY
4000
7.15. Cross-sectional view and mechanical circuit of a loudspeaker of
2 degrees of freedom. In the mechanical network: ml = the mass of the
diaphragm and voice coil. C Ml = the compliance of the diaphragm
suspension system. C M ! = the compliance of the air chamber. rMl =
the mechanical resistance at the throat of the horn. 1M = the driving
force. The graph shows the efficiency frequency characteristic.
FIG.
HIGH fREQUENCY UNIT
LOW fREQUENCY UNIT
CROSS SECTION
LOW FREQUENCY UNIT
H.F.
~
TO
L.f.
WIRING
AMPLifiER
DIAGRAM
ASSEMBLY
FIG. 7.16. A two-channel, theater loudspeaker system
consisting of a folded low-frequency horn unit and a
multicellular horn high-frequency unit. The wiring
diagram shows the electrical filter used to allocate the
power, as a function of the frequency, to the two units.
to 2 miles) under all manner of conditions. This requires acoustical out­
puts of the order of from 500 to 1000 watts. The characteristics of Fig.
7.12 show that it is not practical to build a horn loudspeaker of this capacity
for the reproduction of the lower frequencies. A cross-sectional view of a
high power announce loudspeaker and the simplified mechanical network
is shown in Fig. 7.15. The mechanical network shows a system of two
degrees of freedom. The compliance of the suspension system and the
HORN LOUDSPEAKERS
233
compliance of the air chamber are chosen so that very high efficiency is
obtained over the range required for intelligible speech. A typical efficiency
frequency characteristic of this type of loudspeaker suitable for acoustical
outputs of 500 to 1000 watts is shown in Fig. 7.15. Due to the large audio­
power amplifier requirements, high loudspeaker efficiency is an extremely
important economic factor.
B. Multiple-Horn, Multiple-Channel System.-The two-channel or
"two-way" system, 27 ,28 is the most common example of a multichannel
system. This loudspeaker, Fig. 7.16, consists of a low-frequency folded
~c~
~\. ~
'"
VOICE
COIL
CIRCUIT
MECHANICAL
NETWORK
A
X
SEcnON
A-X
FRONT
VIEW
7.17. Combination horn and phase inverter low-fre­
quency loudspeaker. In the voice coil circuit, e = the
internal voltage of the electrical generator. rEG = the
electrical resistance of the electrical generator. L and rED =
the damped inductance and electrical resistance of the voice
coil. ZEM = the electrical motional impedance. In the
mechanical network: ml = the mass of the diaphragm.
C Jll = the compliance of the diaphragm suspension system.
ZMH = the mechanical impedance at the throat of the horn.
CM2 = the compliance of the air chamber. m2 and rM2 = the
mass and mechanical resistance of the port opening. 1M =
the driving force.
FIG.
horn unit for reproduction from 40 to 300 cycles and a multicellular horn
unit for reproduction from 300 to 8000 cycles.
In order to minimize time delay and phase distortion due to a large path
length difference between the low- and high-frequency horns, the effective
length of the low- and high-frequency horns must be practically the same.
The difference in path length in the system shown in Fig. 7.16 is made
relatively small by employing a short folded horn coupled to a large diameter
dynamic speaker mechanism. A further reduction in path length between
27
28
Wente and Thuras, Jour. A.I.E.E., Vol. 53, No. 1. p. 17, 1934.
Hilliard, J. K., Tech. Bul. Acad. Res. Coun., March, 1936.
234
ACOUSTICAL ENGINEERING
a short, straight axis high-frequency horn may be obtained by shifting the
high-frequency unit backwards.
The high-frequency horn consists of a cluster of relatively small horns
coupled to a common throat, Fig. 7.16. The directional characteristics ofthis
type of loudspeaker were discussed in Sec. 2.20. Fig. 7.16 shows a 12-cell
high-frequency unit. The throat is coupled to one or more mechanisms
depending upon the power requirements.
An electric filter or dividing network is used to allocate the power to the
HIGH
FREQUENCY
UNITS
LOW
FREQUENCY
UNIT
o
FRONT VIEW
FIG. 7.18. A two-channel, theater loudspeaker system consisting of
the combination of a straight axis horn and phase inverter, low­
frequency unit and a two-layer, straight-side horn, high-frequency
unit. (After Volkmann.)
high- and low-frequency units. The filter introduces phase shift as well as
a loss in power of 2 db or more.
The efficiency frequency characteristics of the high- and low-frequency
units of this loudspeaker without the filter are shown in Fig. 7.13, charac­
teristics B. and A.
The low-frequency loudspeaker in the system depicted in Fig. 7.16
employs a short folded horn. Although the horn is short, there is still a
path difference between the low- and high-frequency horns of about 1
wavelength at the overlap frequency of 300 cycles. The path difference
can be obviated by the use of a high- and low-frequency horn of the same
length. In order to conserve space the over-all depth must not be too
HORN LOUDSPEAKERS
235
great. Under these conditions the flare cutoff at the low-frequency horn
will be about 80 cycles. The radiation mechanical resistance can be increased
and the output in the frequency range below the flare cutoff maintained by
the use of a phase inverter system in combination with the horn as shown in
Fig. 7.17. The action of the system may be determined from the mechanical
network of Fig. 7.17. By a suitable choice of constants uniform response
may be maintained in the low-frequency range down to 40 cycles.
A theater loudspeaker system 29 employing a low-frequency loudspeaker
of the type shown in Fig. 7.17 is shown in Fig. 7.18. The low- and high-
FRONT VIEW
FIG. 7.19. A two-channel, theaterloudspeaker system consisting
of a straight-axis horn, low-frequency unit, and horn and lens
high-frequency units. (After Frayne and Locanthi.)
frequency horns are of the type with straight sides and, therefore, exhibit a
reasonably uniform directivity pattern in the horizontal plane. See Sec.
2.19C. This design makes it possible to obtain the directional characteristics
of cellular horns without the complex construction. The low- and high­
frequency horns are of the same length which obviates the transient distor­
tion inherent in two channel systems in which there is a path length difference
between the low and high frequency. This feature also simplifies the
problem of obtaining uniform directivity in the overlap frequency region.
29
Volkmann,
J., Unpublished Report.
236
ACOUSTICAL ENGINEERING
A theater loudspeaker system 30 employing a straight axis, low-frequency
horn and high-frequency horns equipped with diverging acoustic lenses is
shown in Fig. 7.19. The throat area of the low-frequency horn is the same
as the total area of the four cone loudspeaker units. The high-frequency
horns are equipped with slant plate-type diverging lenses which broadens
the directivity pattern. In addition, by suitable design of the lens and the
horn it is possible to obtain reasonably uniform coverage in the theater.
A folded horn loudspeaker31 for operation in the corner of the room for the
reproduction of the low-frequency range is shown in Fig. 7.20. The horn
A
/
-t: \ '
\
- __
\ "-L
\ \
\,
/
I,
___...Y
\\
1/
1/
l'
'\~
,:'
~'v"-;:' -"-"''-.'f.~
B­
-~+~- -
'~'
\
'/
'/
""
I
,.........."':..-:.,
;'"
\oJ
z
\\
\\
a::
o
u
\\
\\\\
I
.t.
Z
:::;
a::
\\
II
1/
II
\oJ
B'
\\
\\
---7' \
, /:'"--­
\
, "
~ I /
I
A'
FRONT VIEW
FIG. 7.20. Perspective and sectional views of a folded horn
for operation in the corner of the room. (After Klipsch.)
30
31
Frayne and Locanthi. Jour. Soc. Mot. Pic. Tel. Eng.• Vol. 63. No.3, p. 82, 1954.
Klipsch. Paul W .• Trans. IRE. Prof. Group on Audio. Vol. AU-I. No.3. p. 16. 1953.
237
HORN LOUDSPEAKERS
is of the folded type. The acoustical radiation resistance presented to a
loudspeaker operating in the corner of a room at the intersection of the
floor and two walls is two times the acoustical radiation resistance presented
to a loudspeaker operating along a wall of a room at the intersection of the
wall and floor. See Sec. 2.2. This increased radiation can only be realized
at the very low-frequency portion of the frequency range.
Some wide frequency range systems divide the frequency range into
three sections, namely, low, mid, and high frequency. Direct radiator,
dynamic and electrostatic, and horn loudspeaker units are used to cover the
high-frequency range. A high-frequency horn loudspeaker for the frequency
range above 10,000 is shown in Fig. 7.21. The diaphragms are of the order
VOICE COIL
MAGNET
FIG.7.21.
Sectional view of a high-frequency horn loudspeaker.
of i inch in diameter. The horns are about two inches in length. The
mouth area of the horn is about three square inches.
C. Compound Horn Loudspeaker. 32-The compound horn loudspeaker
consists of a single diaphragm mechanism with one side of the diaphragm
coupled to a straight axis horn and the other side coupled to a long folded
hom, Fig. 7.22. The equivalent of the system is shown in Fig. 7.22. The
functional acoustical network of the vibrating system is also shown in Fig.
7.22. At the low frequencies the acoustical reactance of the acoustical
capacitance, CA2, is large compared to the throat acoustical impedance,
ZA2, of the low-frequency horn and sound radiation issues from the low­
frequency horn. At the high frequencies the acoustical reactance of the
acoustical capacitance, C A2, is small compared to the acoustical impedances,
ZAl and ZA2, and, therefore, shuts out the low-frequency horn and radiation
issues from the high-frequency horn. In the mid-range, radiation issues
from both horns. The response frequency characteristic, Fig. 7.22, shows
the response range of the two horns. The throats of the two horns may be
chosen so that the efficiency characteristic of this loudspeaker will be the
same as that of the two-channel system discussed in the preceding section.
However, the power handling capacity is somewhat smaller because the
size of the diaphragm must be a compromise between high-and low-fre­
quency requirements.
32
Olson and Massa, Jour. Acous. Soc. Amer., Vol. 8, No. 1, p. 48, 1936.
238
ACOUSTICAL ENGINEERING
SECTION B-B
SYSTEM
WITH
ACOUSTICAL
f:lltlNl
40
FRONT
VIEW
SECTION A-A
NETWORK
100
1000
FREQUENCY
10000
FIG. 7.22. Cross-sectional view of a compound horn loudspeaker, the developed equiv­
alent of the high- and low-frequency horns, and the acoustical network of the acoustical
system. In the acoustical network: M = the inertance of the diaphragm. GAl = the
acoustical capacitance of the diaphragm suspension system. ZAI = the acoustical
impedance at the throat of the small horn. ZA2 = the acoustical impedance at the
throat of the large horn. GA 2 = the acoustical capacitance of the chamber behind
the diaphragm. P = the driving pressure. p = Bli/A. B = the fiux density. I = the
length of the conductor in the voice coil. i = the current in the voice coil. A = the
area of the diaphragm. The sections A-A and B-B refer to the horizontal and vertical
cross sections of the front view. The graph shows the frequency ranges of the high­
frequency and low-frequency horns and the over-all pressure response frequency charac­
teristic.
D. Multiple-Horn, Single-Channel System.-The multiple-horn, single­
channel system consists of a large number of multiple flare horns, each
driven by a diaphragm, Fig. 7.23. A comparison of the efficiency charac­
teristics of a multiflare horn loudspeaker, Fig. 7.23, with a multichannel
system, Fig. 7.23, shows that the efficiencies are of the same order. The
multiple-horn, single-channel system eliminates many of the following
disadvantages of the multichannel system: the phase difference due to the
difference in path length between the two channels, the phase difference
and power loss in the filters and dividing network, the nonuniform direc­
tional characteristics due to the small size of the high-frequency unit, and
the distortion in the relatively small throat of the high-frequency horn.
The space required for the single-channel system is greater than that for
the multichannel system. However, from a technical standpoint the
single-channel system is far superior to the multiple-channel system.
A multiple-horn, single-channel system loudspeaker suitable for high­
power announce systems is shown in Fig. 7.24. This loudspeaker performs
239
HORN LOUDSPEAKERS
the same function as the system shown in Fig. 7.15. The stresses in the
diaphragm and the voice coil system are reduced by the use of a number of
smaller units as contrasted to a single large unit. The possibility of failure
of the system is reduced by the use of a multi-unit driving system. The use
DIAPHRAGM
! {i'"
HORN
TO
AMPLIFIER
WIRING
DIAGRAM
FIG. 7.23.
A multiple-horn, single-channel, wide frequency
range, loudspeaker system consisting of a cluster of multifiare
horns, each coupled to a small diaphragm.
A
/J{
FRONT VIEW
SECTION A-/J{
FIG. 7.24. A multiple-horn, single-channel system for high-power
announce systems.
of a multiple-horn system makes it possible to obtain a greater variety of
directional patterns than is possible in the single-horn system of Fig. 7.15.
E. Horn Loudspeaker for Personal Radio Receivers. 33-The term personal
radio receiver is used to designate a complete radio receiver with self-con­
tained power supply, and of such physical dimensions that it can be easily
33
Olson, Bleazey, Preston, and Hackley, RCA Review, Vol. 11, No. 1, p. 80,1950.
240
ACOUSTICAL ENGINEERING
carried by hand or in the pocket. The performance and compactness of
personal radio receivers are limited by the efficiency with which electrical
power is converted into sound power by the loudspeaker. Since the electrical
power output is limited in the personal receiver, the efficiency of the
loudspeaker is an important factor. The specifications indicate the use
of a horn loudspeaker.
From the considerations in the preceding sections it appears that a
combination horn and phase inverter loudspeaker would be the logical
solution for a high-efficiency loudspeaker for personal receivers.
Perspective and sectional views and the mechanical network of a com­
bination horn and direct radiator loudspeaker for personal radio receivers
are shown in Fig. 7.25. The lid with two sides and the case form a horn
MECHANICAL NETWORK
PERSPECTIVE VIEW
SECTIONAL VIEW
7.25. Perspective and sectional views and mechanical network of a horn
loudspeaker for a personal radio receiver. In the mechanical network : ma = the
mass of the cone and voice coil. 1'MS and C MS = the mechanical resistance and
compliance of the suspension for the cone. C MV = the compliance of the air in
the case volume. ZMH = the quadripole representing the horn. mA and "'MAo =
the mass and mechanical resistance of the air load on the mouth of the horn.
mp and 1'MP = the mass and mechanical resistance of the air in the port and the air
load upon the port. 1m = the driving force developed in the voice coil. (After
Olson, Bleazey, and Preston.)
FIG.
when the lid is open. With the horn collapsed, that is, the lid placed against
the case, the dimensions are the same as those of a direct radiator loud­
speaker system. The experimentally determined response frequency
characteristic is shown in Fig. 7.26. The efficiency is of the order of 25
per cent. It is possible to obtain a sound level of 84 decibels at a distance
of three feet with 10 milliwatts.
F. Folded Horns.-There are innumerable ways of folding or curling a
horn. The different types of folded horns are shown in Figs. 7.16 and 7.22.
The principal purpose of folding or curling a horn is to use the volume
occupied by the horn more efficiently. Three more different types of folding
HORN LOUDSPEAKERS
241
are shown in Fig. 7.27. A simple folded horn is shown in Fig. 7.27A.
A folded horn with a ring-shaped mouth is shown in Fig. 7.27B. The
directional characteristics of a ring-shaped mouth are sharper than those
of the rectangular or circular shapes having equivalent areas (see Sees. 2.9
and 2.10). The horn shown in Fig. 7.27C is used for sending out radiation
o
,..
dI
c
-10
I
~
-15
-20
-25
,
/
~
gj
II:
""
/
I
~
....
l/1\
/
J
-30
200
400
600
1000
2000
4000
FREQUENCY IN CYCLES PER SECOND
FIG. 7.26. Response frequency characteristic of the per­
sonal radio loudspeaker shown in Fig. 7.25.
~{~}e
SIDE
VIEW
FRONT
VIEW
TOP VIEW
~\17~
BOTTOM VIEW
SECTION
A-A'
B
A
FIG. 7.27.
SECTION
A-A
C
Folded horns.
over 360 0 normal to the axis. It is customary to mount this loudspeaker
on a pole.
The high-frequency response is usually attenuated in a folded horn due
to destructive interference incurred by the different path lengths of the
sounds traversing the bends. In order to eliminate destructive interference
the same phase should exist over any plane normal to the axis. This con­
dition is practically satisfied providing the radial dimensions at any bend
242
ACOUSTICAL ENGINEERING
are a fraction of the wavelength. Wide range reproduction of sound requires
a large-mouth horn for efficient reproduction of low-frequency sounds and
small dimensions at the bends of a folded horn for efficient reproduction of
high-frequency sounds. Obviously, it is practically impossible to incor­
porate both of these features into a single folded horn. It is true that
folded horns have been used for years, but, in general, the response at either
or both the low- or high-frequency ranges has been attenuated.
G. Horn Loudspeaker Mechanisms.-The diaphragm, voice coil, magnet
structure, and air chamber of a horn loudspeaker mechanism may be built
in a wide variety of ways. The variations in path length from any part of
the diaphragm to the horn throat should be less than a quarter wavelength
in order to eliminate destructive interference in the air chamber. Several
A
B
c
7.28. Horn loudspeaker driving mechanisms. Mechanisms A, B, C, and
D depict various types of air chambers and diaphragms for coupling to a small
throat horn. Mechanism E depicts a large diaphragm coupled to a large
throat horn.
FIG.
different methods for reducing ihterference in the air chamber are shown in
Fig. 7.28A, B, C, and D and Figs. 7.1, 7.9, 7.15, and 7.21. These expedients
are necessary for efficient reproduction at the high-frequency portion of
the audio range where the wavelength is relatively small. For the low­
frequency portion of the audio-frequency range a large-throat horn may be
coupled to a large diaphragm, as shown in Fig. 7.28E, without incurring
any loss due to interference, notwithstanding the large size, because the
dimensions are small compared to the wavelength.
H. Diaphragms and Voice Coils.-The diaphragms or cones of horn
loudspeaker mechanisms are made of aluminum alloys, molded bakelite
with various bases, molded styrol, fiber, paper, and felted paper. Typical
diaphragm shapes are shown in Figs. 7.1, 7.9, 7.10, 7.15, 7.23, 7.24 and 7.28.
HORN LOUDSPEAKERS
243
Round, square, and ribbon wire voice coil conductors are used as shown in
Fig. 6.77.
1. Field Structures.-Permanent magnet and electromagnetic field
structures used in horn loudspeaker mechanisms are shown in Figs. 7.1,
7.9, 7.10, 7.15, 7.23, 7.24, 7.28, and 6.79. In general, it is customary to
use higher flux densities in the gap in horn loudspeakers than in direct
radiator loudspeakers. Soft iron may be used for the pole tips for flux
densities up to 20,000 gausses (See Fig. 6.82). For flux densities from
20,000 to 23,000 gausses, a special alloy, Permandur 34 (see Fig. 6.82), is
employed for the pole tip material in order to obtain these high densities
with tolerable efficiency.
J. Horn Walls. Vibration and Absorption. 35-In the theoretical analysis
carried out in this chapter it has been assumed that the horn walls are
rigid and nonabsorbing. In the case of certain materials such as wood,
paper, and fiber the absorption of sound by walls of the horn may introduce
an attenuation of several decibels. The absorption may be reduced by the
application of lacquers and varnishes. The attenuation in metallic horns
due to dissipation is negligible. The vibration of the walls of the horn
distorts the response frequency characteristic and introduces "hangover"
and reverberation. The response to transients is usually poor when the
walls of the horn vibrate. This vibration may be reduced by increasing
the thickness of the walls and by suitable bracing. The vibrations and ring
in metallic horns may be reduced by coating the outside of the horn with
deadening material such as asphalt or pitch compounds.
7.5. Throttled Air Flow Loudspeaker.-A throttled air flow loud­
speaker consists of a valve mechanism, actuated by the electrical signal,
which modulates a steady air stream so that the undulations in the throttled
air stream correspond to the variations in the electrical input, see Fig. 7.29.
The throttled air stream is usually coupled to a horn to improve the efficiency
of the system. In order to obtain a constant relationship between the
electrical input and the acoustical output as a function of the frequency,
the ratio of the volume current to the applied current must be independent
of the frequency. This means that the ratio of the amplitude of the valve
to the current must be independent of the frequency. This is, in general,
difficult to accomplish in the high-frequency region in view of the fact that
the valve mechanism must be stiffness controlled. This in turn means that
the mass of the valve must be small and at the same time be sufficiently
rugged to withstand the steady air pressure. The outstanding advantage
of this system is the large acoustical output which can be obtained for a
small electrical input. Efficiencies of more than 100 per cent can be realized
if the ratio of acoustical output to the electrical input is considered. How­
ever, in addition to the electrical power which must be supplied to the
throttling mechanism, there is the power that must be furnished in
supplying the steady stream of air. If this mechanical power is added
34
35
Elmen, G. W., Bell Syst. Tech. Jour., Vol. 15, No. 1. p. 113, 1936.
Phelps, W. D., Jour. Acous. Soc. Amer., Vol. 12, No. 1, p. 68, 1940.
244
ACOUSTICAL ENGINEERING
to electrical input, the efficiency will, of course, be considerably less than
100 per cent.
7.6. Ionophone Loudspeaker. 36-The ionophone loudspeaker consists
of an audio-modulated corona discharge coupled to a hom. Fig. 7.30.
The corona is produced in a specially designed quartz envelope. The corona
HORN
VALVE
AIR INLET
FIG. 7.29. Sectional view of a throttled air flow
loudspeaker.
CYLINDRICAL
SHIELD
FIG. 7.30. Schematic and sectional view of an audio modulated corona dis­
charge loudspeaker. (After Klein.)
is maintained by a radio-frequency power from a high power voltage ampli­
fier. One terminal of the amplifier is coupled to the platinum electrode
located in the quartz envelope and the other terminal is connected to the
cylinder around the quartz envelope. An audio-modulated high-frequency
signal is coupled to a radio-frequency oscillator and an audio signal. The
36
Klein, S., Acustika, Vol. 4, No. 1, p. 77, 1954.
HORN LOUDSPEAKERS
245
intensity of the corona varies in accordance with the amplitude of the
audio signal with the result that the air expands and contracts in correspond­
ing manner. A sound wave is thus produced in the throat of the horn.
Ihe sound generator is of the constant amplitude type, that is, for constant
electrical input the amplitude is independent of the frequency. When this
type of generator is coupled to a horn, the output will be proportional to the
frequency. For practical systems, the amplitude is limited. This limita­
tion restricts the use of this loudspeaker to the upper portion of the audio­
frequency range.
8
MICROPHONES
8.1. Introduction.-A microphone is an electro acoustic transducer
actuated by energy in an acoustical system and delivering energy to an
electrical system, the wave form in the electrical system being substantially
equivalent to that in the acoustical system. A pressure microphone is a
microphone in which the electrical response is caused by variations in pre­
sure in the actuating sound wave. A velocity microphone is a microphone
in which the electrical response corresponds to the particle velocity result­
ing from the propagation of a sound wave through an acoustical medium.
All microphones in use today may be classified as follows: pressure, velocity,
or a combination pressure and velocity. For the conversion of the acoustical
variations into the corresponding electrical variations the following trans­
ducers may be used: carbon, magnetic, dynamic, condenser, crystal, magneto­
strictive, electronic, and hot wire.
Microphones may also be classified as directional or nondirectional.
The particular configuration of the acoustical elements which constitute
the vibrating system determines the directional properties of the micro­
phone. It is the purpose of this chapter to consider the microphones in
most common use today from the standpoint of the above classifications.
8.2. Pressure Microphones.-A. Carbon Microphones.-A carbon mi­
crophone is a microphone which depends for its operation on the variation
in resistance of carbon contacts. The high sensitivity of this microphone
is due to the relay action of the carbon contacts. The carbon microphone
is almost universally employed in telephonic communications where the
prime requisite is sensitivity rather than uniform response over a wide
frequency range. For high-quality reproduction the distortion may be
reduced by employing two buttons in a push-pull arrangement. It is the
purpose of this section to consider single- and double-button carbon micro­
phones.
1. Single-Button Carbon Microphone.-A typical carbon microphone
is shown in Fig. 8.1. The carbon button consists of a cylindrical cavity
filled with carbon granules. The carbon granules are usually made from
anthracite coal. The carbon granules make contact with the diaphragm
and the cylindrical cup. Suitable washers are used to prevent leakage of
the carbon granules between the diaphragm and carbon cup without im­
246
247
MICROPHONES
peding the motion of the diaphragm. A displacement of the diaphragm
produces a change in the pressure between the carbon granules which
changes the electrical resistance from granule to granule. The net result
is a change in the electrical resistance between the diaphragm and the carbon
cup. For small displacements the change in resistance is proportional to
C AZ
C A3
~~tL
~
CA ,
AH
ACOUSTICAL
CM y
NETWORK
MICROPHONE~rE2
ELECTRICAL CIRCUIT
-20
M,
III
fA'
°-30
/\
'"z-40
III
V
0
Q.
:3-50
'"
CROSS - SECTIONAL
VIEW
£
1000
\v ~
10000
FREQUENCY
FIG. 8.1.
Cross-sectional view, the electrical circuit and the acoustical network
of a single-button carbon microphone. In the electrical circuit, rEi = the elec­
trical resistance of the carbon element, rE2 = the electrical resistance of the load,
and e = the polarizing voltage of the battery. In the acoustical network
Ml and 1'.41 = the inertance and acoustical resistance at mouthpiece opening.
ZAH = the acoustical quadripole representing the horn or mouthpiece.
M2 and
1'A2 = the inertance and acoustical resistance of the holes in the mouthpiece.
CAl = the acoustical capacitance of the air chamber in front of the diaphragm.
Ms, 1'A3, and C A 2 = the inertance, acoustical resistance, and acoustical capaci­
tance of the diaphragm. M4, 1'A4, and CAS = the inertance, acoustical resistance,
and acoustical capacitance of the carbon element. CA 4 = the acoustical capaci­
tance of the case. p = sound pressure. The graph shows the open circuit
voltage response frequency characteristic for constant sound pressure in free
space.
the displacement. Consider the electrical circuit of Fig. 8.1, for sinusoidal
motion of the diaphragm, the current, in amperes, in the circuit is given by
.
e
t=
rEO
+ hx sin wt
where e = voltage of the battery, in volts,
rEO = total electrical resistance of the circuit when x = 0, in ohms,
x = amplitude of the diaphragm, in centimeters,
h = constant of the carbon element, in ohms per centimeter,
w = 27Tj, and
f = frequency, in cycles per second.
8.1
248
ACOUSTICAL ENGINEERING
Equation 8.1 may be expanded as follows,
. = -e
L
rEO
(1 -
2 2
• 2
-hx.sm wt + h- -x2 sm
wt ... )
rEO
e ( 1 - -hx.
= sm wt
rEO
rEO
rEO
2x 2
+ -hrEO
22 -
h 2x-2 cos 2wt ... )
2
2
rEO
8.2
Equation 8.2 shows that there is a steady direct current, an alternating
current of the frequency of the diaphragm vibration and harmonics of
this vibration. For a limited frequency range of speech reproduction, the
nonlinear distortion is not particularly objectionable.
The acoustical network of the acoustical system is shown in Fig. 8.l.
The mouthpiece is a short exponential horn and is represented as an acoustical
quadripole, ZAH (see Sec. 5.27). The performance of the system may be
obtained from the acoustical circuit.
The diaphragm of the microphone is a circular plate supported at the
edge (see Sec. 3.5). The effective mass and effective area of the diaphragm
is one-third the total mass and total area of the diaphragm. Below the
fundamental resonant frequency the acoustical capacitance of the diaphragm
CA 2, is the controlling acoustical impedance. Under these conditions the
displacement is proportional to the pressure. Since the change in electrical
resistance of the carbon button and the resultant developed voltage is
proportional to the amplitude, the output for constant sound pressure will be
independent of the frequency below the fundamental resonant frequency of
the system. These observations are supported by the response frequency
characteristic of Fig. 8.1 which depicts uniform response in the low-frequency
range below the fundamental resonant frequency of the system. In the
region of resonance the output is accentuated. In the frequency range
above the fundamental resonant frequency the response falls off rapidly
in a series of peaks which are due to the higher modes of the diaphragm
and the acoustical system.
An improved type of single-button carbon microphone l has been de­
veloped in which the response is quite uniform over a wide frequency range
(Fig. 8.2). The conical diaphragm is made of a thin aluminum alloy. At
low frequencies the diaphragm vibrates as a single unit. However, at the
higher frequencies it is necessary to consider it to be made up of three sepa­
rate masses. These masses consist of the central portion m5, the ribbed
intermediate portion m2, and the outer portion m4. The central portion
includes the mass of the movable electrode and is coupled to the ribbed
portion by the compliance, CM6 , which in turn is coupled to the outer portion
by the compliance, CM2. The paper books which support the edge of the
diaphragm have a compliance, CM4, and a mechanical resistance, rM4.
Their mass is included in the outer portion of the diaphragm, m4. The
internal mechanical resistance of the portions which form the coupling
compliances, CM2 and C M6 , are represented by rM2 and rM6, respectively.
1
Jones,
w.
C., Jour. A.I.E.E., Vol. 57, No. 10, p. 559, 1939.
MICROPHONES
249
A hole is provided in the diaphragm to permit rapid equalization of low­
frequency pressures of high intensity and prevent damage to the diaphragm
and other parts. The mass and the mechanical resistance of this hole,
rn3 and rM3, are so chosen that their effects on the response are confined to
frequencies below 300 cycles. The controlling compliance, CM 3, is that of
the cavity between the diaphragm and the die-cast frame. The carbon
MECHANICAL
NETWORK
0
'"
~"'40
.
,
o
:3"'50
a:
-60
CROSS-SECTIONAL VIEW
:
100
, '1""
-­
B
A
,'/
'"
1000
FREQUENCY
~
i'".J\
,
10000
FIG. 8.2.
Cross-sectional view and the mechanical network of an improved
single-button carbon microphone. The electrical circuit is the same as that
of Fig. 8.1. In the mechanical network, mo and rMO = the mass and mechani­
cal resistance of the holes in the outer grill. C MO = compliance of the air
chamber between the grill and the membrane. ml and YMI = the mass and
mechanical resistance of the waterproof membrane. C Ml = the compliance
of the air chamber between the membrane and the diaphragm. m2, rM2, and
CM2 = the mass, mechanical resistance, and compliance of the central and
outer portion of the diaphragm. ma and r M3 = the mass and mechanical
resistance of the hole in the diaphragm. CM3 = the compliance of the air
chamber behind the diaphragm. C M4 and YM4 = the compliance and mechani­
cal resistance of the paper book suspension. m4 = the mass of the paper book
suspension and the outer part of the diaphragm. ms, YMS, and CMS = the
mass, mechanical resistance, and compliance of the center portion of the
diaphragm and the carbon granules. C M6 and rM6 = the compliance and
mechanical resistance of the center of the diaphragm. JM = the driving
force. JM = pA. A = the area of the diaphragm. p = the sound pressure.
The open-circuit voltage response characteristics are shown in the graph.
A. Response in free space. B. Response for constant sound pressure at the
diaphragm. Dots computed from the mechanical network.
granules are represented by a compliance, CMS, and a mechanical resistance,
rM5.
The mass of the carbon granules is lumped with that of the central
portion of the diaphragm. The holes in the inner grid are sufficiently
large so that there is no reaction upon the response. The holes in the outer
grill add the mass, mo, and the mechanical resistance, rMO. These holes
are coupled to a moisture-proof membrane of mass, mI, and mechanical
250
ACOUSTICAL ENGINEERING
resistance, rMI, by means of the compliance, CMO , of the enclosed cavity.
The cavity compliance, CMI , couples the membrane to the diaphragm.
The response of this microphone computed from the mechanical network
is shown in Fig. 8.2. The response for constant sound pressure on the
diaphragm is also shown in Fig. 8.2. It will be seen that the agreement
between the computed and measured characteristics is very good and sub­
stantiates this type of analysis. The response is very much smoother
than in the case of the plate or disk type of diaphragm.
MECHANICAL NETWORK
.,
-zo
0-30
\oJ
'"z-40
!?
'"~-50
-60
100
CROSS-SECTIONAL VIEW
,./........ ..... i\.
~
1000
FREQUENCY
.0000
FIG. 8.3. Cross-sectional view and the mechanical network of a new type single-button
carbon microphone. The electrical circuit is the same as that of Fig. 8.1. In the
mechanical network, mo and t'MO = the mass and mechanical resistance of the holes in
the outer grill. CMO = the compliance of the air chamber between the inner and outer
grids. ml and t'Ml = the mass and mechanical resistance of the waterproof membrane.
CMl = the compliance of the air chamber behind the diaphragm. C M3 = the compli­
ance of the air chamber behind the waterproof membrane and the diaphragm. m2 and
t'M2 = the mass and mechanical resistance of the cloth.
m3 and t'M3 = the mass and
mechanical resistance of the hole in the diaphragm. m5 = the mass of the diaphragm.
1'M4 and C M4 = the mechanical resistance and compliance of the diaphragm to carbon
cup couples. C M2 = the compliance of the air chamber behind the microphone unit.
1'M5 and C M5 = the mechanical resistance and compliance of the diaphragm suspension
system. m6, 1'M6, and CM6 = the mass, mechanical resistance, and compliance of the
carbon cup and granules. 1M = the driving force. 1M = pA. P = the sound pressure.
A = the area of the diaphragm. The graph shows the open circuit voltage response
frequency characteristic for constant sound pressure on the diaphragm.
The free-space response shown in Fig. 8.2 indicates the diffraction effect
of the microphone as an obstacle in increasing the pressure on the diaphragm
(seeSec.1.11 and Fig. 1.5).
In addition to the smoother response the sensitivity of the new unit is
higher because of the reduction in mass of the vibrating system. Due to
the shape of the carbon chamber the performance of the microphone is less
affected by angular position.
251
MICROPHONES
A new carbon microphone,2 shown in Fig. 8.3, has been developed which
in appearance is similar to the microphone of Fig. 8.2. However, there are
many important features which lead to improved performance. One of the
most important elements that has been added to the microphone of Fig. 8.2
is the mechanical resistance, rM2. The vibrating system is stiffness controlled
below the resonant frequency. At the resonant frequency the mechanical
resistance provides the controlling element. As a consequence, the response
in the frequency region 2000 and 4000 cycles, depicted in Fig. 8.3, is smoother
than that of the microphone of Fig. 8.2. The sensitivity of the microphone
of Fig. 8.3 is somewhat greater than the microphone of Fig. 8.2. This has
been accomplished by an improved design of the carbon cup.
m.~~~
~r-------,J
"'.,
MECHANICAL
NETWORK
MO<'O~B,'j~C
ELECTRICAL
SYSTEM
;~mllilit
1 fIllflllEflI
,~ .~
'00
f'REQUENCY
CROSS-SECTIONAL VIEW
FIG. 8.4. Cross-sectional view. the electrical circuit. and the mechanical
circuit of a double-button. stretched diaphragm, carbon microphone. In the
mechanical circuit. ml and rMI = the mass and mechanical resistance of the
air load. ZMH = the mechanical quadripole representing the cylindrical
cavity or pipe. m2 and CMI = the mass and compliance of the diaphragm.
ma, rM2, and CM2 = the mass, mechanical resistance, and compliance of the
carbon granules. m4. rM3, and C M3 = the mass, mechanical resistance. and
compliance due to the damping plate. JM = the driving force. JM = pA.
A = the area of the diaphragm. p = the sound pressure. The graph shows
the open-circuit voltage response-frequency characteristic for constant sound
pressure at the diaphragm.
2. Double-Button Carbon Microphone.-For applications requmng both
high quality and large power output the single-button carbon microphone
is not suitable due to the large nonlinear distortion. Uniform response
and low distortion may be obtained in a carbon microphone 3 by means of
a system consisting of a stretched diaphragm and two carbon buttons as
shown in Fig. 8.4. The performance of the system may be obtained from
a consideration of the mechanical network of the vibrating system. The
2
3
Inglis and Tuffnel. Bell Syst. Tech. Jour., Vol. 30, No.2. p. 209. 1951.
Jones. W. C.• Bell Syst. Tech. Jour., Vol. 10, No. 1, p. 46. 1931.
252
ACOUSTICAL ENGINEERING
mechanical impedance of a stretched diaphragm, below its resonant fre­
quency, is a stiffness mechanical reactance. Therefore, a constant sound
pressure on the diaphragm will produce substantially constant displace­
ment. Since the change in electrical resistance of the carbon buttons and
the resultant developed voltage is proportional to the displacement, the
voltage output will be independent of the frequency. To provide damping
at the resonant frequency of the diaphragm the damping plate is placed
very close to the back of the diaphragm. As the diaphragm moves, air
is forced through this small space. The high viscosity loss in a small slit
provides the damping (see Sec. 5.4). In order to reduce the stiffness, in
the small space, suitable grooves are provided which reduce the length of
the slit. The rear button is enclosed in the damping plate while the front
button is supported by the bridge. The duraluminum diaphragm is gold­
plated over the area occupied by the carbon buttons to insure contact be­
tween the carbon granules and the diaphragm. The resonant frequency
of the stretched diaphragm is usually placed between 5000 and 8000 cycles
(see Sec. 3.4). In the absence of the damping plate the amplitude for a
constant force at the resonant frequency would be greater than that below
the resonant frequency. By means of the damping plate the amplitude
at the resonant frequency can be reduced to correspond to that of the
remainder of the range. A response frequency characteristic of this micro­
phone is shown in Fig. 8.4.
The electrical circuit diagram for this microphone is shown in Fig. 8.4.
For a sinusoidal motion of the diaphragm the current, in amperes, in one
of the buttons may be written as
.
Zl =
when
e
rEO
8.3
+ hx SIll. wt
e = voltage of the battery, in volts,
= electrical resistance of the circuit, when x = 0, in ohms,
x = amplitude of the diaphragm, in centimeters,
h = constant of the carbon element, in ohms per centimeter,
w = 27Tj, and
j = frequency, in cycles per second.
rEO
The current in the other button is
.
~2
=
e
rEO -
8.4
.
hx SIll wt
The difference between equations 8.3 and 8.4 after expanding is
.
Z2 -
3x 3 • 3
)
+ hrEO
--3 SIll wt ...
x3.
3h 3S
t
h 3x 3 . 3 t
)
=2e
- (hX sin wt + Illw---SIllw . . .
. = - 2e (hX
-
~l
rEO
rEO
. wt
SIll
rEO
rEO
4
rE0 3
4rE0 3
8.5
'
253
MICROPHONES
Comparing equation 8.5 with equation 8.2 shows that the large second
harmonic term has been eliminated by the use of a push-pull two-button
microphone.
One common cause of faulty operation of the carbon microphone is due
to the cohering of the carbon granules caused by the breaking of the circuit
when the current is flowing. The use of electric filters as shown in the
circuit diagram will protect the microphone against cohering.
The frequency range and response of the double-button carbon micro­
phone compares favorably with the condenser microphone. The carbon
microphone is several times more sensitive than the condenser microphone.
However, the limitation is carbon noise.
~~'h
~
MECHANICAL
,,'"wo,
~
NETWORK
'uTu;rJ~ !'' 'I l$
ELECTRICAL
SYSTEM
CROSS-SECTIONAL VIEW
8.5. Cross-sectional view, electrical system, and mechanical circuit of a
condenser microphone. In the electrical system: eo = the polarizing voltage,
YE = the polarizing electrical resistance, YEB = the bias electrical resistance.
CEO = the electrical capacitance of the microphone.
In the mechanical
circuit, ml and YMl = the mass and mechanical resistance of the air load.
SMH = the mechanical quadripole representing the cylindrical cavity or pipe.
m2 and CMl = the mass and compliance of the diaphragm.
YM2, ma, and
CM2 = the mass, mechanical resistance, and compliance of the air film. 1M =
the driving force. 1M = pA. A = the area of the diaphragm. p = the
sound pressure. The graph shows the open-circuit voltage response frequency
characteristics. A. Response for constant sound pressure on the diaphragm.
B. Response for constant sound pressure in free space.
FIG.
B. Condenser Microphone (Electrostatic Microphone).-A condenser micro­
phone, also termed an electrostatic microphone, is a microphone which
depends for its operation on variations in electrical capacitance. The
typical condenser microphone 4 consists of a thin stretched plate separated
from a parallel rigid plate (Fig. 8.5). The electrical system of this micro­
phone is shown in Fig. 8.5.
4
Wente, E. C., Pkys. Rev., Vol. 10, No. 1. p. 39, 1917.
254
ACOUSTICAL ENGINEERING
The electrical capacitance, in statfarads, at any instant is given by
CE = CEO + CEI sin wt
8.6
where CEO = electrical capacitance in the absence of an applied pressure,
in statfarads,
CEI = maximum change in the electrical capacitance due to the
external applied sinusoidal pressure, in statfarads,
w = 24, and
f = frequency, in cycles per second.
From the electrical circuit
1
eo - rEt. = CE
where eo
rE
i
f.d t
8.7
t
polarizing voltage, in statvolts,
electrical resistance of the polarizing resistor, in statohms,
= current, in statamperes, and
i = time, in seconds.
Equation 8.7 assumes that the bias resistor, rEB, and the input electrical
impedance of the vacuum tube is very large compared with rEo Then eo
may be considered to be in series with CEO and rEo Substituting the value
of CE from equation 8.7 in equation 8.6 and differentiating
(CEO
=
=
+ CEI sin wt)rE ~ + (1 + rECElw cos wt)i -
eOCElw cos wt
=
0 8.8
The solution of equation 8.8 is
t
.=
eoCEl
CEO"v(1/CEow)2
+ rE 2
. (t
sm
w
+ '/'1
-I. )
eoCEirE
CE02V[(1/C Eo w)2 + 4rE2] [(1/C Eo w)2
+ terms of higher order
where CPl
+ rE2]
sin (2wt
+ CPl -
CP2)
8.9
tan-1 1/CEOwrE and CP2 = tan-1 1/2CEOwrE.
For small diaphragm amplitudes, the generated voltage, in statvolts, is
=
,.
e = rEt =
CEO
J
eoCEl rO
. (
1
sm wt
___ + r02
CE0 2w 2
-I. )
+ ,/,1
8.10
Equation 8.10 shows that the condenser microphone 5 may be considered
as a generator with an internal open circuit voltage of
e=
eo(~;~) sin (wt + CPl), in statvolts,
and an internal electrical impedance of 1jC Eo w, in statohms.
5
Wente, E. C., Pkys. Rev., Vol. 19, No.5, p. 498,1922.
8.11
255
MICROPHONES
The mechanical network of the mechanical system of the condenser
microphone is shown in Fig. 8.5. The performance of the vibrating system
may be obtained from a consideration of the mechanical network. Equa­
tion 8.11 shows that the voltage is proportional to the amplitude. There­
fore, to obtain a microphone in which the sensitivity is independent of the
frequency, the amplitude, for a constant applied pressure, must be inde­
pendent of the frequency. In the range below the resonant frequency the
amplitude of a stretched membrane for a constant applied force is inde­
pendent of the frequency (see Sec. 3.4). The addition of the back plate
with very close spacing introduces mechanical resistance 6 ,7 due to the
viscosity loss in the narrow slit (see Sec. 5.4). This mechanical resistance
reduces the amplitude at the resonant frequency. The back plate also
introduces stiffness due to the entrapped air. This stiffness can be reduced
without reducing the mechanical resistance by cutting grooves in the back
of the plate. If the damping is made sufficiently large the amplitude at
the fundamental resonant frequency of the diaphragm can be made to
correspond to that of the remainder of the range.
The amplitude of the diaphragm, in centimeters, is given by
x
1M2
=
[rM2
1
1
1
+ jw(m2 + ma) + jw (CMl + CM2) ] jw
8.12
where 1M2 = applied force, in dynes,
1M2 =PA,
P = sound pressure on the diaphragm, in dynes per square centi­
meter,
A = area of the diaphragm, in square centimeters,
rM2 = damping mechanical resistance of air film, m mechanical
ohms,
m2 = effective mass of the diaphragm, in grams,
CMl = compliance due to stiffness of the diaphragm, in centimeters
per dyne,
ma = mass of air film, in grams,
CM2 = compliance due to stiffness of the air film, in centimeters per
dyne,
w = 27T1, and
1 = frequency, in cycles per second.
Equation 8.12 shows that the sensitivity below the resonant frequency
is inversely proportional to the stiffness and the mechanical resistance.
For the same fundamental resonant frequency the stiffness can be reduced
6
7
Crandall, I. B., Phys. Rev., Vol. 11, No.6, p. 449, 1918.
Crandall, " Vibrating Systems and Sound," D. Van Nostrand Company, Princeton,
N.j., 1926.
256
ACOUSTICAL ENGINEERING
by decreasing the mass. This procedure also reduces the amount of mechan­
ical resistance required to damp the fundamental resonance and thereby
obtain uniform response. Aluminum alloys, due to the low density and
high tensile strength, are the logical materials for use in diaphragms. The
minimum diaphragm thickness suitable for the manufacture of condenser
microphones is about .001 inch. The electrical capacitance of a microphone
with a diaphragm diameter of 1% inches and a spacing of from .001 to .002
inch is from 400 to 200 mmfds. Due to the high electrical impedance of this
capacitance it is necessary to locate the microphone near the vacuum tube
amplifier. The electrical capacitance of a long connecting cable reduces
the sensitivity without frequency discrimination because the internal
electrical impedance of the microphone is also an electrical capacitance.
The response frequency characteristics of a condenser microphone for
k\\....,.....,.,~--BACK PLATE
:tI:l"'n'7777a:::::::3lm'77--ri')'-)a\
\~iff--DIAPHRAGM TENSION CONTROL
AND BACK PLATE SUPPORT
CLAMPING
FIG. 8.6. Cross-sectional view of a miniature condenser microphone with a
stretched membrane-type diaphragm.
constant sound pressure on the diaphragm and for constant free wave
sound pressure are shown in Fig. 8.5.
The condenser microphone 8 ,9 shown in Fig. 8.5 employs a diaphragm with
a diameter of 1% inches. The over-all diameter of the condenser microphone
unit is about 3 inches. These microphones were developed about twenty­
five years ago and were employed in the early days of sound reproduction.
The condenser microphone was replaced by the electrodynamic (voice coil
and ribbon) and piezoelectric microphone. During the past decade, smaller
condenser microphones have been developed. A miniaturized version of the
microphone shown in Fig. 8.5 is shown in Fig. 8.6. The over-all diameter of
the microphone unit is a little less than 1 inch. The fundamental resonant
frequency of the diaphragm is about 9000 cycles. The system is highly
damped so that uniform response is maintained to over 15,000 cycles. The
8
9
Harrison and Flanders, Bell Syst. Tech. Jour. , Vol. 11, No.3, p. 451, 1932.
Veneklasen, Paul S., Jour. Acous. Soc. Amer., Vol. 20, No.6, p. 807,1948.
257
MICROPHONES
deviations in response are smooth and can be easily compensated by electrical
means to obtain a response frequency characteristic which is independent of
frequency. The amplifier which may be used with this microphone is
shown in Fig. 8.7. The cathode follower type of operation provides a
DIAPHRAGM
II LOUTPUT
250V
FIG. 8.7. Circuit diagram of a vacuum tube amplifier with a very
large input electrical impedance.
system having a high input electrical impedance. This is necessary for the
small condenser microphones in which the capacitance is only about SO
mmfds in order to maintain the response in the low-frequency region. See
equation 8.11. The condenser microphone shown in Fig. 8.6 is used as a
standard microphone in pressure calibration of laboratory standard micro­
phones. See Sec. 1O.2Ald.
Another miniature condenser microphone 1o is shown in Fig. 8.8. This
microphone employs a plate instead of a stretched diaphragm. See Sec.
CASE
~~~~~~~~~~~~~~
QUARTZ
DIAPHRAGM
FIG. 8.8. Cross-sectional view of a miniature condenser
microphone with plate-type diaphragm.
3.5. The over-all diameter of the microphone unit is about ! inch. The
amplifier used with these microphones is of the type shown in Fig. 8.7.
c. Piezoelectric (Crystal) Microphones.ll.12.13_A piezoelectric micro­
phone is a microphone which depends upon the generation of an electro­
motive force by the deformation of a crystal having piezoelectric properties.
10 Hilliard. J. K.. and Noble. J. J .. Trans. IRE. Prof. Group on Audio. Vol. AU-2.
No.6. p. 168. 1954.
11 Sawyer. C. B .• Proc. Inst. Rad. Eng.• Vol. 19. No. 11. p. 2020.1931.
12 Williams. A. L.. Jour. Soc. Mot. Pic. Eng.• Vol. 18. No.4. p. 196. 1934.
13 Nicolson. U.S. Patent 1.495.429.
258
ACOUSTICAL ENGINEERING
The voltage generated14 due to a deformation of the crystal is proportional
to the displacement. Therefore, to obtain a uniformly sensitive micro­
phone with respect to frequency the displacement for a constant applied
force must be independent of the frequency. Rochelle salt exhibits the
greatest piezoelectric activity of all of the known crystals. For this reason
it is used in audio-frequency microphones. There are two general classi­
fications of crystal microphones-namely, the direct actuated and the
m
r..
~"'I
:::n...
f..
I
ELECTRICAL
CIRCUIT
~
:or
MECHANICAL
CIRCUIT
BIMORPH
ELEMENTS
TERMINALS
TWISTER
BIMORPH
BASE-"~--'I
BENDER
CRYSTAL
BIMORPH
ELEMENTS
CRYSTAL
MICROPHONE
UNIT OR CELL
DIRECT
ACTUATED
CRYSTAL
MICROPHONE
DIAPHRAGM
ACTUATED CRYSTAL
MICROPHONE
FIG. 8.9. Crystal elements and sound cells.
A direct actuated crystal micro­
phone. In the electrical network, CEO = the electrical capacitance of the
crystal. "EO = the electrical resistance of the crystal. Z filL = the electrical
impedance of the load. eo = the open circuit voltage developed by the
crystal. In the mechallical circuit, m, "M, aud C M l = the mass, mechanical
resistance, and compliance of the crystal. C M2 = the compliance of one half
of the air chamber. 1M = the driving force, 1M = pA. A = the effective
area of the crystal. p = the sound pressure. A diaphragm actuated crystal
microphone. In the mechanical circuit, ml, "Mlo and C M1 = the mass,
mechanical resistance, and compliance of the diaphragm. m2, "M2, and
CM2 = the mass, mechanical resistance, and compliance of the crystal.
CM3 = the compliance due to the case volume. 1M = the driving force.
1M = pA. A = the effective area of the diaphragm. p = the sound pressure
at the diaphragm.
diaphragm actuated. In the direct actuated, the sound pressure acts
directly upon the crystal. In the diaphragm actuated, the sound pressure
acts upon a diaphragm which is coupled to a crystal. The crystal element,
Fig. 8.9, is made up of two crystals cut so that a voltage is generated when
forces are applied as shown. The two types of bimorph elements, namely,
"twisters" and "benders," are shown in Fig. 8.9. A bimorph construc­
tion has several advantages over the single crystal, as follows: it lends itself
to a more efficient size and shape; it becomes more sensitive (a gain of 15
times for practical shapes); it reduces the variations of the mechanical
14
1943
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.].,
MICROPHONES
259
and electrical constants of the crystal for changes in temperature. The
temperature limits of bimorph crystals are from -40° F. to 130° F. If
exposed to temperatures in excess of 130° F. the crystal loses its piezoelectric
activity permanently. The sensitivity or voltage output of the crystal
varies with temperature due primarily to a change in the capacitance and
in a lesser degree to a change in the developed voltage. An ADP crystal
with greater temperature and humidity ranges is described in Sec. 13.12.
1. Direct Actuated Crystal Microphone.-In the direct actuated crystal
microphone the sound pressure acts directly upon the crystal. A common
form of sound cell for a direct actuated crystal microphone consists of two
bimorph elements assembled as shown in Fig. 8.9. The cavity formed by
the two crystal elements is completely enclosed so that the application of
an external pressure causes a deformation of the crystal.
The internal voltage, e, developed by the crystal is
where K
x
=
=
e=Kx
8.13
constant of the crystal, and
effective amplitude of the deformation of the crystal by an
applied force.
From the mechanical circuit of Fig. 8.9, the amplitude, in centimeters, is
( +.
1M
1) .
1
+ JW
-:---C + -;---C JW
Ml
JW M2
x=~----------~~------~~
rM
where
rM =
m
CMl
CM2
1M
=
=
=
=
p=
A
=
W
=
1=
Jwm
8.14
effective mechanical resistance of the crystal, III mechanical
ohms,
effective mass of the crystal, in grams,
effective compliance of the crystal, in centimeters per dyne,
compliance·of one half of the air chamber between the crystals,
in centimeters per dyne,
pA, in dynes,
sound pressure at the surface of the crystal in dynes per square
centimeter,
area of the crystal, in square centimeters,
27T1, and
frequency, in cycles per second.
A consideration of equation 8.14 shows that the amplitude will be inde­
pendent of the frequency in the range below the resonant frequency. Under
these conditions the internal voltage developed by the crystal, as given by
equation 8.13, will be independent of the frequency. The resonant frequency
is placed beyond the desired response range of the microphone so that uni­
form response is obtained in the desired frequency range. Uniform response
to 17,000 cycles can be readily obtained.
260
ACOUSTICAL ENGINEERING
A typical direct actuated crystal microphone, shown in Fig. 8.9, consists
of four cells. The internal impedance of a single cell is relatively high.
This high impedance may be reduced by the use of several cells in parallel.
If the crystal element is small compared to the wavelength, the individual
element will be non directional.
2. Diaphragm Actuated Crystal Microphone.-In the diaphragm actuated
crystal microphone the sound pressure acts upon a diaphragm which in turn
drives a crystal. The output of the diaphragm actuated type is considerably
higher than the direct actuated type because the diaphragm acts as a coupling
unit between the relatively low impedance of the air and the high impedance
of the crystal. A cross-sectional view of a diaphragm actuated crystal
microphone is shown in Fig. 8.9. The response frequency characteristic
for constant sound pressure on the diaphragm may be obtained from the
mechanical circuit of Fig. 8.9 and equation 8.13. As shown in Fig. 1.5 the
ratio of the pressure on the face of a cylinder to that in free space increases
as the dimensions become comparable to the wavelength. This effect
accentuates the response in the high-frequency range.
A new, miniature crystal microphone has been developed in which the
diameter is Ii inches. The vibrating system is essentially the same as that
shown in Fig. 8.9 with the addition of a mechanical resistance placed over
the front of the diaphragm. The mechanical resistance controls the response
at the resonant frequency of the diaphragm and crystal combination and
thereby smooths out the response frequency characteristic in the high­
frequency range.
3. Diaphragm-Actuated Barium Titanate Microphones.1 5 ,16-Barium
titanate is a ceramic which exhibits properties similar to that of a piezo­
electric crysta1. Microphones employing barium titanate are constructed
in a manner similar to that of the diaphragm-type crystal microphone of
Fig. 8.9. A strip of barium titanate is used instead of the crystal. The
performance and electrical characteristics are essentially the same as that
of the crystal microphone except the sensitivity is somewhat lower. The
advantage of the barium titanate microphone is that it can be operated at
higher temperatures than Rochelle salt crystal microphones.
D. Moving Conductor Microphones.-A moving conductor microphone is
a microphone in which the output results from the motion of a conductor in
a magnetic field. The conductor may be in the form of a circular coil
which is termed a moving coil microphone or in the form of a straight
conductor which is termed an inductor microphone. These microphones
are also termed dynamic microphones.
1. Moving Coil Microphone (Dynamic Microphone).17,18_A cross-sectional
view of a moving coil microphone is shown in Fig. 8.10. The motion of
Medill, John, Trans. IRE, Prof. Group on Audio.• Vol. AU-l, No.6. p. 7. 1953.
Medill, John. Jour. Acous. Soc. Amer.• Vol. 25. No.5. p. 864.1953.
17 Wente and Thuras. Jour. Acous. Soc. Amer., Vol. 3. No.!, p. 44.1931.
18 Wigginton. L. M.• and Carroll. R. M.• Jour. Audio Eng. Soc .• Vol. 3, No.2. p. 77.
1955.
15
16
MICROPHONES
261
the diaphragm is transferred to a coil located in a magnetic field. The
mechanical circuit of the mechanical system consisting of the diaphragm
coil and suspension system is shown in Fig. 8.1OA.
The velocity, in centimeters per second, of the voice coil is given by
i = _ _ _..::..1M;;;;;;...._--:-_
rMl
where
rMl =
ml =
CM1
=
1M =
.
1
+ JWml
+ JW
-·-C-­
Ml
8.15
mechanical resistance of the suspension system, in mechanical
ohms,
mass of the diaphragm and voice coil, in grams,
compliance of the suspension system, in centimeters per dyne,
and
driving force, in dynes.
The generated internal voltage, in abvolts, is
e
where B
l
i
=
=
=
=
Bli
8.16
flux density in the air gap, in gausses,
length of the voice coil conductor, in centimeters, and
velocity of the voice coil, in centimeters per second.
Equation 8.16 shows that the microphone will be uniformly sensitive
with respect to frequency if the velocity is independent of the frequency.
The characteristics 1 and 2 in Fig. 8.1OA were computed by employing equa­
tion 8.15. These characteristics show that a uniformly sensitive dynamic
microphone, with respect to frequency, must be essentially "resistance
controlled. "
The characteristic marked 2. Fig. 8.1OA shows some falling off in velocity
at the high and low frequencies. This can be corrected by the use of some
additional elements (Fig. 8.lOB). The major portion of the mechanical
resistance is the silk cloth. m2rM2. Mechanical resistance in the case of
silk cloth is due to the high viscosity introduced by the small holes (see Sec.
5.5). Slits have also been used for the resistance element (see Sec. 5.4).
The mass mechanical reactance of the diaphragm is reduced at the higher
frequencies by the compliance. CM2. formed by the volume between the
silk and the diaphragm. The addition of the mechanical elements CM2.
rM2. and m2 changes the characteristic at the high frequencies from that
marked 3 to that marked 5. An increase in response over an octave is
obtained by the addition of these elements. A corresponding increase in
response can be obtained at the low frequencies by means of the case volume.
eM3. and the addition of a tube. marMa.
The mechanical network shows
the action of the additional elements in changing the response from the
characteristic 3 to the characteristic 4-5.
262
ACOUSTICAL ENGINEERING
The most common materials used for the diaphragms of pressure micro­
phones are aluminum alloys, Bakelite, styrol, and paper. In order to obtain
a minimum density-resistivity product, aluminum is almost universally
used for the voice coil (see Table 6.1). Both edgewise wound ribbon and
round wire have been used for the voice coil (see Sec. 6.27 and Fig. 6.76).
p
DIA~SION
Cw,Tw,
A
I
V
IY
GOIL
CROSS-SECTIONAL
I
VIEW
fN 5
C
I~
~
./
Z
'"
""­
~
-...;:::
MECHANICAL CIRCUIT
.oo~o
100
1000
10000
fREQUENCY
~
~J
ELECTRICAL CIRCUIT
I
B
CROSS - SECTIONAL VIEW
I
>­
!:
u
S
4
~
g.OI/( A
!oJ
>
.oo~o
100
1000
1
10000
fREQUENCY
MECHANICAL
FIG. 8 .10. A. Cross-sectional view and mechanical circuit of a diaphragm,
coil, and suspension. In the electrical circuit, rEG = the electrical resistance
of the coil. L = the inductance of the coil. ZEL = the electrical impedance
of the load. eG = the open circuit voltage developed by the coil. In the
mechanical circuit, ml, rMl. and CMI = the mass, mechanical resistance, and
compliance of the vibrating system. 1M = the driving force. 1M = pA.
A = the area of the diaphragm. P = the sound pressure. The velocity
frequency characteristic for a unit force and a mechanical resistance of 1
mechanical ohm is indicated as curve 1 on the graph. The same for a mechani­
cal resistance of 60 mechanical ohms. B. Cross-sectional view and mechanical
circuit of a dynamic microphone. In the mechanical network, ml, rMI, and
CMI = the mass, mechanical resistance, and compliance of the diaphragm ann
suspension. C M 2 = the compliance of the air chamber behind the diaphragm.
m2 and r M2 = the mass and mechanical resistance of the silk cloth.
ma and
rMa = the mass and mechanical resistance of the air in the tube. CMa = the
compliance of the case volume. 1MI and 1M2 = the driving forces. 1MI =
PIA and 1M2 = P2A. A = the area of the diaphragm. PI = the sound
pressure at the diaphragm. P2 = the sound pressure at the tube. Curve 3
on the lower graph is the same as curve 2 on the upper graph. Curve 4 is the
response with the tube ma, rM3 added. Curve 5 is the response with the com­
pliance C M 2 added.
263
MICROPHONES
An examination of the diffraction of sound as a function of the angle of
the incident sound by various objects shows that the sphere exhibits the
most uniform directional pattern. A spherical case with the diaphragm
located on the surface of the sphere seems to be the logical starting point
for a nondirectional pressure microphone. Referring to Fig. 1.5, it will be
seen that the microphone will show excess response over the range from
0° to 60° and will be lacking in response from 120° to 160. This nonuniform
response can be corrected by placing a disk, of semi-transmitting charac­
teristics and of diameter equal to the spherical case, directly above the
diaphragm and spaced one-fourth inch. Employing this expedient, a non­
directional characteristicI9 is obtained over the response frequency range.
r[G
LG
~
ELECTRICAL CIRCUIT
m
C-..,
~'
u'~:"r:':=!!!!!!!;k~.",;B~~T
m3
r.
loll
r"3
SILK
f.., f"l!
I
I
MECHANICAL
CROSS - SECTIONAL VIEW
NETWORK
ijMI~IIIIIII~I~~
a: 30
100
1000
fREQUENCY
10000
FIG. 8.11. Cross-sectional view, electrical circuit and mechanical network
of an inductor microphone. In the electrical circuit, rEG = the electrical
resistance of the conductor. L = the inductance of the conductor.
ZEL = the
electrical impedance of the load. eG = the open circuit
voltage developed in the conductor. In the mechanical network, mI,
rMI, and CMI = the mass, mechanical resistance, and compliance of the
diaphragm and conductor, m2 and rM2 = the mass and mechanical resis­
tance of the bolt of silk. C M2 = the compliance of the air chamber behind
the diaphragm. ma and rMa = the mass and mechanical resistance of the
air in the tube. CMa = the compliance of the case volume. IMI and
1M2 = the driving forces. IMI = PIA, 1M2 = P2A. A = the area of the
diaphragm. PI = the pressure at the diaphragm. P2 = the pressure at
the tube. The graph shows the free space, open-circuit, voltage response
frequency characteristic.
2. Inductor Microphone 2o (Straight-Line Conductor).-The inductor micro­
phone is another example of a moving conductor microphone. A cross­
sectional view of this microphone is shown in Fig. 8.11. The diaphragm,
rMIC MImI, of this microphone is "V" shaped with a straight conductor
19
20
Marshall and Romanow, Bell Syst. Tech. Jour., Vol. 15, No.3, p. 405, 1936.
Olson, H . F .• U.S. Patent 2.106.224.
264
ACOUSTICAL ENGINEERING
located in the bottom of the "V." The mechanical network of this micro­
phone is the same as that of the dynamic microphone in the preceding section
and the action is the same. A transformer, housed in the magnet structure,
is used to step up the low electrical impedance of the conductor to that
suitable for transmission over a line of several hundred feet.
3. Ribbon Microphone.-The pressure ribbon microphone 21 ,22 consists
of a light metallic ribbon suspended in a magnetic field and freely accessible
to the atmosphere on one side and terminated in an acoustical resistance
on the other side. The essential elements are shown schematically in Fig.
8.12. These elements may take various forms as, for example, the pipe is
usually coiled in the form of a labyrinth (see Fig. 8.14).
The acoustical network 23 of the pressure ribbon microphone is shown in
Fig. 8.12.
The inertance and acoustical capacitance of the ribbon are designated
by MR and CAR.
The acoustical resistance and mass of the air load upon the ribbon are
designated by r AA and M AA. The expression for the air load upon the
ribbon will now be derived. The pressure, in dynes per square centimeter,
at a distance a in centimeters, from an elementary source is (see Sec. 2.2)
p = dS
4-rra
where dS
jwpumaxEiwtE-ika
8.17
area of the source, in square centimeters,
maximum velocity of dS, in centimeters per second,
p = density of air, in grams per cubic centimeter,
=
Umax =
w =
2",j,
j
frequency, in cycles per second,
velocity over the surface dS, in centimeters per second,
time, in seconds,
2",/>.., and
wavelength, in centimeters.
=
U =
t =
k
>..
=
=
The pressure at any point on the ribbon due to a velocity umaxEiwt of the
ribbon is
8.18
where al = radius vector having the shortest air distance from the point
1 to the surface element dS. To compute the total force, the above integra­
tion must be performed and then the resulting pressure integrated over the
surface of the ribbon.
21
22
23
Olson, H. F., U.S. Patent 2,102,736.
Olson, H. F., Jour. Soc. Mot. Pic. Eng., Vol. 27, No.3, p. 285, 1936.
Olson, H. F., Broadcast News, No. 30, p. 3, May, 1939.
265
MICROPHONES
The total force is
JMA =
jwPU:;X€iwtJJdSfJ ~~ €i
8.19
ka 1
dS'
where
= surface element at 1.
The acoustical impedance due to the air load is
.
ZAA=r AA +JXAA =
[lf
MsrAS
_
M~
RIBBON
:
to MA
=
rAA
=-
CAR
R~:rNZ[2 i~i·
JMA
A2
LINE
Umax€
it
i~(i) :~[:TI[J
ELEC'==T=-R-IC-A-L-S"":Y=-STEM
"RONT
8.20
w
ELEC. EQUIV.
PIPE
VIEW
FIG. 8.12.
Schematic view, electrical system and its equivalent and
acoustical network of a pressure ribbon microphone. In the electrical
circuit, rEI = the electrical resistance of the ribbon. ZE2 = the external
electrical impedance load presented to the ribbon. eo = the open circuit
voltage developed by the ribbon. In the acoustical network, MA and rAA =
the inertance and acoustical resistance of the air load on the ribbon.
MR and CAR = the inertance and acoustical capacitance of the ribbon.
Ms and rAS = the inertance and acoustical resistance of the slit. ZAE =
the acoustical impedance due to the electrical circuit. ZAP = the acousti­
cal impedance of the pipe. p = the sound pressure.
The ribbon is spaced from the pole pieces of the magnetic structure to
allow freedom of motion. This slit or aperture, r AS and MAS, gives rise to
an acoustical impedance (see Sec. 5.4),
ZAS =
rAS
+ jwMs
8.21
where r AS = acoustical resistance of the slit, in acoustical ohms, and
Ms = inertance of the slit, in grams per (centimeter).4
The back of the ribbon is terminated in an acoustical resistance in the
form of a finite pipe damped with tufts of felt. The acoustical network of
the pipe shows that for the mid- and high-frequency range the acoustical
impedance is an acoustical resistance.
The acoustical resistance of the pipe referred to the ribbon is
42
rAP =
where Ap
=
Ap
area of the pipe, in square centimeters.
8.22
266
ACOUSTICAL ENGINEERING
The acoustical impedance due to the electrical circuit may influence the
motion of the ribbon. The acoustical impedance due to the electrical
circuit is
(El)2
ZAE= - - ­
8.23
AR2ZET
where
ZET =
AR =
total electrical impedance in the ribbon circuit, in abohms,
area of the ribbon, in square centimeters,
B = flux density in gausses, and
I = length of the ribbon, in centimeters.
The acoustical impedance, ZAE, due to the electrical circuit, and the acous­
tical impedance, ZAS , due to the aperture between the ribbon and pole
pieces, are in general small compared to the other impedances in the system
save at the very low frequencies .
The acoustical impedance characteristics of the elements of a pressure
ribbon microphone are shown in Fig. 8.13.
The volume current of the ribbon, III cubic centimeters per second, is
given by
p
u= rAP + rAA + JXAR
.
.
+ JXAA
where
rAP =
r AA =
XAR =
-
.
8.24
JXAP
acoustical resistance of the pipe, in acoustical ohms,
acoustical resistance of the air load upon the ribbon, in acous­
tical ohms,
acoustical reactance of the inertance and acoustical capaci­
tance of the ribbon, in acoustical ohms,
XAA =
acoustical reactance of the air load upon the ribbon, in acous­
tical ohms,
XAP =
acoustical reactance of the pipe, in acoustical ohms, and
sound pressure, in dynes per square centimeter.
p=
The volume current of the ribbon and the phase angle between the volume
current and pressure computed from equation 9.24 is shown in Fig. 8.13.
The velocity of the ribbon, in centimeters per second, is
.
u
x=AR
8.25
The voltage, in abvolts, generated in the ribbon is given by
e
where B
=
I =
=
Eli
flux density, in gausses, and
length of the ribbon, in centimeters.
8.26
267
MICROPHONES
The shape of the voltage curve will be the same as that of Uv in Fig. 8.13.
This assumes that the pressure is the same for all frequencies. However,
due to the obstacle effect (see Sec. 1.11), the pressure on the ribbon increases
at the higher frequencies and the output is practically independent of the
frequency.
An example of an unobtrusive microphone 24 employing the vibrating
system of Fig. 8.12 is shown in Fig. 8.14. The ribbon is terminated in a
damped pipe. The upper portion of the microphone is equipped with a pipe
1000
20
~IOO
o
~I 6
9o
18
r,.
~
~
!
u
~
::>
o
u
-<
x_.
10
--
I
I
A.OI4 0
100
'"
"
5 o~
~I 0
o
I.---
o
8
I--'""
X;:o
x_£
/~
r--
7o
6 0-'
«
r..
.......
Uv
~14
~I 2
x_.... I---'
x__
1--"
......... I--"
So
'"~6
~4
FREQUENCY
4>,
.........
2
~~
1000
II
-'
10000
B 040
100
r-
+v
./
1000
2
Io
0
10000
FREQUENCY
8.13. A. The acoustical impedance characteristics of the elements of the ribbon
pressure microphone. XAR = the ribbon acoustical reactance. XAA = the air load
acoustical reactance. r AA = the air load acoustical resistance. rAP = the acoustical
resistance of the pipe. XAP = the acoustical reactance of the pipe. XAE = the
acoustical reactance due to the electrical system. p = the sound pressure. B. U y =
the volume current of the ribbon for a sound pressure of 1 dyne per square centimeter.
4> = the phase angle between the ribbon volume current and the driving pressure.
4>1 = leading. .p2 = lagging.
FIG.
coupled to the ribbon. This provides a small pickup area and, therefore, a
nondirectional characteristic. The surge acoustical impedance of the pipe
is the same as the acoustical impedance of the damped pipe. Under these
conditions a smooth response frequency characteristic is obtained because
the sound flows into the pickup pipe, past the ribbon and then into the
damped pipe without reflections. A small horn is used at the pickup point
to accentuate the response in the frequency region above 5000 cycles.
4. Probe Microphone. 25 ,26-In some acoustical measurements, a micro­
phone equipped with a small sound pickup system with a high acoustical
impedance so that it will not disturb the sound field is a useful tool. A
probe-type microphone shown in Fig. 8.15 satisfies these requirements.
The probe-type microphone consists of a tube about 4 inches in length and
an inside diameter of from .020 to .1 inch coupled to a small condenser
microphone. The consideration of the attenuation of sound in tubes as a
function of the frequency and diameter will be found in Sec. 5.32. The
24
25
26
Olson and Preston, Audio Engineering, Vol. 34, No.7, p. 18, 1950.
Hilliard, John K., Trans. IRE, Prof. Group Audio, Vol. AV-2, No.6, p. 168, 1954.
Benson, Robert W., Jour. Acous. Soc. Amer., Vol. 25, No. 1. p. 128, 1953.
268
ACOUSTICAL ENGINEERING
frequency discrimination introduced by the probe can be compensated in
the microphone amplifier. The large attenuation in small tubes reduces the
effects of resonance in the tube with the result that the response frequency
characteristic is smooth and free of peaks and dips.
It
PICKUP HORN
CONNECTOR
RIBBON
MAGNET
DAMPED LABYRINTH
CONDENSER
MICROPHONE
PROBE TUBE
COUPLING
CHAMBER
FIG. 8.14. Cross-sectional view of a ribbon-type pressure microphone.
FIG. 8.15.
Cross-sectional view of a
probe-type microphone consisting of a
tube of small bore coupled to a condenser
microphone.
5. Comparison of Electrodynamic Microphones.-At this point it appears
appropriate to digress and examine the characteristics of the two most
common electrodynamic microphones, namely, the ribbon type and dia­
phragm-voice coil type.
The high-frequency response of any microphone is an inverse function
of the mass of the vibrating system. The use of negative acoustical reactance
elements to reduce the effective acoustical impedance over a limited frequency
range can be applied to any system and does not alter the fundamental
relationship between mass and high-frequency response. These facts can be
deduced by the application of the Reactance Theorem to acoustical networks.
In a moving conductor system, there are three fundamental parameters,
namely, the acoustical impedance, the flux density, and the conductor. The
ultimate flux density is limited by magnetic materials and, therefore, can
be made the same for any system. In the ribbon system, the entire system
acts as both conductor and diaphragm. In the moving coil system, the mass
269
MICROPHONES
must be divided between two parts, namely, the diaphragm and the con­
ductor. The graph of Fig. 8.16 depicts the sensitivity of a moving con­
ductor microphone. This data shows that the highest sensitivity is obtained
when the mass of the diaphragm
o
is zero, namely, the conductor also In
0 _3
acts as the diaphragm. There­
~
fore, the ultimate sensitivity in ...
~ -6
the high-frequency region will ...
::I
always be higher in a ribbon 0-9
r-.....
...
system than in a diaphragm and CI
of
.....
r-­
~-12
moving coil system.
g
A diaphragm-type dynamic
-15
.5
o
I
1.5
2
2.5
microphone requires a rigid dia­
3
MASS OF THE DIAPHRAGM
phragm system in order to pre­
MASS OF THE COIL
vent spurious responses due to
FIG.
8.16.
The
voltage output of a dynamic
relative motion of different parts
as a function of the ratio of the
of the diaphragm. Since the microphone
mass of the diaphragm to the mass of the coil.
ribbon serves a dual purpose of The ribbon microphone corresponds to the
diaphragm and voice coil, rigidity value zero for this ratio.
is not required. Therefore, the
response frequency characteristic of a ribbon system is smoother than the
diaphragm-voice coil system.
Another consideration is the low-frequency response. In well-designed
ribbon-type microphones, the resonant frequency of the ribbon can be
placed below the audible range. Therefore, above the audible range, the
diaphragm and conductor system is mass controlled. This simple mass
characteristic makes it a particularly simple task to develop suitable
phase shifting networks for ribbon transducers.
The transient response is another important characteristic of a microphone.
It can be shown that in a well-designed system, the one with the lowest
inherent mass will exhibit the most faithful response to transients. There­
fore, the ribbon system will exhibit the best transient response. This has
been substantiated by large- and small-scale explosion tests.
Closely allied to transient response is wind noise. In order to determine
the wind response of different microphones, a wind generator, described in
Sec. 1O.2G, was developed. This wind machine delivers a combination of a
steady and a fluxuating air stream and, therefore, simulates wind under
actual conditions. Tests have been made comparing the wind-noise response
of different microphones obtained with natural wind and the wind machine.
These results show that there is practically a perfect correlation. The
advantages of the wind machine are as follows: wind response can be obtained
at any time and the wind noise produced by the machine remains constant.
Using the wind machine, tests have shown that with the same screening
and the same response frequency characteristic, the response to wind of all
microphones is directly related to the sensitivity, which means the signal to
wind noise is the same.
'" "
""
----
1
270
ACOUSTICAL ENGINEERING
As regards the relationship of weight and over-all sensitivity of micro­
phones, the following conclusion can be drawn: for the same response
frequency characteristic, the weight of a ribbon microphone does not differ
from that of a moving coil microphone.
E. Magnetic Microphones. 27 ,28-A magnetic microphone consists of a
diaphragm acted upon by sound waves and connected to an armature which
varies the reluctance in a magnetic field surrounded by a coil. Fig. 8.17.
Two different types of magnetic transducers used in magnetic micro­
phones are shown in Fig. 8.17. These two transducers are the most common
types used in magnetic microphones.
PERSPECTIVE VIEW
PERSPECTIVE VIEW
DIAPHRAGM
COIL
~W~_MAGNET
SECTIONAL VIEW
SECTIONAL VIEW
A
B
8.17. Perspective and sectional views of magnetic
microphones of the balanced armature type.
FIG.
In the magnetic system of Fig. 8.18, let the armature be deflected a
distance ~x, in centimeters, from the center position. The flux through
the armature will be
~cp
where cp
M
A
=
=
=
a=
=
MA ~x
4a 2
8.27
flux, in maxwells,
magnetomotive force of the steady field, in gilberts,
area of the pole pieces in square centimeters, and
spacing between the pole pieces and the armature.
In this consideration it is assumed that the area of the top and bottom pole
pieces and the spacing between the armature and the top and bottom pole
27
28
Bauer, B. B., Trans. IRE, Prof. Group on Audio, Vol. AU-l, No.6, p. 4, 1953.
Bauer, B. B., Jour., Acous. Soc. Amer., Vol. 25, No.5, p. 867, 1953.
MICROPHONES
271
pieces are all the same. It is also assumed that all the reluctance resides
in the air gap.
The same result may be obtained from the magnetic network of Fig.
8.18. When the armature is deflected a distance Ax, there will be a change
in the reluctance AR in the upper branches 1 and 2 of the magnetic network.
The flux Acp through the branch 3 which is the armature when the armature
is deflected a distance AR is given by
Acp
where R
=
=
MAR
4R2
8.28
MAAx
4a 2
8.29
a
A' and
AR_ AX
-A
Equation 8.28 may be written
Acp
=
Equation 8.29 is the same as equation 8.27.
DIAPHRAGM
CDIL­
N TURNS
e
tP
SPACER
MM
MAGNET
RM ---t::t!:_ _---1...J
SCHEMATIC VIEW
MAGNETIC NETWORK
FIG. 8.18. Schematic view and magnetic circuit of a
magnetic generator transducer. R = magnetic reluc­
tance of each of the four air gaps between the armature
and pole pieces. t:.R = the change in the reluctance R
due to a displacement of the armature. M = the
magnetomotive force of the magnet. q, = the magnetic
flux in the armature.
Employing equation 8.27 or 8.29, the change in flux with respect to time is
Acp
At
=
MAAx
4a 2 At
8.30
MAdx
4a 2 dt
8.31
or
dcp
dt
272
ACOUSTICAL ENGINEERING
AUDIO
OUTPUT
SECTIONAL
VIEW
T
ELECTRICAL SYSTEM
~
m T
rMT
rMO
mO
C UO
!
m,
r M3
m3
C M3
f~~~---'
T
·~MC_L ~~c
mz
M2
I
_
i
Me
ML
'
1.11
~------~----~,----~-----+----------~~~
MECHANICAL
NETWORK
FIG. 8.19.
Sectional view, electrical system, and mechanical
network of an electronic microphone. In the mechanical net­
work, ZME = the mechanical impedance of the electronic trans­
ducer. ml, YMI, and C M1 = the effective mass, mechanical
resistance, and compliance of the outer portion of the bar.
m2, YM2, and C M2 = the mass, mechanical resistance, and com­
pliance of the inner portion of the bar or anode. m3, YM3, and
C M3 = the mass, mechanical resistance, and compliance of the
diaphragm of the electronic transducer. mD, YMD, and C MD =
the mass, mechanical resistance, and compliance of the diaphragm
and suspension. mp and YMP = the mass and mechanical resis­
tance of the air in the tube. YML and C ML = the mechanical
resistance and compliance of the link connecting the diaphragm
and transducer. C MO = compliance of case volume. 1MI and
1M2 = the driving forces. 1MI = PIA and 1M2 = P2A.
A = the
area of the diaphragm. PI = the sound pressure at the dia­
phragm. P2 = the sound pressure at the tube opening. The
electrical system shows the wiring diagram for a diode type
electronic transducer.
The electromotive force e, in abvolts, generated in the coil is given by
- N de/> - NMA .
e-
where
tit -
4a 2 x
832
.
x = velocity of the armature, in centimeters per second, and
N = number of turns in the coil.
Equation 8.32 shows that the open circuit voltage is proportional to the
velocity of the armature. Therefore, to obtain constant output for constant
MICROPHONES
273
sound pressure on the diaphragm, the velocity of the armature must be
independent of the frequency. Therefore, the system must be resistance
controlled to obtain a constant relationship between the voltage output
and the impinging sound pressure. This can be accomplished by means of
an acoustical resistance behind the diaphragm similar to that of the dynamic
microphone described in Sec. S.2Dl.
F. Electronic Micyophone. 29-An electronic microphone is a microphone
in which the output results from the motion of one of the elements in a
vacuum tube.
A schematic view of an electronic microphone is shown in Fig. 8.19.
The voltage output of an electronic transducer is given by
e = Kxa
where K
8.33
constant of the system, and
=
Xa = amplitude of the element.
The output of the electronic microphone may be computed from the
mechanical network of Fig. 8.19.
The two driving forces 1Ml and 1M2, in dynes, are equal and opposite in
phase. The driving force 1Ml is given by
8.34
where An = area at the diaphragm, in square centimeters, and
PI = sound pressure at the diaphragm, in dynes per square centi­
meter.
The driving force 1M2 is given by
1M2
where An
=
P2
=
=
hAn
8.35
area of the diaphragm, in square centimeters, and
sound pressure at the port, in dynes per square centimeter.
At the high frequencies the mechanical reactance due to the compliance,
is small compared to the mechanical impedance of the port, mT, rMT.
Under these conditions, the system is driven by 1Ml. At the extreme low
frequencies the mechanical reactance of the compliance, CMe , is large
compared to the mechanical impedance of the port, mT, YMT. Since 1Ml
and 1M2 are of opposite phase, the net driving force is practically zero. In
the region where the mechanical reactance due to the compliance, CMe,
and the mechanical reactance due to the port, rMT, mMT, are comparable,
the addition of this mechanical network introduces a phase shift of such
magnitude that both forces,fMl and1M2, contribute in driving the mechanical
system.
CMe,
29
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 19, No.2, p. 307. 1947.
274
ACOUSTICAL ENGINEERING
The response may be obtained from a consideration of the mechanical
network. The mechanical network of Fig. 8.19 may be reduced to the
mechanical network of Fig. 8.20 in which
+ jwmT
8.36
ZMl =
YMT
ZM2 =
jwC MC
ZM3 =
YMD
.
1
+ JWmD
+ JW
-.­C-­
MD
8.38
ZM4 =
YML
1
+ JWCML
-.-­
8.39
1
ZM5
where
YMT =
mT =
CMC
=
YMD =
mD =
CMD
=
YML =
CML
=
ZM5 =
=
8.37
8.40
ZME
mechanical resistance of the tube, in mechanical ohms,
mass of the air in the tube, in grams,
compliance of the case volume, in centimeters per dyne,
mechanical resistance of the diaphragm, in mechanical ohms,
mass of the diaphragm, in grams,
compliance of the diaphragm, in centimeters per dyne,
mechanical resistance of the coupling link, in mechanical
ohms,
compliance of the coupling link, in centimeters per dyne, and
ZME the mechanical impedance of the electronic transducer,
in mechanical ohms.
The mechanical impedance of the electronic transducer is
ZME =
ZM5
ZM6(ZM7+ZMS) (ZM9+ZMIO)+ZM9ZMIO(ZM6+ZM7)+ZM7ZMS(ZM9+ZMIO)
(ZM7+ ZMS) (ZM9+ ZMlO) +ZM9ZMIO
where
8.41
ZM6 = jwml
1
ZM7=YMI+~C
JW Ml
ZMS =
.
YM3
ZM9 = YM2
1
+ J W 3 + JW
-,---c
M3
1
+ JW
-;--C
M2
The amplitude, in centimeters, due to the driving force IMI is
Xl =
1Ml
-....,...----......::.=~-----....,...
. ( +
JW ZM3
ZMIZM2
zMl
zM2
+
+
ZM4ZM5 )
zM4
zM5
+
8.42
MICROPHONES
275
The amplitude, in centimeters, due to the driving force !M2 is
-
jw[ZMI (ZM2+ ZM3) (ZM4
!M2ZM2(ZM4+ ZM5)
+ZM5) +ZM4ZM5 (ZMI +ZM2) +ZM2ZM3(ZM4 +ZM5)]
The amplitude, in centimeters, of
X3
8.43
is
m2
=
!MZM7 ZM9
jW[ZM6 (ZM7 +ZMS) (ZM9+ ZMlO)
+ZM9ZMIO(ZM6 +ZM7) +ZM7ZMS(ZM9+ZMIO)]
8 44
.
where
r
JM =
+ X2) .
+ ZM5 JW
(ZM4ZM5) (Xl
ZM4
The amplitude response charac­
teristic can be obtained from equa­
tion 8.44 and the constants of the
system. The voltage output can be
obtained from the amplitude and
equation 8.33.
The electrical connections for the
electronic microphone are shown in
Fig. 8.19.
8.3. Velocity Microphones. -
MECHANICAL
FIG. 8.20. Mechanical network of Fig.
8.10 in terms of the following:
First-Order Gradient Microphones.
-A pressure gradient microphone is
a microphone in which the electrical
response corresponds to the difference
in pressure between two points in
space. In general, when the distance
between these two points is small
compared to the wavelength, the
pressure gradient corresponds to the particle velocity. A velocity micro­
phone is a microphone in which the electrical response corresponds to the
particle velocity resulting from the propagation of a sound wave through an
acoustical medium. The acoustical and electrical elements which form the
coupling means, between the atmosphere and the electrical system, for
transforming the sound vibrations into the corresponding electrical variations,
may be arranged in innumerable ways to obtain pressure gradient or velocity
microphones. I t is the purpose of this section to consider pressure gradient
and velocity microphones.
A. Pressure Gradient Microphone.30,31,32,33,3~The response of a pressure
gradient microphone, as the name implies, is a function of the difference
Olson, H. F., Jour. Soc. Mot. Pic. Eng., Vol. 16, No.6, p. 695, 1931.
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 3, No. 1. p. 56, 1931.
32 Olson, H. F., Proc. Inst. Rad. Eng., Vol. 21. No.5, p. 655,1933.
33 Massa, F., Jour. Acous. Soc., Amer., Vol. 10, No.3, p.173,1939.
34 Anderson and Wigginton, Audio Engineering, Vol. 34, No. 1. p. 12, 1950.
30
31
276
ACOUSTICAL ENGINEERING
in sound pressure between two points. Obviously, a pressure gradient
microphone may be built in a number of ways. One type of pressure
gradient microphone consists of two pressure actuated units, separated by
a very small distance, with the electrical outputs connected in opposition.
Figure 8.21 schematically depicts the essential elements of a pressure gra­
dient microphone. A cylinder of mass m is coupled to a conductor located
in a magnetic field. The cylinder is assumed to be the only portion of the
system which will be influenced by sound waves. The diameter of the
cylinder is assumed to be small
compared to the wavelength.
0 -1
Therefore, the average intensity
will be the same for all points on
the surface of the cylinder. The
vibrating system is assumed to
be constrained so that the only
motion possible is one in a direc­
tion parallel to the longitudinal
FIG. 8.21.
Pressure gradient microphone.
axis of the cylinder. Under these
conditions the vibrating system is
driven by the difference between the forces on the two ends of the cylinder
due to the impinging sound wave.
Assume a plane sound wave, from equation 1.22, the pressure, in dynes
per square centimeter, at x = 0 may be written
r--
P = kcpA sin (kct)
P = Pm sin kct
8.45
where c = velocity of sound, in centimeters per second,
k = 27TjA,
A = wavelength, in centimeters,
p = density, in grams per cubic centimeter,
A = amplitude of cp,
cp = velocity potential, and
Pm = maximum sound pressure, in dynes per square centimeter.
The pressure at the end of the cylinder Xl = - D..xj2 for a direction of
propagation 0 is
PI = Pm sin k (ct + ~x cos 0)
The pressure at the other end of the cylinder X2
P2
=
Pm sin k (ct -
=
8.46
D..xj2 is
~x cos 0)
8.47
The difference in pressure between the two ends of the cylinder is
D..p =
PI - P2 = 2Pm cos (kct)
sin
(k~X cos 0)
8.48
277
MICROPHONES
The driving force, in dynes, available for driving the cylinder along the
x axis is
III;!
where S
=
SD.p
=
=
2SPm cos (ket) sin
(k~X cos e)
8.49
area of the end of the cylinder, in square centimeters.
If D.x is small compared to the wavelength the driving force is
JM =
27TJ
S - pmD.x cos
e
e cos ket
8.50
A comparison of equations 8.45 and 8.50 shows that for a wave of constant
sound pressure the driving force is proportional to the frequency.
The velocity of the mechanical system, for D.x small compared to the
wavelength, is
i =
where m
=
w =
!M
Jwm
S,?m D.x cos
=
Jem
e cos ket =
SPm D.x cos
em
e sin ket
8.51
mass of the cylinder, in grams, and
27TJ,f = frequency, in cycles per second.
This quantity is independent of the frequency and as a consequence the
ratio of the generated voltage to the pressure in the sound wave will be
independent of the frequency.
The velocity of the mechanical system for any value of D.x is
i
=
i
=
2SPm
. (k et ) sm
. (kD.x
mw sm
2 cos uD)
8.52
C~ cos e)
8.53
2!m sin (ket) sin
where D = distance between the two ends of the cylinder.
The voltage output, in abvolts, of the conductor is
e
where E
=
l
i
=
=
=
Eli
8.54
flux density in the field in which the conductor moves, in
gausses,
length of the conductor, in centimeters, and
velocity of the conductor, in centimeters per second.
The response frequency characteristic of a mass controlled, dynamic
pressure gradient microphone computed from equations 8.53 and 8.54 is
shown in Fig. 8.22.
The directional characteristics of a pressure gradient system of the type
shown in Fig. 8.21 and computed from equation 8.53 are shown in Fig. 8.23.
It will be seen that when the ratio D is greater than Aj4 the directional
pattern becomes progressively broader as the frequency increases. In the
case of the baffle-type ribbon microphone, the directional characteristics
278
ACOUSTICAL ENGINEERING
r---...
1.0
.9
.8
"\
r\
\
t­
{
!; .3
o
.2
IL'1
II
.1
o
.01
2
34$6789
.1
2
3
o
X
4
~
«578.
I
~
I \ If\.
2
3
"""
45&78'
10
FIG. 8.22. Computed open-circuit voltage response frequency charac­
teristic of a pressure gradient, mass-controlled, electrodynamic micro­
phone.
D=kA
'0
D=.!A
'.0
H-++3~E'E'+'f-t-1"
270H-++3~!iE'+--1f-t--jo'
D=A
'00
'00
FIG. 8.23. Directional characteristics of a pressure gradient microphone as a function of
the dimensions and the wavelength. The polar graph depicts the output, in volts, as a
function of the angle, in degrees. The maximum response is arbitrarily chosen as unity.
279
MICROPHONES
first become sharper than the cosine pattern and then broader as the dimen­
sions become comparable to the wavelength. In other words, the doublet
theory is not in accord with the observed results. Of course, deviations
would be expected when the dimensions of the baffle become comparable
to the wavelength because of variations in both intensity and phase due to
changes in the diffraction of sound by the baffle.
The above considerations have been concerned with a plane wave.
From equation 1.40 the pressure component in a spherical wave is
p
=
kcpA sin k(ct - r)
8.55
r
Let the distance on the axis of the cylinder between the sources and point
and Xl on the cylinder be r - !:::..xj2 and r
!:::..xj2 (Fig. 8.12). The
difference in pressure between the two ends of the cylinder is
+
X2
2rcos k(ct - r) sin
I:lp
=
(~) + 2D sin k(ct -
r) cos
(~)l
r2 _ (~r
kcpA [
8.56
If D is small compared to rand kD is small compared to unity, equation
8.56 becomes approximately
!:::..p
=
kcpAD [kr cos k(ct - r~2+ sin k(ct - r)]
8.57
This equation is similar to equation 1.42 for the particle velocity in a
spherical sound wave. Therefore, the voltage output of this microphone
corresponds to the particle velocity in a sound wave. The response of a
pressure gradient microphone as a function of the distance from a point
source and the frequency is shown in Fig. 8.40A.
B. Velocity Microphone. 35 ,36,37-Free-ribbon microphones are used for
all types of sound collection. Essentially these microphones consist of a
loosely stretched ribbon suspended in the air gap between two pole pieces
(Fig. 8.24). In addition to supplying the flux to the air gap, the pole
pieces serve as a baffle for acoustically separating the two sides of the ribbon.
The configuration and dimensions of the baffle determine the effective sound
path between the two sides of the ribbon. Under the influence of a sound
wave the ribbon is driven from its equilibrium position by the difference in
pressure between the two sides. The motion of the ribbon in the magnetic
field induces a voltage between the two ends of the ribbon. The electrical
output of this system under certain conditions corresponds to the particle
velocity in a sound wave. Accordingly, the term velocity microphone has
been applied to the free-ribbon microphone. In past analysis it has been
customary to treat the system as an acoustical doublet. This method is
essentially accurate when the effective dimensions of the baffle are small
35
36
37
Olson, H. F., Jour. Soc. Mot. Pic. Eng., Vol. 16, No.6, p. 695,1931.
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 3, No. 1, p. 56, 1931.
Olson, H. F., Proc. Inst. Rad. Eng., Vol. 21, No.5, p. 655, 1933.
280
ACOUSTICAL ENGINEERING
compared to the wavelength. When the effective dimensions are com­
parable to the wavelength, there is considerable discrepancy between the
simple doublet theory and the actual performance. It is the purpose of
"'\
/
~
MAGN
0
1 =1
=
=
POLE -
=
R IBBON
=
RIBBON
=
··
-~
,
,,
·
I
=
1= 1
0
0
TERMIN AL- f--­
'­
FIG. 8.24.
phone.
..I
fRONT
VIEW
END VIEW
The essential elements of a velocity micro­
this section to develop the theory of the conventional baffle-type velocity
microphone.
Approximate solutions for the diffraction of sound by a circular and
square plate have been obtained. 3s These analyses may be applied to the
problem of the baffle-type ribbon microphone.
The ratio of the pressure at the center of a circular plate for any angle
of the incident sound is
Po
P=
where IL
=
IOU
=
IOu
=
(J =
R
=
I
+ vI
cos (J
_ sin2
I - vI - sin 2 (J
.
SIll
(J
[
(J
8.58
I
,
1 when u = 0,
2 when u =F 0,
angle of the incidence,
radius of the plate, in centimeters,
k = 27T/>',
>. = wavelength, in centimeters, and
] u = Bessel function, of the order
38
u.
Sivian and O'Neil. Jour. Acous. Soc. Amer., Vol. 3, No.4. p. 483.1932.
281
MICROPHONES
The pressure at the center on the front and back of a circular plate for
normal incidence 8 = 0° or 180°, from equation 8.58, is
l~ol = vis IP~801 =
8.59
4 cos kR
8.60
1
The pressure frequency characteristic on the front and back of a circular
baffle for normal incidence computed from equations 8.59 and 8.60 is shown
in Fig. 8.25. It will be seen that the pressure at the front rises to a value
3.5
3.0
/
11
I
/
/
2.0
...ii!
o·
0.
..
~
1.5
V
-
1.0
'/
/
~
180·
.5
o
.01
a
3
.. ~
e
7 89.
1
2
2
~
FIG. 8.25. Computed pressure frequency characteristic,
at the center, on the front, and the back of a circular
baffle for normal incidence of the impinging sound wave.
of three times that in free space at RIA = .5, then falls back to the same
as the free space pressure at RIA = 1, and repeats for RIA = 1.5 and
RIA = 2, etc. The pressure at the back is the same as the free space pres­
sure for all frequencies. The measured pressure at the center on the front
and back of a circular baffle is shown in Fig. 8.26. In order to reduce
errors in measurement to a minimum, baffles of different diameters were
used. In addition, several different pressure measuring arrangements
were used. The results shown in Fig. 8.26 represent an average of these
determinations. It will be seen that theory and experiment are in fairly
good agreement. Some of the deviation may be attributed to finite size
of the pressure measuring system.
282
ACOUSTICAL ENGINEERING
The phase angles at the front and back of a circular baffle computed from
equation 8.58 are shown in Fig. 8.27. A point in the plane wave cor­
responding to the plane of the baffle is the reference plane for the phase.
It will be seen that for RIA less than .5 the phase of the pressure at the front
of the baffle leads that of the pressure in the wave. For values of RIA less
3.5
1"'\
3
II
2.5
n
/
2 .0
If
c..
I
+
21.5
~V i-" -r--,
I
c:f
;;;...
~
5
0
.01
2
3.
s,e78!l
2
.1
2
R
1\
FIG. 8.26. Measured pressure frequency characteristic
at the center, on the front, and the back of a circular
baffle for normal incidence of the impinging sound wave.
than .1 the phase on the front leads by the same amount as the phase on the
back lags the pressure in the wave.
Equation 8.58 may be used to compute the difference in pressure between
the two sides of a relatively small ribbon located in a large baffle (Fig. 8.28).
The difference in pressure between the two sides of the ribbon in a circular
baffle, Fig. 8.28, is
8.61
t1p = Pe - PB 180
+
+
where PB and Po
180 may be obtained from equation 8.58.
impedance of the ribbon, Fig. 8.28, is given by
. MR
ZAR =JW
where M R
=
CAR =
j
--­
WCAR
inertance of the ribbon, and
acoustical capacitance of the ribbon.
The acoustical
8.62
283
MICROPHONES
From equation 8.28 the total force of the air load upon the ribbon is
f MA
=
jWpUmaxEJwt
47T
If Ifa;:-E
ds ika
1
dS'
8.63
The above integration extends over both sides of the ribbon and cog­
nizance must be taken of the 180 0 difference in phase between the front
90
Cf'
r--.
o
~
-90
V\
V
~
""",
\
-180
\
In
'"
Ir
'"
Cl
~-Z70
~
~
~-360
-<
\
\
\
-540
-630
-720
.01
\
2
3
4
SITal
.1
2
3
4
5 6789
R
I
2
X
FIG. 8.27. Computed phase frequency characteristic
at the center, on the front, and the back of a circular
baffle for normal incidence of the impinging sound wave.
and back when integrating between the two surfaces. The integration of
equation 8.63 may be carried out by dividing the ribbon into small elements
and carrying out the indicated integration.
The acoustical impedance of the air load is
ZAA
=
r AA
.
+ JXAA =
JMA
A R 2Umax€ jwt
8.64
284
ACOUSTICAL ENGINEERING
The acoustical impedance, ZAS, of the slit between the ribbon and pole
pieces is given by equation 8.21. The acoustical impedance due to the
electrical circuit is given by equation 8.23.
The resonant frequency of the ribbon is usually placed below the audible
limit. Therefore, the acoustical capacitance of the ribbon may be neglected.
The acoustical resistance, r AA, of the air load
is negligible save at the very high frequencies.
When the fundamental resonant frequency of
the ribbon is located below the audible­
frequency range, the negative reactance term
in equation 8.62 may be neglected. Under
these conditions the acoustical impedance of
the vibrating system is
ZAT =
FIG. 8.28. A velocity micro­
phone with a large circular
baffle.
jwMR
+ jwMAA
.
I:1p
X=--­
ARZAT
where
AR =
8.65
where M AA = inertance of the air load.
The velocity, in centimeters per second, of
the ribbon is
8.66
area of the ribbon, in square centimeters.
The voltage output in abvolts is
e = Eli
where B
l
=
=
i =
8.67
flux density, in gausses,
length of the ribbon, in centimeters, and
velocity of the ribbon, in centimeters per second.
The response characteristic of a mass-controlled ribbon located in a large
circular baffle, Fig. 8.28, computed from equation 8.67 is shown in Fig. 8.29.
The experimental response of a ribbon microphone with a circular baffle is
shown in Fig. 8.30. The agreement between the measured response and
the computed response is quite good. There is some deviation between
RIA = .5 and RIA = .8. There is also some discrepancy in this region
between computed and measured pressures (Figs. 8.25 and 8.26). It is
interesting to note that the theoretical response of the pressure gradient
microphone and the ribbon in a baffle is practically the same, Figs. 8.22
and 8.29.
The measured directional characteristic of the ribbon microphone with ,
a circular baffle is shown in Fig. 8.31. It will be seen that, for small values
of RIA, the directional characteristic corresponds to a cosine function. Be­
tween RIA = i and t the directional pattern is sharper than a cosine char­
acteristic. Then for RIA larger than t the characteristics broaden and
assume irregular shapes. The theoretical directional characteristics em­
ploying equations 8.58, 8.61, 8.66, and 8.67 are shown in Fig. 8.32. It will
be seen that the agreement with the experimental results of Fig. 8.31 is
285
MICROPHONES
quite good. There is some deviation for DIA = t. It is in this region that
deviations occurred between the theoretical and experimental results for
the pressure, Figs. 8.25 and 8.26, and for the response, Figs. 8.29 and 8.30.
The theoretical directional characteristic for a doublet, Fig. 8.23, becomes
progressively broader for RIA =i, t, and i and does not agree at all with the
experimental results. For RIA = !, t, and 1 the shape of the theoretical
directional characteristic of the doublet does not correspond with the
experimental results. Summarizing, the theoretical directional characteris­
tics of a ribbon microphone with a circular baffle agree within a few per
--.....
0
.9
""­
r\
.e
~
'" .7
~
~ .6
1\
...J
...J
~.5
Z
-,4
f­
~
~.3
o
-
.2
I\"
.1
o
.01
1/
2
3
4
5 • 7 a8
.I
2
3
4
56789
1
W \~ 'I\~
2
3
4
5
e
788
10
FIG. 8.29.
Computed open-circuit voltage response frequency charac­
teristic of a mass-controlled, electrodynamic ribbon located in a large
circular baffle.
cent of the measured directional characteristics. However, the discrep­
ancy between the measured directional characteristics of a ribbon in a
circular baffle and the theoretical directional characteristics of a doublet or
pressure gradient system is very large for values of RIA greater than f.
The phase between the actuating force, equation 8.61, and the particle
velocity in a plane wave, for a ribbon microphone with a circular baffle, is
shown in Fig. 8.33. It will be seen that this force leads the particle velocity
by 90° for small values of RIA. The phase angle between the voltage out­
put of the ribbon and the particle velocity is also shown in Fig. 8.33. For
small values of RIA the voltage output of a mass-controlled, dynamic ribbon
microphone with a baffle corresponds to the particle velocity in the sound
wave.
286
ACOUSTICAL ENGINEERING
- ..... ~
1.0
.9
i\
.8
.2
II \\II,,..
.1
o
.01
II
234S178'
.1
23451789
I
2
hn
3
4
5 . 788
10
FIG. 8.30. Measured open-circuit voltage response frequency charac­
teristic of a mass-controlled, electrodynamic ribbon located in a large
circular baffle.
I-t+HMtCH++-+oo
FIG. 8.31. Measured directional characteristics of a ribbon microphone with a large
circular baffle (see Fig. 8.28) as a function of the radius of the baffle and the wavelength.
The polar graph depicts the output, in volts, as a function of the angle, in degrees.
The maximum response is arbitrarily chosen as unity.
287
MICROPHONES
180
180
100
FIG. 8.32. Computed directional characteristics of a ribbon microphone with a large
circular baffle (see Fig. 8.28) as a function of the radius of the baffle and the wavelength.
The polar graph depicts the output, in volts, as a function of the angle, in degrees. The
maximum response is arbitrarily chosen as unity.
0
r-r-.
............
60
'"
30
0
6
\
\
1\
30
-I
1
9 I
2
3
4
$,
67891
.01
2
3
-4567811
.1
2
3
4
FIG. 8.33. The phase angle, in degrees, between the actuating force and the
particle velocity for a mass-controlled ribbon with a circular baffle as a function of
RIA. The phase angle between the voltage output of a mass-controlled, electro­
dynamic ribbon located in a magnetic field as a function of RIA.
The above analysis has been concerned with a ribbon located in a cir­
cular baffle. Irregular baffles instead of circular baffles are used in com­
mercial microphones for two reasons: first, a suitable magnetic field results
in an irregular baffle and, second, the sound path lengths between the two
sides of an irregular baffle differ and, as a consequence, it is possible to
obtain uniform directional response characteristics over a wide frequency
288
ACOUSTICAL ENGINEERING
range. An analytical solution of the irregular plate is difficult. However,
the graphical method may be used and is very effective.
In well-designed velocity microphones which have been built in the past,
the effective sound path introduced by the baffle has been made less than
one-half wavelength for all frequencies within the useful range. There
are two reasons for this selection of sound path: first, the response up to
this frequency is quite uniform, while above this frequency the response
falls off rapidly with increase of the frequency; second, in the case of an
irregular baffle the directional characteristics are of the cosine type to
within a few per cent of this frequency limit. A commercial microphone
is shown in Fig. 8.24. It will be seen that the effective baffle is irregular
in shape. The directional characteristics of the microphone of Fig. 8.24
4000~
20-2000'"
1.0
270'H-++~~+-r4-JgO
HORIZONTAL
7000­
1.0
270
11000~
1.0
270f-+-+-t--al1!E-+-t-H
180
180
180
FIG. 8.34. The directional characteristics of the velocity microphone shown in Fig. 8.24.
The polar graph depicts the output, in volts, as a function of the angle, in degrees. The
maximum response is arbitrarily chosen as unity.
are shown in Fig. 8.34. Further, the deviation from a cosine characteristic
is very small.
The above considerations have been concerned with a plane wave. As in
the case of the pressure gradient microphone, it can be shown that the
output of a baffle-type velocity microphone corresponds to the particle
velocity in a spherical wave. The response of a baffle-type velocity micro­
phone as a function of the distance from a point source and the frequency
is shown in Fig. 8AOA.
The response of the baffle-type velocity microphone may be obtained
from the acoustical network of Fig. 8.35 and the acoustical impedance
characteristics of the acoustical elements of the system of Fig. 8.36. The
ribbon is 2.2 inches in length and .2 inch in width. The flux density is
9000 gausses. The open-circuit voltage generated of the ribbon is given
by equation 8.26. The computed voltages are indicated by the dots in
289
MICROPHONES
RIBBON
RIBBON
'
TRANS.
LINE
TRANS.
V.T.
r[1811~,=-----=,III~
TERMINAL
M::~~E
ELECTRICAL
SYSTEM
M .rA2
~
~
MsfAS
MRCAR
~J
A'
,RONT
VIEW
SECTION
A-I(
ACOUSTICAL
NETWORK
FIG. 8.35. Front and sectional views of the vibrating system of a velocity
microphone. In the electrical circuit, e = the open circuit voltage. rEI =
the electrical resistance of the ribbon. ZE2 = the electrical impedance of the
load upon the ribbon due to the transformers and vacuum tube. ZEA = the
electrical impedance due to the acoustical system. In the acoustical network,
rAI. M 1, rA2, and M 2 = the acoustical resistances and inertances due to the
air load on the front and back of the ribbon. rAS and Ms = the acoustical
resistance and inertance due to the slit between the ribbon and the pole
pieces. MR and CAR = the inertance and acoustical capacitance of the
ribbon. ZAE = the acoustical impedance due to the electrical system. PI and
P2 = the sound pressure at the front and back of the ribbon.
IOOr---~-----r----'-'-----'-----~---'-'-----r----~----'-~---'
10
'"
z
u
«0
'"
Q.
~
..J
«
.Q.
iI:
<l
~
f-
(/)
=>
u
0
«
01
~--~----~-----L-IOLO~~~~~L-~~LIO~OLO~--L-----L---~'~O~00~O~-200~~~
FREQUENCY
IN
CYCLES
PER
SECOND
FIG. 8.36. The acoustical impedance characteristics of the velocity microphone.
the positive and negative acoustical reactances due to the inertance and
the acoustical capacitance of the ribbon. XAA = the acoustical reactance due to
the air load. rAA = the acoustical resistance due to the air load. XAE = the
positive and negative acoustical reactances due to the electrical system. !1P/P =
the ratio of the difference in pressure between the front and back of the ribbon and
the free-field pressure.
XAR =
290
ACOUSTICAL ENGINEERING
Fig. 8.37. The experimental response frequency characteristic of the micro­
phone of Fig. 8.24 is shown in Fig. 8.37. The agreement between the
theoretical computed characteristic and the experimental determined
characteristic is very good. A transformer is used to step up the electrical
impedance of the ribbon to 250 ohms which is suitable for transmission
over a line. The characteristics shown in Fig. 8.37 depict the open-circuit
voltage from the terminals of the 250-ohm transformer of Fig. 8.35. The
line electrical impedance is stepped up to 50,000 ohms at the grid of the
vacuum tube. The input to the vacuum tube of Fig. 8.35 is 23 db above
., 15
..J
'" 10
m
8o ~
:!O 0
v
0
50
I 00
FREQUENCY
IN
500
1000
CYCLES
PER
SECOND
5000
10000
20000
FIG. 8.37. Open-circuit voltage response frequency characteristics of a velocity
microphone at the terminals of the 250-ohm output. 0 db = 600 microvolts per
dyne per square centimeter. When the impedance is stepped up to 50,000 ohms for
the input to the grid of a vacuum tube, the voltage is 8.4 millivolts per dyne per
square centimeter. The dots represent the computed response and the solid line, the
measured response.
the voltage in the line. This is .0085 volt per dyne per square centimeter
at the grid of the vacuum tube.
In general, the electrical load, ZE2, of the transformer and vacuum tube
is large compared to the electrical resistance of the ribbon and the motional
electrical impedance, ZEA. Under these conditions the voltage delivered at
the grid is the same open-circuit voltage developed by the ribbon multiplied
by the step-up ratio of the transformers. However, if the electrical resist­
ance of the ribbon and the motional electrical impedance are comparable in
magnitude to the electrical load impedance, cognizance must be taken of the
electrical resistance of the ribbon and the motional electrical impedance
when computing the voltage developed across the load. In the equivalent
electrical circuit of Fig. 8.35, the motional electrical impedance, in abohms,
is given by
(Bl)
8.68
ZEA = A'RZAP
where
the total acoustical impedance,
AR = area of the ribbon, in square centimeters,
B = the flux density, in gausses, and
1 = the length of the ribbon, in centimeters.
The total acoustical impedance ZAT is the acoustical impedance at the
point ZAE of Fig. 8.35 with the acoustical impedance due to the electrical
ZAP =
MICROPHONES
291
circuit considered to be zero. In the same way, in computing the acoustical
impedance ZAE, of the acoustical network of Fig. 8.35, from equation 8.23
and the electrical impedance at e in the electrical circuit, the electrical
impedance ZEA due to the acoustical system is considered to be zero.
8.4. Unidirectional Microphones.-A unidirectional microphone is
a microphone with a substantially unidirectional pattern over the response
frequency range. Unidirectional
microphones may be constructed
MAGNET
by combining a bidirectional
SILK
CLOTH
microphone and a nondirectional
VELOCITY
microphone or by combining a
..L_o.---- SECTION
single element microphone with
PRESSURE
SECTION
an appropriate acoustical delay
1-+0-+-'___RIBBON
system. It is the purpose of
this section to consider combina­
tion unidirectional microphones
and single-element unidirectional
microphones.
rOLDED
A. Combination Unidirectional
PIPE
Microphones.-The combination
unidirectional microphone 39 ,40,41
consists of a bidirectional micro­
phone and a nondirectional
microphone.
A unidirectional
microphone consisting of a ribbon
velocity element (see Sec. 8.3B
and Fig. 8.24) and a ribbon pres­
sure element (see Sec. 8.2D3 and
Fig. 8.12) is shown in Fig. 8.38.
The damped pipe terminating the
back of the pressure ribbon is
folded in the form of a labyrinth
FIG. 8.38.
Unidirectional microphone with
and enclosed in a case. The the
screen removed. Ribbon type pressure
velocity and pressure sections are and velocity elements.
formed from a single continuous
ribbon. A common magnetic structure is used for both the velocity and
pressure sections. Due to a finite length of pipe for the pressure section the
velocity of the pressure ribbon leads the pressure in the sound wave at the
low frequencies (see Sec. 8.2D3 and Fig. 8.13). The acoustical resistance
(silk cloth) introduces a corresponding shift in the velocity section. At the
high frequencies the phase shifts in the two elements are made the same by
suitable geometrical configurations of the field structure.
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 3, No.3, p. 315, 1932.
Weinberger, Olson, and Massa, Jour. Acous. Soc. Amer., Vol. 5, No.2, p. 139, 1933.
41 Olson, H. F., Jour. Soc. Mot. Pic. Eng., Vol. 27, No.3, p. 284,1936.
89
40
292
ACOUSTICAL ENGINEERING
A unidirectional microphone 42 consisting of a ribbon velocity element and
a dynamic pressure element is shown in Fig. 8.39. Equalizers are used to
correct the amplitude and phase of the dynamic element to conform with
the velocity element.
VELOCITY
ELEMENT
OUTPUT
PRESSURE
ELEMENT
A'
fRONT VIEW
CROSS- SECTION A-A
8.39. Unidirectional microphone consisting
of a ribbon-type velocity element and a dynamic­
type pressure element.
FIG.
1. The Response of the Unidirectional Microphone as a Function of the
Distance and the Frequency.43-The low-frequency response of the velocity
microphone is accentuated when the distance between the source and the
microphone is less than a wavelength. The same effect occurs to a smaller
extent in the unidirectional microphone. It is the purpose of this section
to consider the response of the unidirectiona~ microphone as a function of
the frequency and distance from a point source.
The voltage output of a nondirectional microphone as a function of the
distance, r, is given by
Rl .
eND = SIll wt
8.69
r
where Rl = sensitivity constant of the microphone,
w = 27ff,
f = frequency, in cycles per second,
r = distance, in centimeters, from a point source of sound, and
t = time, in seconds.
42
43
Marshall and Harry, Jour. Acous. Soc. Amer., Vol. 12, No.4, p. 481. 1941.
Olson, H . F., Broadcast News, No. 30, p. 3, May, 1939.
293
MICROPHONES
The voltage output of the bidirectional velocity microphone as a function
of the distance and the wavelength A, in centimeters, is
eBD =
where R
=
r =
8=
R2
(~ sin wt - 2~2 cos wt)
cos 8
8.70
sensitivity constant of the microphone,
distance, in centimeters from a point source of sound, and
angle between the direction of the incident sound and the
normal to the ribbon.
If the output of the unidirectional microphone as a function of the angle 8
is to be a cardioid of revolution for plane waves, then Rl must be made
equal to R2. The ratio of the output of the unidirectional microphone as
a function of the distance and frequency as compared to a pressure micro­
phone is
Response Ratio
=
j(~r + cosr 8)2 + (A cos 8)2
21T1'2
(~r
8.71
This ratio for 8 = 0, 30°, 60°, 90 120°, 150 , and 180 0 for 1, 2, and 5 feet
is shown in Fig. 8.40. The same ratio for a conventional velocity micro­
phone for 1, 2, and 5 feet is shown in Fig. 8.40. It will be seen that the
accentuation in the unidirectional microphone is smaller than in the case
of the velocity microphone.
2. Efficiency of Energy Response to Random Sounds of the Unidirectional
Microphone as a Function of the Relative Sensitivities of the Bidirectional
and N ondirectional Microphones. 44_The unidirectional microphone consists
of the combination of a bidirectional microphone, in which the output is a
function of the cosine of the angle of incidence, and a nondirectional micro­
phone. In general, it is customary to make the output of the bidirectional
microphone, for 8 = 0, equal to the nondirectional microphone. For this
condition the directional characteristic is a cardioid of revolution. In the
case of both the bidirectional and the cardioid unidirectional microphones,
the ratio of energy response to generally reflected sound is one-third that of a
nondirectional microphone. It is interesting to investigate the efficiency
of response to random sound of other ratios of sensitivity of the bidirectional
to the nondirectional unit.
The voltage output of a microphone consisting of a bidirectional and non­
directional unit is given by
0
eUD =
0
Rl
+ R2 cos 8
where Rl = voltage putput of the nondirectional microphone, and
R2 = voltage output of the bidirectional unit for 8 = O.
44
Olson, H. F., Broadcast News, No. 30, p. 3, May, 1939.
8.72
294
,
~
w 41\
II)
,
~
w3
a:
A
i\
,
\\
:33 \ r-..'
~
I FT.
rr'
i\\
w
~
..t!. 0·
I­
~
30~YI
2:2
«...J
I
B~
2 fEET
UNIDIRECTIONAL
If
l'l...~0 90· I
Ill~ ~ .I,r~~o·
w
a: I
100o
00
FREQUENCY
0
5
I FOOT
:Jl4 \ l'~
....fJF . r--- ....... ....
a: I
~~
a:
w
>2
W
UNIDIRECTIONAL.
5
0.
\
~ "...J
VELOCITY
1\
\
;Z
ACOUSTICAL ENGINEERING
-rI'#1000
100
FREQUENCY
UNIDIRECTIONAL.
5
5 FEET
:;j4
;Z
0
l'~
\' ~~
,
2:2 \.' ~'
w
I­
«
...J
l1 3
o·
~
w
a: I
U;O·
cOzo
0.
II)
I
"-30
w
180·
90"
~
120"...
~
100
fREQUENCY
-";::t-.
O·
2:2
« ~
.....30" ..-I6C1'
I-
1'60~
...J
w
a:
1000
I~ "-
) f::::!=I.
DO20
IsO"
9O·
I
.....
,
120"
100
FREQUENCY
°
FIG. 8.40. A. The relative voltage output of a velocity (or pressure
gradient) microphone as compared to a nondirectional pressure micro­
phone for distances of 1, 2, and 5 feet. E, C, D, the relative voltage
output of a unidirectional microphone as compared to a nondirectional
pressure microphone for distances of 1, 2, and 5 feet and for various
angles of the incident sound.
The efficiency of energy response of the unidirectional microphone as com­
pared to a nondirectional microphone for sounds originating in random
directions, all directions being equally probable, from equation 8.117, is
..
.
21T fo (RI + R2 cos 8)2 sin 8 d8
DIrectIOnal efficIency =
41T(Rl + R2)2
1 (Rl
6
+ R2)3 - (RI (Rl + R2)2Rz
R z)3
8.73
For the standard velocity microphone Rl = 0, R2 = 1, and the ratio is
t. For the cardioid unidirectional Rl = 1, R2 = 1, and the ratio is t.
However, for other values the ratio is different. For example, between
Rl/R2 =
to RI/R2 = 1 the efficiency is less than t and becomes .25 for
°
295
MICROPHONES
Rl/R2 = .33. The efficiency for various values of the ratio Rl/R2 is
shown in Fig. 8.41. The data in Fig. 8.41 show that it is not important
that the two microphones be of the same sensitivity. It is important,
however, that the ratio Rl/R2 be equal to 1 or less than 1.
The same results are shown in Fig. 8.42 by means of polar diagrams.
Fig. 8.41 shows that the energy response of the bidirectional microphone
I)
9
8
.7
/"
c .6
z
-I­'" _5
/"
3,V
V
--­
V
- -
V
~8 ~e-=
/'
c
'" .4
V
~
~
I-­ II'"
V
T--!-'
-
-
I
.1
0
o
2
3
4
5
R,+ R2
6
7
8
9
10
20
30 40
<Xl
FIG. 8.41.
The ratio of energy response to random sounds of a
directional microphone consisting of a bidirectional and a nondirec­
tiona! unit as a function of the ratio of the outputs of the elements,
as compared to the nondirectional microphone. END = energy
response of a nondirectional microphone. ED = energy response of
a directional microphone. Rl = voltage output of the nondirec­
tional unit. R2 = voltage output of the directional unit.
and the cardioid unidirectional is the same. However, for 0 < Rl/R2 < 1
the response to random sounds is less than in the case of either of these two
microphones.
3. Efficiency of Energy Response to Random Sounds of a Unidirectional
Microphone as a Function of the Phase Angle between the two Units. 45 -The
preceding discussions have assumed that the phase angle between the out­
puts of the two units did not change with frequency. There are two
principal sources of phase shift between the two units, namely, a phase
shift due to a finite separation, and a phase shift due to a difference in the
phase frequency characteristics.
Consider the case in which there is a phase shift c/> between the output
of the bidirectional and nondirectional units. The output of each separate
unit is eo volts. The output of the combination is
e = eoV(COs 8
45
+ cos c/»2 + (sin c/»2
Olson, H. F., Broadcast News, No. 30, p. 3, May, 1939.
8.74
296
ACOUSTICAL ENGINEERING
The efficiency of the energy response of the above system to that of a
nondirectional microphone is
Directional efficiency
27Te02
fo" [(cos 28 + 2 cos 8 cos cP +
cos 2 cp)
+ sin 2 CPJ sin 8 d8
8.75
167Te0 2
Directional efficiency
27Te o2
J:
[cos 2 8
+ 2 cos 8 cos cP + IJ sin 8 d8 = ~
8.76
3
167Te02
The efficiency is the same as in the case of no phase angle shift.
BIDIRECTIONAL-~~~----L-7>~--.--71"'~-----r.~-----"'tt---
NONDIRECTIONAL------"tt-----i1H-f--.+--+-t------''--t--t_-f---'Lf------\--+_
COMBINATION
--~-o<:---
"=io__~---'Ic-*~~-+-t_--+-+----\--+_
R,
R,
-=00
-=1
R2
ENERGY RESPONSE
TO RAfilDOM SOUND
R2
.L
3
FIG. 8.42. Directional diagrams of various combinations of bidirectional and non­
directional microphones and the energy response to random sounds.
If the units are separated by a finite distance d, then there will be a
phase difference between the units which is
cP =
d
X360 cos 8
8.77
where d = distance between the units, in centimeters,
A = wavelength, in centimeters, and
8 = angle between the direction of the incident sound and the
normal to the ribbon.
Note that this separation is in line with the units.
(d/A) 360 cos 8 = K cos 8 in equation 8.74 the output is
e = eoV[cos 8 + cos (K cos 8)]2
Substituting cP
+ [sin (K cos 8)J2
=
8.78
MICROPHONES
297
The efficiency of the energy response of the above system to a non­
directional system is given by
Directional efficiency
27Te02
Jo" {[cos 8 + cos (K cos 8)]2 + [sin (K cos 8]2} sin d8
16n-e0 2
1
3
8.79
That is, the efficiency is independent of the separation between the units.
Of course, for very large distances the separation disturbs the response for
8 = O. However, in the conventional microphone this does not occur.
Therefore, the effect of finite size has no effect on the efficiency of energy
response to random sounds.
4. Distortion of the Directional Pattern in the Unidirectional Microphone.
-Deviations from the cardioid characteristic in the unidirectional micro­
phone are due to
1. Phase shift in the velocity microphone due to deviation from a pure
mass reactance
2. Phase shift in the velocity microphone due to diffraction
3. Phase shift in the pressure microphone due to deviation from resistance
control
4. Phase shift in the pressure microphone due to diffraction
5. Deviation in the output from a cosine directional characteristic in the
velocity microphone
6. Deviation in output with angle in the pressure microphone
7. Unequal sensitivity of the two elements
The phase angle between the output of a velocity microphone and the
particle velocity in a plane wave has been considered in Sec. 8.3B. The
phase shift in a pressure ribbon microphone has been considered in Sec.
8.2D3. It is possible to adjust these phase shifts and those due to diffrac­
tion so that the cancellation for 180 0 will be of the order of -30 db up to
10,000 cycles. In the case of the dynamic pressure unit the problem of
maintaining appropriate phase shifts is more difficult.
B. Single-Element Unidirectional Microphones. 46 -Unidirectional micro­
phones consisting of the combination of a nondirectional and a bidirectional
microphone have been described in Sec. 8.4A. It is the purpose of this
section to describe single-element unidirectional microphones in which a
unidirectional pattern is obtained by combining a single-element electro­
acoustic transducer with a phase-shifting network.
1. Phase-Shifting Unidirectional Microphone.-A unidirectional micro­
phone consisting of a nondirectional and bidirectional microphone has been
described in the preceding section. It is the purpose of this section to
describe other means for obtaining directional response.
The elements of a phase-shifting microphone are shown in Fig. 8.43.
The open ends of the pipes are separated by a distance D. A bend of
46
Olson, H. F., Jour. Soc. Mot. Pic. Eng., Vol. 52, No.3, p. 293, 1949.
298
ACOUSTICAL ENGINEERING
length d is placed in the shorter pipe. The ribbon element measures the
difference in pressure between the two pipes. The difference in pressure
between the two pipes is given by
D..p
where po
=
D =
d=
,\. =
8=
I
=
. (d-D
2po SIll
-A-
7T
+ D7T
T cos 8)
8.80
sound pressure, in dynes per square centimeter,
separation between the receiving ends of the pipes, III centi­
meters,
acoustic path introduced by the bend, in centimeters,
wavelength, in centimeters, and
angle the incident pencils of sound make with the aXIS of
the system.
r--
D
-4
/'
RIBBON
DAMPED
t
PI PES
....... .............
d=2D
d=D
-
,"
110
180
180
180
180
FIG. 8.43. A directional microphone employing a phase shifting system. The polar
graphs show the directional characteristics for various ratios of diD. The polar graph
depicts the output, in volts, as a function of the angle, in degrees. The maximum
response is arbitrarily chosen as unity.
If the distances D and d are small compared to the wavelength, D..p will
be proportional to the frequency. If a mass-controlled, electrodynamic
element is used, the output will be independent of the frequency.
A series of directional characteristics for various ratios of D to d is shown
in Fig. 8.43
2. Polydirectional Microphone.-The single element polydirectional
microphone47 is shown in Fig. 8.44. The ribbon is located in the air gap
formed by the pole pieces. A permanent magnet supplies the flux to the
air gap. The entire one side of the ribbon is covered by the labyrinth
connector. The connector, in turn, is coupled to a damped pipe or labyrinth.
The type of directional characteristic is governed by the size of the aperture
in the labyrinth connector.
The action of this microphone can be obtained from Fig. 8.45 which
shows the schematic view of the microphone and the acoustical network.
The sound pressure acting on the open side of the ribbon may be written
Pi =
47
POlEi(wt+4>l)
Olson, H. F., Proe. Insf. Rad. Eng., Vol. 32, No.2, p. 77, 1944.
8.81
299
MICROPHONES
where POI = amplitude of the pressure, in dynes per square centimeter,
w =
27Tj,
j = frequency, in cycles per second,
t = time, in seconds, and
q,1 = phase angle with respect to a reference point, in radians.
FIG. 8.44. The elements of a single-ribbon poly­
directional microphone.
The sound pressure acting on the aperture in the labyrinth connector may
be written
8.82
where P02
=
q,2 =
amplitude of the pressure, in dynes per square centimeter, and
phase angle with respect to a reference point, in radians.
M.
r.s
Ms TAS
MArAA
MRCAR:
P,
r
rA •
~2rA2
ZA2
z.,
r' 3
FRONT
VIEW
1.12
TA •
~
CROSS-SECTIONAL
VIEW
ACOUSTICAL
r' 3
NETWORK
FIG. 8.45. Front view, cross-sectional view, and the acoustical network of
a polydirectional microphone. In the acoustical network, ME and CAR =
the inertance and acoustical capacitance of the ribbon. MA and 1'AA = the
inertance and acoustical resistance of the air load on the front of the ribbon.
M sand r AS = the inertance and acoustical resistance of the slit between the
ribbon and pole pieces. M 2 and r A 2 = the inertance and acoustical resis­
tance of the aperture in the pipe. l'A3 = the acoustical resistance of the
damped pipe. ZAE = acoustical impedance due to the electrical system.
PI = the sound pressure at the front of the ribbon. P2 = the sound pres­
sure at the back of the connector.
The reference point for the phase may be changed so that
PI =
POlEj(wt)
8.83
and
8.84
ACOUSTICAL ENGINEERING
300
The phase angle 1>3 is a function of the angle of the incident sound as
follows:
8.85
1>3 = 1> cos (J
where
(J =
1>
1>
=
=
angle between the normal to the surface of the ribbon and the
direction of the incident sound,
phase angle for (} = 0, and
function of the frequency.
The volume current, in cubic centimeters per second, of the ribbon due to
the pressure PI is
Xl
ZAl = rAA
r AS -
wZrAsMRC AR
ZA2 =
rA2
+
PI(ZA2
ZA3)
ZA1ZA2
ZA1ZA3
ZA2ZA3
+
+
8.86
+
+ jwMA +
where
1-
=
+
jwMs - jw 3M sMRC AR - wMsC ARZAE
jwC ARrASZAE
2
W C AR(MR
Ms)
jwrASC AR
jwCARZAE
+ jwM2
+
+
+
ZA3 = rA3
r AA
acoustical resistance of the air load on the ribbon, in acoustical
ohms,
M A = inertance of the air load on the ribbon, in grams per (cent i­
meter)4,
r AS = acoustical resistance of the slit between the ribbon and the
pole pieces, in acoustical ohms,
M s = inertance of the slit between the ribbon and the pole pieces,
in grams per (centimeter)4,
MR = inertance of the ribbon, in grams per (centimeter)4,
CAR = acoustical capacitance of the ribbon, in (centimeter)5 per dyne,
r A2 = acoustical resistance of the aperture, in acoustical ohms,
M2 = inertance of the aperture, in grams per (centimeter)4,
r A3 = acoustical resistance of the damped pipe, in acoustical ohms,
and
ZAE = acoustical impedance due to the electrical circuit in acoustical
ohms (see equation 8.23).
=
Since the acoustical impedance due to the inertance and acoustical re­
sistance of the slit between the ribbon and pole pieces is very large com­
pared to the acoustical impedance of the ribbon, these two elements may
be neglected. Further, since the resonant frequency of the ribbon is placed
below the audible range, the acoustical impedance due to the acoustical
capacitance of the ribbon may be neglected for the audible-frequency range.
Then,
301
MICROPHONES
The volume current, in cubic centimeters per second, of the ribbon due to
the pressure P2 is
X2 =
P2(ZA3)
ZAlZA2
+ ZAIZA3 + ZA2ZA3
8.87
The resultant volume current, in cubic centimeters per second, X R , of the
ribbon is the difference between equations 8.86 and 8.87,
XR
Xl
=
~
X2
8.88
The value of the phase angle, C/>, can be determined from the geometry of
the microphone. The values of the impedance can be determined from the
mass and dimensions of the ribbon, the area of the damped pipe or labyrinth,
and the diameter of the aperture in the labyrinth connector.
The directional characteristics of the microphone are controlled by varying
the area of the aperture in the labyrinth connector. The effect of varying
the aperture can be obtained from the schematic view and the acoustical
network (see Fig. 8.46).
In Fig. 8.46A the aperture is so large that the back of the ribbon is
effectively open to the atmosphere. In this case the acoustical impedance
ZA2 is zero.
Therefore, the acoustical resistance, r AS, of the labyrinth is
effectively short-circuited. The action then is exactly the same as that of
the velocity microphone. From equations 8.86, 8.87, and 8.88 the volume
current of the ribbon is
XR
=
Xl ~ X2 = (PI ~ Pz)
8.89
ZAI
If the amplitudes of PI and
XR
pz are equal,
=
(POI
~
then
POI €irJ>COS8) €iwt
ZAI
If the angle
c/>
8.90
is small
X R = PIc/> cos 8 = I:1p cos 8
ZAI
ZAI
8.91
where I:1p = Pic/> the difference in pressure between the two sides of the
ribbon. Equation 8.91 will be recognized as that of the velocity micro­
phone. The directional characteristic is bidirectional.
In Fig. 8.46E the aperture is closed. In this case the acoustical im­
pedance ZA2 is infinite. Under these conditions the pressure pz is ineffective.
From equations 8.86, 8.87 and 8.88 the volume current of the ribbon is
given by
XR
=
Xl ~ X2
=
Xl
PI
=
ZAI
+ ZA3
8.92
Equation 8.92 will be recognized as that of the pressure ribbon microphone.
The directional characteristic is nondirectional.
Using an aperture which may be varied, it is possible to obtain any
limacon characteristic between the cosine bidirectional Fig. 8.46A and the
302
ACOUSTICAL ENGINEERING
non directional characteristic Fig. 8.46E, as depicted by Fig. 8.46, parts B,
e, and D. The directional characteristic of Fig. 8.46C is given by
e= R
R cos 8
8.93
This is a cardioid characteristic which is obtained in the two-element uni­
directional microphone by making the output of the bidirectional element
+
"u
r"' l
rA2
M,
·'n'
·'n'
TA2
rAI
ZA.
~.
r.,
ZA'
r.,
rAI
ZAI
~
A
M,
"'""""'"
ZA'
!..
~.
rA,
ZA. r..
'W
rAJ
Z. .
rAJ
rAI
rA2
B
p.
M,
ZA'
TA,
M.
lA.
TA•
rAJ
lA'
TA•
·'tr
rAI
1"2
C
M,
rAI
IA.z
TA,
lA. TA•
M.
p.
TAJ
ZAl
TA•
., 0·'
lA'
D
M,
ZA'
Pz - -
ZAZ=tD
TAl
ZAl
rA•
TA,
E
ACOUSTICAL
SYSTEM
ACOUSTICAL
NETWORK
DIRECTIONAL
CHARACTERI STIC
FIG. 8.46. The acoustical system, the acoustical
network, and the directional characteristics of
the polydirectional microphone for various values
of the aperture M2, 1'A2.
equal to the nondirectional unit.
is given by
The directional characteristic of Fig. 8.46B
R
e = '2
+ 3R
2 cos 8
8.94
For a wider directional pickup angle the characteristic of Fig. 8.46D may
be more desirable. This characteristic is given by
e
8R
=
7
+ 6R
7 cos 8
8.95
MICROPHONES
303
The energy response to random sounds as compared to that of a non­
:iirectional microphone is t for the bidirectional characteristic, Fig. 8.46A,
3.lld the cardioid characteristic, Fig. 8.46C. The energy response for the
characteristic of Fig. 8.46B is 1. This is the maximum value of discrimina­
tion obtainable in this microphone. That is, the energy response varies
~rom t to 1 and back again to t in going from the bidirectional charac­
teristic, Fig. 8.46A, to the cardioid characteristic of Fig. 8.46C. The
~nergy response of the characteristic of Fig. 8.46D given by equation 8.95
is 0.39. The energy response varies from t to 1 in going from the cardioid
characteristic of Fig. 8.46C to the nondirectional characteristic of Fig.
8.46E. The general expression 48 for the directional characteristics obtain­
able with this microphone is
e
=
Rl
+ R2 cos ()
8.96
The ratio of the energy response of this microphone as compared to a non­
directional microphone for any ratio of Rl to R2 is shown in Fig. 8.4l.
3. Uniaxial Microphone. 49 ,50-A unidirectional microphone in which
the maximum directivity corresponds to the axis of the microphone is
termed a uniaxial microphone. A first-order gradient uniaxial microphone
is shown in Fig. 8.47. The transducer consists of a ribbon terminated in a
damped pipe coupled with phase-shifting networks. The action of the
system may be deduced from a consideration of the acoustical network.
The two holes in the labyrinth connector form the essential portion of the
phase-shifting network so that the directional pattern will be of the uni­
directional type. The front face of the microphone is equipped with two
lobes. The lobes perform the following functions: the reduction of the
:ieleterious effects of diffraction, the accentuation of the high-frequency
response, and the support of the blast baffies. The front of the microphone
is equipped with two blast baffies and the side holes are equipped with
:lingle blast baffies. There is an additional element that contributes to
increased directivity, namely, the damped cavity between the magnets.
The directivity pattern of the uniaxial microphone for a certain set of
constants may be expressed as
e = eo (.3
where eo
=
() =
()
+ .7 cos () cos 3)
8.97
sensitivity constant of the microphone, and
angle between the normal to the plane of the ribbon and the
direction of the incident sound wave.
48 A limacon is a curve defined by e = a + b cos 0.
When a ~ 0, e = b cos 0,
a bidirectional characteristic. When b = 0, e = a, a nondirectional characteristic.
When a = b, e = a + a cos 0, a cardioid characteristic. For other values of a and b
any type of characteristic of this family may be obtained.
49 Olson, Preston, and Bleazey, RCA Review, Vol. 14, No. 1. p. 47,1953.
50 Olson, Preston, and Bleazey, Trans. IRE, Prof. Group Audio, Vol. AU-I, No.4,
p. 12, 1952.
304
ACOUSTICAL ENGINEERING
ACOUSTICAL NETWORK
8.47. Sectional view and acoustical network of a uniaxial
microphone. In the acoustical network, PI = the sound pressure
on the front of the microphone. MAl and r AAI = the inertance and
acoustical resistance of the air load on the front of the microphone.
M Bl, r ABI, M' BI, and r'ABI = the inertances and acoustical resis­
tances of the blast baffies on the front of the microphone. C AcI and
CAcI = the acoustical capacitances of the volumes between the
blast baffies. M sand r AS = the inertance and acoustical resistance
of the slit between the ribbon and pole pieces. M B , rAR, and
CAR = the inertance, acoustical resistance, and acoustical capaci­
tance of the ribbon. ZAE = the acoustical impedance due to the
electrical circuit. P2 = the sound pressure at the apertures in the
labyrinth connector. M A2 -and rAA2 = the inertance and acoustical
resistance of the air load at the apertures of the labyrinth connector.
M B2 and r AB2 = the inertance and acoustical resistance of the blast
baffies on the side on the microphone. C Ac2 = the acoustical
capacitance of the volume behind the blast baffie. M2S and rA2S =
the inertance and acoustical resistance of the screen covering the
hole in the labyrinth connector. M2 and rA2 = the inertance of the
hole in the labyrinth connector. rAP = the acoustical resistance of
the labyrinth. P3 = the sound pressure at the damped cavity
behind the magnets. M A3 and rAA3 = the inertance and acoustical
resistance of the air load upon the damped cavity. M B3 and
r AB3 = the inertance and acoustical resistance of the blast baffie over
the damped cavity. C Ac3, rAG3, and r'Ac3 = the acoustical capaci­
tance and acoustical resistances of the cavity between the magnets.
M = the coupling between the cavity and the apertures.
FIG.
MICROPHONES
305
The directional pattern of the first-order gradient uniaxial microphone is
shown in Fig. 8.48. It will be seen that the directivity is greater than that
of a cardioid pattern obtained with a conventional unidirectional microphone
in that the response is down 10 db at 90° and 26 db at 180°. This increased
directivity is due to the damped cavity between the magnets. The response
of the uniaxial microphone to random sounds is 1. The corresponding
response for a microphone with a cardioid pattern is t. This means that
the response of this microphone to random sounds is 60 per cent that of a
microphone with a cardioid directional characteristic. From the standpoint
of sound pickup distance, the uniaxial microphone will operate at 30 per
FIG. 8.48.
Directional characteristic of the uniaxial micro­
phone of Fig. 8.47.
cent greater distance than the unidirectional microphone with a cardioid
directional characteristic for the same reverberation, undesirable sounds, or
noise. Tests of the blast proofing shows that the microphone will stand the
firing of a 45 caliber pistol firing blanks at a distance of 5 feet indoors with
the direction of firing at right angles to the microphone. It will withstand
the same firing at smaller distances outdoors.
4. Uniphase Dynamic Microphone.-The uniphase dynamic micro­
phone 51 ,52 is a unidirectional microphone employing a diaphragm-voice
coil transducer unit and a phase-shifting acoustical network to obtain uni­
directional characteristics. Schematic views and the acoustical network of
the unidyne microphone are shown in Fig. 8.49. The diaphragm and voice
coil assembly is mounted on two spiders. The clearance between the voice
coil and pole piece is used as one of the phase-shifting elements, M 2rA2.
51
52
Bauer, B. B., Jour. Acous. Soc. Amer., Vol. 13, No.1, p. 41, 1941.
Bauer, B. B., Electronics, Vol. 15, No. 1, p. 31, 1942.
306
ACOUSTICAL ENGINEERING
The volume behind the diaphragm forms an acoustical capacitance, C A2.
The volume in the magnet structure forms another acoustical capacitance,
C.43. C A3 is coupled to C A2 by means of the silk cloth which forms the
acoustical resistance element, r A2. The performance of the system may
be determined from a consideration of the acoustical network. The con­
stants of the acoustical network are selected so that the difference in pressure
between the two sides of the diaphragm is proportional to the frequency.
Under these conditions uniform response with respect to the frequency will
ACOUSTICAL
NETWORK
A
FRONT
VIEW
SECTION A-A
FIG. 8.49. Front view, sectional view, and the acoustical
network of a unidyne unidirectional microphone. In the
acoustic circuit, M l • r Al, and CAl = the inertance, acousti­
cal resistance, and acoustical capacitance of the diaphragm
and suspension system. M 2 and r.t2 = the inertance and
acoustical resistance of the slit between voice coil and pole.
C A 2 = the acoustical capacitance of the air space between
the diaphragm and pole. M3 and YA3 = the inertance and
acoustical resistance of the silk cloth. C A3 = the acousti­
cal capacitance of the air space in the magnet. Pl = the
pressure at diaphragm. P2 = the pressure at the voice
coil.
be obtained if the diaphragm system is a mass reactance. In a particular
model of the microphone the constants were selected so that the directional
pattern is that corresponding to a ratio of Rl/R2 = .5 in Fig. 8.49.
A unidirectional microphone 53 employing a ribbon conductor and essen­
tially the same acoustical system as that of the microphone of Fig. 8.49
is shown in Fig. 8.50. The back of the ribbon is terminated in an acoustical
capacitance in shunt with an acoustical resistance and inertance in series.
The phase-shifting network shown in Fig. 8.50 provides unidirectional
response with a single ribbon transducer.
53
Bauer and Medill, Conv. Record, IRE, Part 6, Audio, p. 12, 1954.
MICROPHONES
307
MAGNET
RIBBON
POLE
ACOUSTICAL NETWORK
FRONT VIEW
FIG. 8.50. Front and sectional views and the acoustical network of
a ribbon-type unidirectional microphone with lumped acoustical
elements. In the acoustical network, Pl = the sound pressure on
the front of the microphone. MA = the inertance of the air load
upon the ribbon. Mil and CAll = the inertance and acoustical
resistance of the ribbon. CAl = the acoustical capacitance of the
cavity behind the ribbon. M 1 and I'Al = the inertance and acousti­
cal resistance of the cloth covering the chamber. pz = the sound
pressure on the back of the microphone.
5. V ariable-Distance Unidirectional Microphone. 54_A unidirectional
gradient microphone in which the front to back distance varies approxi­
mately inversely as the frequency over a major portion of the audio-frequency
range is shown in Fig. 8.51. There are four actuating pressures as follows:
h acting on the front of the diaphragm of the microphone and P2, h, and
P4 acting through acoustical impedances upon the back of the diaphragm.
The networks are designed so that a unidirectional pattern is obtained as
outlined in the preceding sections. In addition, the effective difference in
pressure at the low-frequency range involve hand P4, through intermediate
steps of hand P3 in the mid-frequency range and hand P2 in the high­
frequency range. With the variable type of pickup the proximity effect
is reduced.
6. Directional Condenser Microphone. 55 -A directional microphone em­
ploying a condenser unit as the transducer is shown in Fig. 8.52. The
transducer unit consists of two damped diaphragms. The vibrating system
consists of the two separate and spaced diaphragms. Each diaphragm is
spaced at a small distance from the back plate. The damped space provides
an acoustical capacitance and acoustical resistance for the diaphragm. The
54
55
Wiggins, A. M., Jour. Acous. Soc. Amer., Vol. 26, No.5, p. 687, 1954.
Bauch, F. W.O., Jour. Audio Eng. Soc., Vol. 1. No.3, p. 232,1953.
308
ACOUSTICAL ENGINEERING
r
Mo
SECTIONAL VIEW
Co
rA2
PI
I
I
CA2
M3
MI
fA4
M4
rA3
r1' r '1' f'i
ACOUSTICAL NETWORK
FIG. 8.51.
Sectional view and acoustical network of a variable
distance microphone. PI = the sound pressure on the front
of the microphone. P2, Pa, and P4 = the sound pressures
acting at different parts on the back of the microphone.
MD = inertance due to the mass of the diaphragm.
CAD =
the acoustical capacitance of the diaphragm suspension
system. CAl = the acoustical capacitance of the volume
back of the diaphragm. rAI = the acoustical resistance of
the shortest path. M 4, C A4, and rA5 = the inertance, acousti­
cal capacitance, and. acoustical resistance of the diaphragm
in the circuit of the medium path. M I , M3, CA3 , rA3, and
rA4 = the inertances, acoustical capacitance, and acoustical
resistances involved in the largest path. M2, rA2, and C A 2 =
the inertance, acoustical resistance, and acoustical capacitance
in a side branch.
two vibrating systems are placed back to back. The cavities behind the
diaphragm are interconnected by small holes. The phase shift in the
vibrating system combined with the electrical polarizing system makes it
possible to obtain a variety of directional characteristics as well as a non­
directional characteristic. With the potentiometer at the full negative
position a bidirectional pattern is obtained. With the potentiometer set
at the zero position a cardioid type of directional pattern is obtained. With
the potentiometer set at the full positive position a nondirectional pattern is
obtained.
7. Dipole Microphone. 56-A dipole microphone is a microphone in which
the response is a function of the sound pressure between two distinct points.
56 Olney, Slaymaker, and Meeker, Jour. Acous. Soc. Amer., Vol. 16, No.3, p. 172,
1945.
309
MICROPHONES
FRONT
DIAPHRAGM
..-__--ll-­
CIRCUIT DIAGRAM
FRONT VIEW
SECTION A-A'
FIG. 8.52. Front and sectional views and circuit diagram of the vacuum tube
amplifier of a directional condenser microphone.
A schematic view of the acoustical system is shown in Fig. 8.53. The
transducer is a carbon element. The use of the two tubes makes it possible
to remove the microphone transducer from a location directly in front of
IAI
ACOUSTICAL
Z"a
M,
CAl
NETWORK
MlcAlrAI
~'"
z..
~
ZA4
ZA2
Pz
ACOUSTICAL
SYSTEM
CAl
8.53. Schematic view and acoustical network of the dipole
microphone. In the acoustical circuit, ZAi and ZA2 = the acoustical
impedances of the silk cloth terminating the ends of the pipes.
ZA3 and ZA4 = the acoustical quadripoles representing the cylindri­
cal pipes. C A 2 and CAS = the acoustical capacitances of the air
chambers on the two sides of the diaphragm. M i , CA , and rAl =
the inertance, acoustical capacitance, and acoustical resistance of the
diaphragm and carbon button. Pl and P2 = the sound pressures at
the pipes.
F1G.
the talker's mouth and yet retain the acoustical advantage of a close­
talking microphone. The microphone and telephone receiver are made
an integral unit in a telephone operator's set. A disk of silk cloth covers
the end of each tube. The acoustical resistance termination practically
eliminates the resonance in the tubes.
The performance of the system may be determined from a consideration
of the acoustical network. The two pipes are represented as acoustical
310
ACOUSTICAL ENGINEERING
quadripoles. The performance of a cylindrical pipe has been considered
in Sec. 5.25.
The dipole microphone is a first-order gradient microphone. The
directional pattern is of the cosine type. The performance is essentially
the same as that of the phase-shifting microphone with ribbon element, save
that with the carbon element the output will be proportional to the fre­
quency in a plane wave. However, as a close-talking microphone the output
will be independent of the frequency. It also possess the antinoise charac­
teristics of a close-talking, first-order, gradient microphone.
C. 3
CA4
~~~
~
I
C
T
.'
ACOUSTICAL
C. a
~2
NETWORK
A
,---..,
/'
,
\
I
\
I
I
\
\
,
I
"
/
........
I
-,=:,--:",-C. 2
M2r.a 1>2
_-/ /
A'
FRONT VIEW
SECTION
A-I<
FIG. 8.54. Front view, sectional view and acoustical
network of the differential microphone. In the
acoustical network, M I , rAI, and M 2, rA2 = the
inertances and acoustical resistances of the two holes
in the case. C,i1 and CA2 = the acoustical capaci­
tances of the air chambers on the two sides of the
diaphragm. M3, rA3, and CA3 = the inertance, acous­
tical resistance, and acoustical capacitance of the
diaphragm. M 4 , rA4, and CA 4 = the inertance, acous­
tical resistance, and acoustical capacitance of the carbon
elements. PI and P2 = the sound pressures at the two
holes in the case.
8. Differential Microphone. 57 Lip Microphone.-A differential micro­
phone is a gradient type microphone used for close talking. In general,
it is held in place on the upper lip by a strap arrangement. A schematic
view of the acoustical system is shown in Fig. 8.54. The transducer is a
carbon element. The performance of the system may be determined from
a consideration of the acoustical network of Fig. 8.54. The differential
microphone is a first-order gradient microphone. The directional pattern
is of the cosine type. Employing a carbon element the output will be
proportional to the frequency in a plane wave. However, as a close-talking
microphone the output will be independent of the frequency. It also
possesses the anti-noise characteristics of a close-talking, first-order,
gradient microphone.
57
Ellithorn and Wiggins, Proc. lnst. Rad. Eng., Vol. 34, No.2, p. 84P, 1946.
311
MICROPHONES
8.5. Higher Order Gradient Microphones. 58 ,59-First-order pressure
gradient microphones have been described in Sec. 8.3. The response in a
first-order gradient microphone corresponds to the gradient of the sound
pressure. The response of higher order gradient microphones corresponds
to the order of the gradient of the sound pressure. The directional charac­
teristics of gradient microphones are cosine functions; the power of the
cosine is the order of the gradient. I t is the purpose of this section to
consider higher order gradient microphones.
A. Second-Order Gradient Microphones. 60 ,61_A gradient microphone of
order two is a microphone in which the response corresponds to the pressure
gradient of the pressure gradient.
The actuating force in a second-order gradient microphone is the dif­
ference in pressure between two two-point systems, and may be written
!::.(!::.P)
=
PMD1D2 [ -
k2r2 sin k(c - r)
+ 2kr cos
k(ct r3
r)
+ 2 sin k(ct -
r)]
cos
28
8.98
where Dl
D2
=
=
distance between the points in the pairs of points, and
distance between the two pairs.
A second-order gradient microphone may be made up of two oppositely
phased first-order gradient microphones as shown in Fig. 8.55.
The acoustical network of the acoustical system of one of the units in
the second-order microphone is shown in Fig. 8.55. The controlling element
in the system is an acoustical resistance. The transducer is of the dynamic
type. Therefore, in a plane wave, the voltage output of a single unit will be
proportional to the frequency. Connecting two of the units in opposition
the voltage output of the second-order gradient microphone will be pro­
portional to the square of the frequency. However, as a close-talking
microphone the output will be independent of the frequency. It possesses
the antinoise characteristics of a close-talking, second-order, gradient
microphone. The directional characteristics of the second-order gradient
microphone, as equation 8.98 shows, are bidirectional and proportional to
the square of the frequency.
B. Gradient Microphones of Any Order.62-The general expression for
the actuating pressure for a microphone of any order n for any two points
separated by a distance 8r is is
I5np
Olson,
Olson,
60 Olson,
61 Olson,
62 Olson,
58
59
H.
H.
H.
H.
H.
=
F., U. S.
F., Jour.
F., U. S.
F., Jour.
F., Jour.
r
onp n
0
.PM
or n 8r =orn (- J
Eik(ct-r») (I5r cos 8)n
Patent 2,301.744.
Acous. Soc. Amer., Vol. 17, No.3, p. 192, 1946.
Patent 2,301,744.
Acous. Soc. Amer., Vol. 17, No.3, p. 192, 1946.
Acous. Soc. Amer., Vol. 17, No.3, p. 192, 1946.
8.99
312
ACOUSTICAL ENGINEERING
CAl
M3f A3 C A3
~
rAI
101,
UNIT I
eA21t~rtli~~~~~~
VOICE
COILS
DIAPHRAGM
UNIT 2
VOICE
COIL
SECTIONAL
VIEWS
ELECTRICAL
CONNECTION
8.55. Sectional view, electrical connection, and acoustical network
of a second-order gradient microphone. The acoustical network applies to
a single unit. The electrical connection shows the two units connected in
opposition. In the acoustical network, r Al and M I = the acoustical
resistance and inertance at the front of the unit. YA2 and M2 = the
acoustical resistance and inertance at the back of the unit. CAl and
CA2 = the acoustical capacitances on the two sides of the diaphragm.
M 3 , rA3, and CA 3 = the inertance, acoustical resistance, and acoustical
capacitance of the diaphragm. PI and P2 = the sound pressures at the
front and back of the unit.
FIG.
(GRADIENT)·
(GRADIENT)'
cos e
(GRADlENT)2
(GRADIENT)'
cos"e
(GRADIENT)'
cos4 e
FIG. 8.56.
The directional characteristics of gradient microphones of order zero, one,
two, three, and four.
Equation 8.99 shows that the pressure available for driving the micro­
phone is proportional to the nth power of the frequency. The directional
characteristics are bidirectional cosine functions, the power of the cosine
is the order of the gradient. The directional characteristics for gradient
microphones of orders zero, one, two, three, and four are shown in Fig. 8.56.
C. Noise Discrimination of Gradient Microphones. 63-Gradient micro­
phones of order one and higher are directional. Therefore, these micro­
63
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 17, No.3, p. 192, 1946.
313
MICROPHONES
phones discriminate against sounds from random directions. The magnitude
of the discrimination is given by the expression in equation 8.117 as follows,
271"1" Rn 2 cos 2n sin 8 d8
Directional efficiency
where Rn
=
n
=
8=
Ro
=
=
8.100
4 71R 02
0
response of the gradient microphone on the axis,
order of the gradient,
angle between the axis of the gradient microphone and the
direction of the incident sound, and
response of the gradient microphone of order zero.
50
40
if!
...J
W
III
Li
w
o
30
'"'"
~20 b-,.
w
if!
ti
10
~
~
""~"
SECOND
~DER
FIR~
~
Q.
if!
W
0::
ORDE
~
r­ I---­
ORD
o
.2
.8
.4
I
2rrr
--­
2
4
8
~::
280
IYz
140
5&0
280
1120
560
2800
1400
5600
2800
11200
5600
11200
70
140
280
700
1400
2800
5600
3"
10
T
DISTANCE
FREQUENCY
IN
CYCLES
PER
SECOND
FIG. 8.57.
Response of zero-, first-, and second-order gradient
microphones to a small source as a function of 2TTr/>' where r =
distance and >. = wavelength. The response frequency charac­
teristics of all three are assumed to be independent of the frequency
for a plane wave, that is, 2TTr/>' = 00. The frequency scales below the
graph apply to three distances, namely, 3, I!, and! inches.
If the sensitivity of the gradient microphone of order zero is the same as
that of a gradient microphone or order n, equation 8.100 becomes
·
.
I effi'
D lrectlOna
Clency
=
2n 1+ 1
8.101
The above equation assumes that the distance between the origin of the
sound and the microphone is greater than nA, where n is the order of the
gradient, and A is the wavelength.
314
ACOUSTICAL ENGINEERING
A further increase in discrimination against noise and other undesired
sounds may be obtained if a gradient microphone is used as a close-talking
microphone. The response of gradient microphones of order zero, one, and
two to a small source as a function of the wavelength and distance from a
small sound source are shown in Fig. 8.57. The response frequency charac­
teristics of all three are assumed to be independent of the frequency for
a plane sound wave. Referring to Fig. 8.57 it will be seen that the response
of a gradient microphone is accentuated when the distance between the
sound source is less than n'\'. This feature of a gradient microphone may
be used to obtain high discrimination against unwanted sounds. If the
microphone is used as a close-talking microphone and the noises originate
ZERO
ORDER
o
III
..J
~
!oJ
m
FIRST
U -to
V
!oJ
o
!!:
./
-20
!oJ
III
~
A­
-30
rt)
V
/'
V
!oJ
a:
-40
-50
10 2
2
/
5~g~~~
/'
V
~
~
V
...­
.... ~
I­
V
4
FREQUENCY
8
IN
103
CYCLES
4
PER
8
104
SECOND
FIG. 8.58.
Response frequency characteristics of zero-, first-,
and Second-order gradient microphones to a plane wave. The
microphones are compensated so that the responses of all three
are the same and independent of the frequency when operating at
a disfance of ! inch from a small sound source.
at a distance from the microphone, considerable discrimination against the
noise can be obtained. For example, assume that the distance between
the mouth and the microphone is ! inch which is the average distance for
a close-talking microphone, the response frequency characteristics of zero-,
first-, and second-order gradient microphone, as function of the frequency
are shown in Fig. 8.57. The response of the gradient microphones is
accentuated at the low frequencies. If compensation is introduced so that
the response of all three becomes uniform with respect to frequency for
the i-inch distance from the small sound source, the response frequency
characteristics for distant sounds will be as shown in Fig. 8.58. These
characteristics show the discrimination against distant axial sounds by the
first- and second-order gradient microphones as compared to a pressure or
315
MICROPHONES
zero-order gradient microphone. These characteristics apply to all first­
and second-order gradient microphones.
In general, noise and unwanted sounds originate in random directions.
Under these conditions additional discrimination will be introduced by the
directional pattern. The response of zero-, first-, and second-order gradient
microphones, compensated for uniform response at i-inch distance, to
distant sound originating in random directions is shown in Fig. 8.59. First­
order gradient antinoise microphones have been described in Secs. 8.4B8
and 8.SB. The characteristics for a first-order gradient microphone
apply to these microphones. 64 A second-order gradient microphone has
ZERO
ORDER
o
III
..J
fiRST
ORD;!Y'
~ -20
w
III
Z
~ -30
III
w
V
/
" -40
-50
102
­
r-­
l--- -----/
W
III
8o -10
,/
/
SECO~
ORDER
V
2
,..,. ~
4
fREQUENCY
IN
/
a 103
CYCLES
2
4
PER
SECOND
8
10"
FIG. 8.59.
Response frequency characteristics of zero-, first-, and
second-order gradient microphones to random sounds originating at a
distance. The microphones are compensated so that the responses
of all three are the same and independent of the frequency when
operating at a distance of ! inch from a small sound source.
been described in Sec. 8.SA. The discrimination of the second-order
gradient microphone is tremendous. This has been substantiated by
actual tests in which it is impossible to drown out speech in a second-order
gradient microphone for any noise which the normal ear can withstand
without pain.
D. Higher Order Unidirectional Gradient Microphones. 65-It has been
shown in the preceding sections that the directivity of a gradient microphone
increases with increasing powers of the pressure gradient. The directional
characteristics of these systems are of the bidirectional type. In many
applications unidirectional characteristics are more desirable. Unidirec­
tional microphones employing first-order gradient units have been considered
64
65
Anderson and Wigginton, Audio Engineering, Vol. 34, No.4, p. 16, 1950.
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 17, No.3, p. 192, 1946.
316
ACOUSTICAL ENGINEERING
in Sec. 8.4. It is the purpose of this section to consider higher order com­
bination gradient microphones with unidirectional characteristics.
A higher order unidirectional gradient microphone may be obtained by
combining two first-order gradient microphones with a delay system as
shown in Fig. 8.60. The voltage output of this system is given by
ez
where eo
=
Dl
=
Dz
=
= eo(Dz + Dl cos 8)
8.102
cos 8
reference voltage output,
distance between the first-order gradient elements, and
path length of the delay.
Equation 8.102 holds for the frequency range in which Dl and Dz are small
compared to the wavelength. The reference voltage output is a function
EBCXJ
e,,=e~
(1+3cose)cose
FIG. 8.60. Higher order unidirectional microphone consisting of two similar
gradient units of order one and a delay system. The directional characteristics
of the gradient microphone of order one are shown as well as the directional
characteristics of the higher order gradient microphone for two different delay
conditions.
of the frequency and the type of electro acoustical generating system. The
maximum discrimination against random sounds occurs when Dz = iDl .
For this condition the energy response to random sounds is one-eighth
that of a nondirectional microphone. This is a very high order of directivity.
The directional characteristics for two different conditions are shown in
Fig. 8.60.
The system of Fig. 8.61 consists of two combination pressure and pressure
gradient microphones, described in Sec. 8.4, and a delay system. A number
of combinations are possible in this system, as for example, combination
units with various delays and dissimilar combination units with various
delays. The directional characteristics for two different conditions are
shown in Fig. 8.61.
E. Second-Order Gradient Uniaxial Microphone. 66-A second-order
gradient microphone with a unidirectional directivity pattern consisting of
the combination of two unidirectional microphones each consisting of two
66
Olson and Preston, RCA Review, Vol. 10, No.3, p. 339, 1949.
317
MICROPHONES
e.a=e~(I
+ cose)cose
e.a=e~<'1
+ 3COSe)Cose
FIG. 8.61. Higher order unidirectional microphone consisting of two unidirec­
tional elements described in Sec. 8.4 and a delay system. The directional charac­
teristics of two different unidirectional units are shown and the combination
higher order gradient for D2 >0 and Dl = O.
uniaxial, first-order gradient microphones as described in Sec. 8.4B3 is
shown in Fig. 8·.62.
The upper limit of the useful frequency range of a second-order gradient
microphone made up of two first-order gradient microphones, as shown in
Fig. 8.62 and schematically in Fig. 8.63, is determined by the distance
between. the units. This upper frequency limit is given by
c
ic = lJ
8.103
where ic = upper frequency limit, in cycles per second,
c = velocity of sound, in centimeters per second, and
D = distance between the units, in centimeters.
e. eo l.3 + .7cos6cos ~ I cose
FIG. 8.62. The elements of a second-order gradient uniaxial microphone
consisting of two first-order gradient uniaxial microphones.
318
ACOUSTICAL ENGINEERING
The voltage output of a gradient microphone of the type shown in Fig.
8.62 in the low-frequency range, that is, in the range for which D< <A is
given by
D
8.104
e= 2eo X 7T
where e =
eo =
D =
A=
voltage output of the combination, in volts,
voltage output of an individual unit, in volts,
distance between the units, in centimeters, and
wavelength, in centimeters.
Equations 8.103 and 8.104 establish the frequency range of operation
of a gradient microphone. A consideration of the requirements for a second­
order microphone indicated that a combination of two first-gradient uniaxial
~
=
=
c::=> =
==
=
==
=
==
c:=:::=:::::::: =
=
=
=
=
==
=
o c::::==::::::=>
0
FRONT SCREEN
c:::=::::J.
CONNECTOR SCREEN
UNIT NO.1
CABLE TO REAR
UNIT NO.2
CABLE
8.63. Schematic view of a second-order gradient microphone consisting
of two first-order gradient uniaxial microphones and the front and connector
screens.
FIG.
microphones of the type described in the preceding section could be used for
the elements. It also appeared that a high order of directivity was not
required in the high-frequency range. The microphone which was developed
operates as follows: The system is of the second-order gradient type up to
1000 cycles; there is a transition from second- to first-order gradient from
2000 to 4000 cycles, and above 4000 cycles the directivity pattern is the
same as the uniaxial microphone. As described in the preceding section,
the directional pattern of the uniaxial microphone is sharper than a cardioid.
Specifically, the directivity pattern up to 2000 cycles is given by
e = eo (.3
+ .7 cos 8 cos ~)
cos 8
8.105
where eo = sensitivity constant of the microphone, and
8 = angle between the axis of the microphone and the direction
of the incident sound wave.
It is in the region below 2000 cycles that practically all of the difficulty
due to reverberant and other undesirable sounds occurs. Since operation
319
MICROPHONES
shifts from the two microphones to the single microphone in the front in
the high-frequency region, it would be a comparatively simple task to
develop a microphone with a sharper directivity pattern in the high­
frequency region for use as the front microphone if this appeared to be
desirable.
The electrical system used with the two first-order gradient uniaxial
units is shown in Fig. 8.62. This consists of a network which transfers
from the two units in series opposition in the frequency range below 2000
cycles to the single unit above 4000 cycles. Suitable compensation circuits
are also included in the electrical system.
The polar directional patterns for 200, 1000, and 4000 cycles are shown in
Fig. 8.64. These patterns show a high order of discrimination for the sides
and rear hemisphere in the mid- and low-frequency range.
180
200 CYCLES
180 0
1000 CYCLES
180'
4000 CYCLES
8.64. The directional characteristics of a second-order gradient uniaxial micro­
phone for 200, 1000, and 4000 cycles.
FIG.
The application for the second-order gradient uniaxial microphone is
for the pickup of sound over large distances or under acoustically difficult
conditions where a high degree of directivity is desired, as for example, in
sound motion pictures and television. The size and weight of the microphone
are such that it may be mounted on a conventional boom. The response of
the second-order gradient microphone to random sounds is t that of a non­
directional microphone. The increased directional efficiency makes it
possible to use a pickup distance of more than 3 times that of a nondirectional
microphone and 1.8 times that of a unidirectional microphone with a cardioid
directional pattern.
8.6. Wave Type Microphones.-Directional microphones may be
divided into two classes as follows: first, wave-type microphones which
depend for directivity upon wave interference, and second, gradient-type
microphones which depend for directivity upon the difference in pressure or
powers of the difference in pressure between two points. In the first class
of microphone, in which the directivity depends in some way upon wave
interference, to obtain any semblance of directivity the dimensions of the
microphone must be comparable to the wavelength of the sound wave.
320
ACOUSTICAL ENGINEERING
Typical microphones of this classification are reflector, lens, and line micro­
phones. The second class of microphones has been considered in Secs.
8.3, 8.4, and 8.5. The dimensions of gradient microphones, as contrasted
to wave-type microphones, are small compared to the wavelength. It is
the purpose of this section to consider two examples of wave microphones,
namely, the parabolic reflector microphone and the line microphone.
A. Parabolic Rejlector. 67 ,68,69,70,71-Reflectors have been used for years
for concentrating and amplifying all types of wave propagation. The
surface of the parabolic reflector is shaped so that the various pencils of
incident sound parallel to the axis are reflected to one point called the
focus (Fig. 8.65). To obtain an appreciable gain in pressure at the focus,
the reflector must be large compared to the wavelength of the incident
sound. This requirement of size must also be satisfied in order to obtain
200 '"
SECTIONAL
1000'"
VIEW
FIG. 8.65. Cross-sectional view of a parabolic reflector for a microphone.
The polar graphs show the directional characteristics. The polar graph
depicts the pressure, in dynes, at the microphone as a function of the angle,
in degrees. The maximum response is arbitrarily chosen as unity.
sharp directional characteristics. If this condition is satisfied at the low
frequencies the size of the reflector becomes prohibitive to be used with
facility.
A cross-sectional view of a parabolic reflector and a pressure microphone
located at the focus is shown in Fig. 8.65. When the microphone is located
at the focus the gain at the high frequencies is considerably greater than at
the mid-frequency range. The accentuation in high-frequency response
may be overcome by moving the microphone slightly out of focus. This
expedient also tends to broaden the sharp directional characteristics at the
high frequencies.
The directional characteristics of a parabolic reflector 3 feet in diameter,
used with a pressure microphone, are shown in Fig. 8.65. It will be seen
67
68
69
70
71
Olson and Wolff, Jour. Acous. Soc. Amer., Vol. 1, No.2, p. 173, 1930.
Hanson, O. B., Jour. Acous. Soc. Amer., Vol. 3, No. 1, Part 1, p. 9, 1931.
Hanson, O. B., Jour. Acous. Soc. Amer., Vol. 3, No. 1, Part 1, p. 81, 1931.
Dreher, Carl, Jour. Soc. Mot. Pic. Eng., Vol. 16, No. 1, p. 29, 1932.
Olson and Wolff, Jour. Acous. Soc. Amer., Vol. 1, No.3, p. 410,1930.
MICROPHONES
321
that the directivity increases with frequency. For example, the system
is practically nondirectional at 200 cycles. On the other hand, the direc­
tional characteristiC is very sharp at 8000 cycles.
B. Lens Microphone. 72, 73_A lens microphone consists of an acoustic
lens arranged so that the pencils of sound arrive in phase at a common
point termed the focus, and a microphone located at focus. A schematic
sectional view of a pressure microphone located at the focus of an acoustic
lens is shown in Fig. 8.66. The directivity pattern is determined by a
INCIDENT SOUND
ACOUSTIC
LENS
FIG.
8.66.
Schematic view of a lens microphone.
relationship between the dimensions of the lens and the wavelength similar
to that of the parabolic reflectormicrophone.
C. Large-Surface Microphone.-A large-surface microphone,74 in the form
of a large number of dynamic microphone units arranged on a spherical
surface, is shown in Fig. 8.67. A curved surface source has been considered
in Sec. 2.20. The microphone shown in Fig. 8.67 is based upon the funda­
mental principles of a curved surface radiator or receiver. The angular
spread of the microphone shown in Fig. 8.67 is about 50°. The diameter is
four feet. The directivity pattern follows that of a curved surface system
of this diameter. The directivity pattern is reasonably uniform above 300
cycles. The low-frequency limit of uniform directivity could be extended
by employing a microphone of a larger diameter.
A large-surface microphone 75 in the form of a condenser microphone
consists of a nonstretched diaphragm designed to vibrate in phase over its
entire surface. The operating elements consist of an aluminum foil front
electrode cemented to a 3J32-inch foam rubber sheet mounted on an
aluminum back plate. The whole assembly is fitted in a picture frame
18 inches square and protected by a perforated metal face plate. The
directivity pattern is that of a square plate, considered in Sec. 2.17.
Clark, M. A., Jour. Acous. Soc. Amer., Vol. 25, No.4, p. 829, 1953.
Clark, M. A., Trans. IRE, Prof. Group on Audio, Vol. AU-2, No. 1. p. 5, 1954.
74 Olson, Preston, and May, Unpublished Report.
75 Aamodt and Harvey, Jour. Acous. Soc. Amer., Vol. 25, No.4, p. 825, 1953.
72
73
322
ACOUSTICAL ENGINEERING
DYNAMIC PRESSURE
MICROPHONE UNITS
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
FRONT VIEW
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
~
0
0
0
0
0
0
0
0
0
0
SECTIONAL VIEW
FIG. 8.67. Large surface microphone consIsting of a large
number of dynamic microphone units arranged upon a spheri­
cal surface.
D. Line Microphones. 76 ,77,78-A line microphone is a microphone con­
sisting of a number of small tubes with the open end, as pickup points,
equally spaced along a line and the other end connected to a common junc­
tion to a transducer element for converting the sound vibrations into the
corresponding electrical variations. In the line systems to be considered,
the transducer will be a ribbon element located in a magnetic field and ter-
FIG. 8.68. Line microphone.
Useful directivity on the line
axis. This microphone consists of a large number of small
pipes arranged in a line with the distance from the opening of
each pipe to the common junction decreasing in equal steps.
The system is terminated in a ribbon element and an acoustical
resistance.
Olson, H. F .• Jour. Inst. Rad. Eng.• Vol. 27. No.7. p. 438. 1939.
Mason and Marshall. Jour. Acous. Soc. Amer., Vol. 10. No.3. p. 206. 1939.
78 Olson. H. F .• Broadcast News. No. 28. p. 32, July. 1938.
76
77
MICROPHONES
323
minated in an acoustical resistance. Under these conditions the output of
the pipes can be added vectorially.
1. Line Microphone: Useful Directivity on the Line Axis. Simple Line.­
This microphone consists of a number of small pipes with the open ends,
as pickup points, equally spaced on a line and the other ends joined at
a common junction decreasing in equal steps (Fig. 8.68). A ribbon element,
connected to the common junction and terminated in an acoustical resistance
in the form of a long damped pipe, is used for transforming the acoustical
vibrations into the corresponding electrical variations.
The contribution, in dynes per square centimeter, by any element n at
the common junction of the microphone may be expressed as
P n -_ B n
cos 2'TI"
+ J'B n sin• 2
'TI"
(it
(it
J' -
J' -
Xn -
Xn
Xn -
Xn
Pn = Bn£2711/t£21Tj(xn-xn cos
where f
=
t
=
Xn =
,\. =
8=
Bn =
,\.
,\.
8)
cos 8)
cos
8.106
8.107
frequency, in cycles per second,
time, in seconds,
distance of the element n from the center of the line, in centi­
meters,
wavelength, in centimeters,
angle between axis of the line and the incident sound, and
amplitude of the pressure due to element n, in dynes per square
centimeter.
8)/;'
In the case of a uniform line, with the strength a constant, the resultant
when all the vectors are in phase is Bnl, where 1 is the length of the line.
The ratio, Re, of the response for the angle 8 to the response for 8 = 0 is
Re = -1Bnl
IJ' /
2
B n £21Tj[/t+(X-x cos 8)/;'] dx
-1/2
I
8.108
The absolute value of the term on the right is given by
Re =
~
1
IJ
12
/
£21Tj(X-X cos e)/A
-1/2
. ~ (l - 1 cos 8)
sin 1\
~ (l - l cos 8)
dx
I
8.109
8.110
The directional characteristics of the microphone of Fig. 8.68 for various
ratios of length of the line to the wavelength are shown in Fig. 8.69. These
characteristics are surfaces of revolution about the line as an axis. This
microphone is useful for collecting sounds arriving from directions making
small angles with the microphone axis.
324
ACOUSTICAL ENGINEERING
Lf:NGTH '"
~
LENGTH
=
%
180
LENGTH =2).
LENGTH
'.0
1.0
LENGTH =)..
180
4A
LENGTH'" 8;>,.
II)
'80
FIG. 8.69. The directional characteristics of the microphone shown in
Fig. 8.68 as a function of the ratio of the length of the line to the wave­
length. The polar graph depicts the output, in volts, as a function of the
angle, in degrees.
2. Line Microphone: Useful Directivity on the Line Axis. Line with
Progressive Delay.-As in the case of Fig. 8.68 this microphone consists of
a number of small pipes with the open ends, as pickup points, equally
spaced on a line and the other ends joined at a common junction. In
addition, there is inserted a delay which is proportional to the distance
from the end of the line or the pickup point nearest the common junction
(Fig. 8.70).
1 I
R8 =
2 B n f 2nj [(x-X cos 8IA)+d/Aj dx I
8.111
-I III
En l
-112
FIG. 8.70. Line microphone. Useful directivity on the
line axis. This microphone differs from Fig. 8.68 in that
a delay is inserted in each small pipe. The amount of
delay is proportional to the distance from the pipe opening
to the pickup point nearest the common junction.
325
MICROPHONES
where d is the path length of the delay introduced for the point furthest
removed from the common junction.
sin
Re=
i (l -
X(l -
I cos
I cos
(J
(J
+ d)
8.112
+ d)
The directional characteristic of the microphone of Fig. 8.70 for various
ratios of the length of the line to the wavelength, and for a delay path of
one-fourth times the length of the line is shown in Fig. 8.71. Comparing
LENGTH =~
LENGTH =~
LENGTH =A
2
LENGTH
=2}.
LENGTH
=4.>.
w.•
'00
.
,
•
FIG. 8.71. The directional characteristics of the microphone shown in Fig. 8.70 for a
time delay of one-quarter the length of the line as a function of the ratio of the length of
the lines to the wavelength. The polar graph depicts the output, in volts, as a function
of the angle, in degrees.
Fig. 8.71 with Fig. 8.69, it will be seen that the same directional characteristic
can be obtained with a shorter line by introducing appropriate delay. In
the case of a delay path comparable to the wavelength, loss in sensitivity
occurs.
3. Line Microphone: Useful Directivity on the Line Axis. Two Lines
and a Pressure Gradient Element.-This microphone consists of two lines
of the type shown in Fig. 8.70 arranged so that the ribbon element measures
the difference in the pressures generated in the two lines (Fig. 8.72). The
centers of the two lines are displaced by a distance D. In the line nearest
the element, a bend of length D is inserted between the junction and the
ribbon element.
To show the action of the pressure gradient system, assume that the
length of all the small pipes is the same and the openings between the two
sets are separated by a distance D. Under these conditions the line systems
are nondirectional.
The difference between the forces on the two sides of the ribbon, assuming
326
ACOUSTICAL ENGINEERING
that the mass mechanical reactance of the ribbon is large compared to the
mechanical resistance of the damped pipes, may be expressed as
fM
. (7TD A
cos fJ)
A cos (27Tft) sm
=
8.113
where A = constant, including the pressure of the impinging sound wave
and dimensions of the microphone.
If D is small compared to the wavelength, equation 8.113 becomes
fM
=
A
7TD
T
cos (27Tft) cos fJ
8.114
Equation 8.113 shows that the force available for driving the ribbon is
proportional to the frequency and the cosine of the angle fJ.
Employing mass-controlled ribbon of mass m r , the velocity is given by
x = j2~mr (7T~)
=
cos (27Tft) cos fJ
-zA (1TD) sin Z7Tft cos fJ
7Tmr c
8.115
This quantity is independent of the frequency and, as a consequence,
the ratio of the generated voltage to the pressure in the sound wave will be
independent of the frequency.
The above discussion assumes that the lines are nondirectional. The
<li'FrTI~~act"istia; of the individual lines of Fig. 8.72 Me given by
~~
DELAY
~",Q
8.72. Line microphone. Useful directivity on the line axis. This microphone
consists of two lines of the type shown in Fig. 8.70 displaced by a distance D along the
axis. In the line nearest the ribbon element a bend is inserted which introduces a path
length D. The ribbon element measures the difference in pressure in the two lines.
FIG.
equation 8.111. The directional characteristics of the microphone, shown
in Fig. 8.72, for D small compared to the wavelength are the product of
equations 8.112 and 8.115. The directional characteristics may be written
as
sin ~ (I - I cos 8 + d)
-------------------cos8
(I - I cos fJ + d)
X
8.116
327
MICROPHONES
The directional characteristics of the microphone shown in Fig. 8.72 for
various ratios of the length of the line to the wavelength for a delay of
one-quarter times the length of the line are shown in Fig. 8.73. A measure
of the value of a line with progressive delay and a pressure gradient element
for improving the directivity may be obtained by comparing Fig. 8.73 with
LENGTH=~
LENGTH
=1-
LENGTH'"
A
LENGTH
= 2A
'8.
LENGTH =4).
-.
'80
FIG. 8.73.
The directional characteristics of the microphone shown in Fig. 8.72 for a
time delay equivalent to one-quarter the length of the line as a function of the ratio of the
length of the line to the wavelength. The polar graph depicts the output, in volts, as a
function of the angle, in degrees.
Fig. 8.69. Employing these expedients approximately the same directivity
can be obtained with a line of one-quarter the length of the simple line shown
in Fig. 8.68.
4. Ultradirectional Microphone. 79-Directional microphones employing
lines of various types have been considered in the preceding section. These
directional characteristics indicated considerable variation with frequency.
Experience gained from work on reflectors indicated that a directional
characteristic which varies with frequency is undesirable, principally due to
the introduction of frequency discrimination for points removed from the
axis. In addition, the response to reflected sound is a function of the
frequency which alters the reverberation characteristics of received sound.
From the results of experiments upon directional systems, it appears that
a microphone with a small solid angle of pickup would be useful in recording
sound motion pictures, in television pickup, in certain types of sound broad­
cast as, for example, symphony and stage productions, and in many applica­
tions of sound reinforcing. However, the directional characteristics must
be independent of the frequency. This can be accomplished by employing
a number of separate lines, each covering a certain portion of the frequency
range. It is the purpose of this section to describe an ultradirectional
microphone consisting of five separate lines.
The ultradirectional microphone shown schematically in Fig. 8.74 consists
of five units. Units 1, 2, and 3 are of the type shown in Fig. 8.68. Units
4 and 5 are of the type shown in Fig. 8.72. An electrical filter system is used
to allocate the outputs of the units to their respective ranges. The response
characteristics of the units with the filter systems are shown in Fig. 8.75.
79
Olson, H. F., Jour. Inst. Rad. Eng., Vol. 27, No.7, p. 438, 1939.
328
ACOUSTICAL ENGINEERING
FIG. 8.74. Ultradirectional microphone consisting of five units.
Units 1, 2, and 3 are of the type shown in Fig. 8.68. Units
4 and 5 are of the type shown in Fig. 8.72. An electrical filter
system is used to allocate the output of the units to their respec­
tive ranges.
H
VI
'"zlOO
o
~
I­
~
80
l'\.
'\7
eo J
~40
u
a: ZO
J;'
<oJ
0.
®
@
!>
/
7
A
/
'\
~
[7
3
~
"-I"; F-
X
'"
.5
'"
/
,,/
IA
J
P\
6
FREQUENCY
7
0
®
r-..
~v
/
8 ' IO~
I N CYCLES
/
J
l)\.
~
...
PER
r-.....
,
6
7
8.'0.. .
SECOND
FIG. 8.75. Voltage response frequency characteristics of the units and electrical
filter system shown in Fig. 8.74.
Fig. 8.76 illustrates the principles used in obtaining uniform directional
characteristics. Fig. 8.76A is the directional characteristic of line 3 at
700 cycles. Fig. 8.76B shows the directional characteristics of lines 2 and
3 at 950 cycles. The resultant of these characteristics is also shown in
Fig. 8.76B. The same is shown in Fig. S.76C for 1250 cycles. In Figs.
8.76B and 8.76C the directional characteristic of line 2 is broader than Fig.
8.76A while the characteristic of line 3 is narrower. The resultant of
1.0
700
1.0
1.0
CYCLES
A
FIG. 8.76. A. The directional characteristic of line 3 of Fig. 8.74 at 700 cycles. B. The
directional characteristics of lines 2 and 3 and the resultant at 950 cycles. C. The direc­
tional characteristics of lines 2 and 3 and the resultant at 1250 cycles. D. The
directional characteristics of the microphone shown in Fig. 8.72 for the range from 85
to 8000 cycles fall within the shaded area.
329
MICROPHONES
lines 2 and 3 is a directional characteristic very close to Fig. 8.76A. The
directional characteristics of the microphone shown in Fig. 8.76 for the range
from 85 to 8000 cycles, except for the small lobes for angles greater than 90°,
fall within the shaded area of Fig. 8.76D. Considering that this microphone
has a frequency range of 6l octaves, it is a remarkably uniform directional
characteristic.
S.7. Throat Microphone. 8o ,81,82,83-The throat microphone is a micro­
phone actuated by direct contact of the diaphragm with the throat. A
perspective view and a sectional view of a carbon-type throat microphone
are shown in Fig. 8.77. Since the acoustical impedance of the flesh of the
UNIT
~~=~~
COMPLETE THROAT
MICROPHONE
SECTIONAL VIEW
OF UNIT
FIG. 8.77.
Complete throat microphone and sectional view of the carbon­
type throat microphone unit.
throat is very large compared to acoustical impedance of air, the acoustical
impedance of the vibrating system of the throat microphone can be made
correspondingly larger than the conventional air-type microphone. Since
the vowel sounds originate in the throat and the consonants in the head
the vowel sounds are predominant in the output. Furthermore, the high­
frequency consonant sounds are attenuated in passing through the throat.
Therefore, the high-frequency response must be accentuated to obtain
intelligible speech. The units shown in Fig. 8.77 are of the carbon type.
Other types of transducers as, for example, the magnetic type are also used
in the throat microphone.
S.S. Lapel, Lavalier, and Boom Microphones. 84 ,85-For certain
applications, particularly in public address and announce systems, a micro­
phone which can be hooked in the button-hole has been very useful. For the
Shawn, J., Communications, Vol. 23, No. 1, p. 11, 1943.
Martin, D., Jour. Acous. Soc. Amer., Vol. 19, No. 1, p. 43, 1947.
82 Greibach and Pacent, Elec. Eng., Vol. 65, No.4, p. 187, 1946.
83 Erickson, J. R., Bell Lab. Record, Vol. 23, No.6, p. 193, 1945.
84 Olson and Carlisle, Proc. Inst. Rad. Eng., Vol. 22, No. 12, p. 1354, 1934.
85 Jones and Bell, Jour. Soc. Mot. Pic. Eng., Vol. 19, No.3, p. 219, 1932.
80
81
330
ACOUSTICAL ENGINEERING
same applications a microphone mounted upon a small light boom supported
in a variety of ways has also been used. An example of a boom-type
microphone is the telephone operator's set in which the small carbon micro­
phone is supported on a boom attached to the headband. A lavalier micro­
phone is a term used to designate a small microphone supported by means of
a small band around the neck in the form of a pendant. The principal
purpose of the lapel, lavalier, and boom-type microphones is to allow the
person to walk and turn freely without introducing any appreciable change
in the output as would be the case if a stationary microphone were used.
It also allows the talker to use his hands as contrasted to a hand-held micro­
phone. Carbon, crystal, dynamic, and velocity microphones have been
used for these applications. The general design of lapel, lavalier, and boom
microphones is the same as the conventional microphones described in this
chapter except that the size is smaller.
8.9. Hot-Wire Microphone.-The hot-wire microphone consists of a
fine wire heated by the passage of an electri­
MOTION OF
AIR -----...:
z
cal current. Fig. 8.78. The cooling action
E2
due to the motion of air past the wire causes
FINE WIRE
1'1'1'
a change in electrical resistance of the wire.
ELECTRICAL CIRCUIT
In a sound wave the particle velocity cools
FIG. 8.78. The elements of a
the wire. There are also some other minor
cooling effects produced by the pressure in
hot-wire microphone.
a sound wave. The change in resistance due
to the passage of a sound wave may 'Je used to detect the presence of a sound
wave. However, the frequency of the electrical output is twice the frequency
of the sound wave because the wire is cooled equally by both positive and
negative particle velocities. The use of a direct current air stream for
=0
LOOP
ANTENNA
FIG. 8.79.
receiver.
Schematic diagrams of a radio transmitter microphone and radio
MICROPHONES
331
polarization appears to be impractical. Therefore, this microphone cannot
be used for the reproduction of sound.
8.10. Radio Microphone. 86-A radio microphone consists of a small
radio transmitter and microphone assembly and a radio receiver consisting
of an antenna and receiver. The advantage of the system is the elimination
of the microphone cable. Schematic diagrams of the radio transmitter
microphone and the radio receiver are shown in Fig. 8.79. The radio
transmitter is a conventional amplitude transmitter. The microphone,
transmitter, batteries, and antenna are housed in a case about eight inches
in length and one and one-half inches in diameter. In the system shown in
Fig. 8.79, a loop antenna is coupled to the receiver. The loop consists of a
single turn around the pickup area. A very wide range, automatic gain
control system is required in the receiver to compensate for the variations
in transmission due to standing waves.
8.11. Directional Efficiency of a Sound Collecting System. 87 ,88,89,
90,91,9~The ratio of energy response of a directional microphone as com­
pared to a nondirectional microphone, all directions being equally probable,
is termed the directional efficiency. The directional efficiency of a micro­
phone is given by
Directional efficiency
where 1(if;)
=
dn", =
=
~ fo47r ]2(if;) dn",
8.117
ratio of the voltage output for incidence at the angle if; to that
for if; = 0, and
element of solid angle at the angle if;.
The directional efficiency of a microphone is a measure of the energy
response to reverberation noise and other undesirable noise.
In many systems in which the directional pattern cannot be expressed
in simple terms which can be integrated, the determination of the direc­
tional efficiency must be carried out by numerical integration. The direc­
tional efficiencies of cosine functions are easily determined. Directional
patterns which are powers of the cosine function are plotted in Fig. 8.80.
The directional efficiency for these patterns is also given. For the same
signal to random noise, reverberation, etc., the directional microphone
may be operated at Ilv'directional efficiency times the distance of a non­
directional microphone.
86 Phinney, Thomas A., Trans. IRE, Prof. Group Audio, Vol. AU-2, No.2, p. 44,
1954.
87 Olson, H. F., Jour. Soc. Mot. Pic. Eng., Vol. 16, No.6, p. 695, 1931.
88 Olson, H. F., Jour. Acous. Soc. Amer., Vol. 3, No. 1, p. 56, 1931.
89 Olson, H. F., Proc, Inst. Rad. Eng., Vol. 21, No.5, p. 655, 1933.
90 American Standards Association Sectional Committee Z-24, Report on Calibra­
tion of Microphones, Jour. Acous. Soc. Amer., Vol. 7, No.4, p. 300, 1936.
91 Baumzweiger, B., Jour. Acous. Soc. Amer., Vol. 11, No.4, p. 447, 1940.
92 Bauer, B.B., Trans. IRE, PGA 8, July, 1952.
332
ACOUSTICAL ENGINEERING
fUNCTION
1+
cose
-2-­
COS
e
~~~::C~~~~L
fUNCTION
+2
COS e
-I -­
I
3"
I
6"
I
60
DISTANCE
GAIN
1.7
case
2.4
60
cos 2e
TO
cos 2e
3.2
cos e
..L
cos3 e
3.7
cos4 e
I
cos4 a
4.2
casee
~.I
3
coste
14
18
..L
26
90
150
150
8.80. The directional efficiency of microphones having directional characteristics,
which are various cosine functions. The ratio of energy response of a directional micro­
phone to the energy response of a nondirectional microphone for sounds originating in
random directions is termed directional efficiency. The ratio of a distance at which a
directional microphone may be operated as compared to a nondirectional microphone is
also shown. All characteristics are considered to be unidirectional-that is, one lobe.
FIG.
By means of the characteristics shown in Fig. 8.80, the efficiency of other
characteristics may be obtained by comparing with the cosine function
which has approximately the same shape.
The directional efficiency is also termed random efficiency and the inverse
the directivity index.
8.12. Wind Excitation and Screening of Microphones.-There are
three possible sources of excitation to which a microphone is subject when
placed in a wind. There may be pressure fluctuations due to velocity
fluctuations present in the wind even though the microphone is absent.
There may be pressure fluctuations due to turbulence produced by the
microphone in a wind otherwise free from pressure fluctuations, that is, in a
wind of uniform velocity. There may be radiation from the first two
sources. The effect of the first source may be reduced by screening which
takes advantage of the wind pressure distribution over the microphone, the
effect of the second by streamlining the microphone, and the third is mini­
mized by reductions in the first and second sources.
The customary wind screen consists of a frame covered with silk en­
closing the microphone (Fig. 8.81B). Very sheer silk reduces the response
to wind without appreciable attenuation of the sound. A spherical shape
has been found to offer the best shielding properties. The shielding proper­
ties increase with the volume of the shield.
In general. the response to wind is much higher for directions normal to
the diaphragm by applying the principles of hydrodynamics. A wind
screen has been developed which reduces the wind response of the micro­
333
MICROPHONES
phone. The Bernoulli93 wind screen is shown in Fig. 8.81A. The wind
pulses travel through the screen and exert a pressure on the diaphragm.
These same pulses cause a reduction in pressure at the periphery 1. These
two effects tend to balance each other and, therefore, the response to wind
is reduced. This type of screen reduces the wind response about 12 db.
FLOW
LINES
A
B
FIG. 8.81. Wind screens for microphones. A. Bernoulli wind screen applied to a
dynamic microphone. B. Wind screen consisting of a wire frame covered with sheer
silk.
8.13. Nonlinear Distortion in Microphones.-The sources of distortion
in microphones are, in general, the same as in the case of loudspeakers.
The two principal causes are due to nonlinear mechanical or acoustical
elements and nonuniform magnetic field in dynamic types. The latter
type of distortion can be made negligible in well-designed units. For
example, in a velocity microphone the amplitude of the ribbon for a plane
wave of 100 dynes per square centimeter at 30 cycles is less than a milli­
meter. The distortion due to a variation in the field over this distance is
less than fO' of one per cent. In the case of the velocity microphone the
system is mass controlled and there are no nonlinear elements. The measured
distortion (see Sec. 1O.2C) in a velocity microphone for sound pressures up
to 1000 dynes per square centimeter is less than t of one per cent at 80 cycles.
The most common source of nonlinear distortion in dynamic microphones
originates in the suspension system. In some cases at the lower frequencies
the harmonic distortion for a sound pressure of 100 dynes per square centi­
meter may be several per cent. This very high distortion is usually caused
by instability of certain portions of the suspension due to dissymmetry of
the corrugation and inhomogeneity of the material. As already pointed
out, the distortion in carbon microphones is very high due to the nonlinear
characteristics of granular contacts. Considerable improvement has been
93
Phelps, W. D., RCA Review, Vol. 3, No.2, p. 203, 1.938.
334
ACOUSTICAL ENGINEERING
made in carbon materials in recent years and the distortion, although still
high, has been materially reduced.
8.14. Transient Response of Microphones.-The subject of transient
response of vibrating systems, together with applications to loudspeakers
has been considered in Sec. 6.25. The measurement of transient response
of loudspeakers will be considered in Sec. 1O.3G. The transient response of
a microphone may be predicted from the mechanical or acoustical network
of the vibrating system.
In the case of the vibrating system of the mass-controlled velocity micro­
phone the response to transients is very good. The acoustical circuit of
Fig. 8.35 may be reduced to the simplified acoustical circuit of Fig. 8.82A,
provided the elements, M A and M R, the inertances due to the mass of the
A
ACOUSTICAL
8.82.
FIG.
c
B
CIRCUITS
Acoustical circuits of a velocity microphone.
MA and
!1P =
!1p = the difference in pressure between the two sides of the
ribbon. w = 21Tf. f = the frequency. P' = a sound pressure pro­
M
R
= the inertance due to the air load and the ribbon mass.
jwp'.
portional to the free-field sound pressure. Under the conditions
depicted A is equivalent to B. B is equivalent to C.
air load and the mass of the ribbon are the controlling elements. For the
audio-frequency range, the microphone may be designed so that the dif­
ference in pressure, I:!..p, between the two sides of the ribbon is proportional
to the frequency (see Sec. 8.3). Under these conditions,
I:!..p
=
jwp'
8.118
where w = 27Tj,
j = frequency in cycles per second, and
P' = a sound pressure proportional to the free-field sound pressure,
in dynes per square centimeter.
Equation 8.118 shows that the acoustical circuit 8.82B is equivalent to
acoustical circuit 8.82A. From acoustical circuit 8.82B the volume current
lS
u__
jwp'
-jwMR+jwM A
P'
8.119
Equation 8.117 shows that in Fig. 8.82 the acoustical circuit 8.82B may be
reduced to acoustical circuit 8.82C. Since, in acoustical circuit 8.82C, an
acoustical resistance is driven by a constant sound pressure, the response to
335
MICROPHONES
transients is perfect. This has been substantiated by actual tests 93 ' in which
it is possible to obtain square waves from the output of a velocity microphone
actuated by a loudspeaker with a very smooth, wide range response fre­
quency characteristics. In multiresonant systems with nonuniform re­
sponse frequency characteristics it is impossible to obtain any semblance
of a square wave from a loudspeaker microphone combination.
8.15. Noise in a Sound Pickup System.-Noise usually determines
the lower limit of reproduction in a sound translating system. The sources
of noise in a sound pickup system, depicted in Fig. 8.83, follow: The ambient
I
STUDIO
NOISE
MOLECULE
THERMAL
NOISE
I ATOM AND
I ELECTRON
I THERMAL
NOISE
I
I
I
I
I BARKHAUSEN
NOISE
I
I
I
: SHOT, IONIZA- I
I TION, THERMAL,'
SECONDARY E- I
I MISSION, ETC. I
NOISE
ELECTRON
THERMAL
NOISE
0 0
AIR
STUDIO
FIG. 8.83.
RIBBON
I
I
Ti
11
11
.,..
I
I
I
I
I
I
I
I
I
I
I
I
ITRANSFORMERS I VACUUM TUBE I PLATE AND I
I
I
I
I GRID RESISTORSI
I
I
I
I
MICROPHONE
AMPLIFIER
Sources of noise in a sound pickup system.
noise in the studio. The noise due to the random pressures upon the
diaphragm caused by the thermal agitation of the air molecules. The noise
due to the thermal agitation of the atoms in the diaphragm. The noise due
to the thermal agitation of the electrons in the conductor. The noise
due to the Barkhausen effect in the core of the transformer. The noise due
to shot effect, secondary emission, ionization, hum, etc., in the vacuum
tube. The noise due to the thermal agitation of the electrons in the plate
resistor.
A. Ambient Noise in the Studio.-The ambient noise in the studio is
usually one of the most important factors in determining the lower limit
of reproduction from the standpoint of the pickup system. The general
amhient noise level in a studio varies from 10 db for a very quiet studio
to 35 db for a noisy studio, as in the case of an audience. The spectrum
of room noise is shown in Fig. 12.38. It will be seen that room noise is not
uniform with respect to frequency. In the case of thermal noise the
generated voltage is proportional to the square root of the width of the
frequency band regardless of the position in the frequency spectrum.
B. Noise Due to Thermal Agitation of the Air Molecules.-Superposed
on the average atmospheric pressure there are fluctuations caused by the
distribution of thermal velocities of air molecules. The rms thermal sound
93'
Olson and Preston, RCA Review, Vol. 7, No.2, p. 155, 1946.
336
ACOUSTICAL ENGINEERING
pressure, p, in dynes per square centimeter, in the frequency interval
between fI and h may be obtained from the equation
- Jf!2
J!l Pt
P=
2 df =
Jf!2J!
4kTr A df
8.120
where Pt 2 df = square of the thermal acoustic pressure in the interval df.
df = frequency interval, in cycles per second,
rA = acoustical radiation resistance, in acoustical ohms,
k = Boltzmann's constant, 1.37 X 10-16 ,
T = absolute temperature, in degrees Kelvin.
In the case of a diaphragm-type microphone the acoustical resistance, r A,
can be obtained from Sec. 8.2Dl and Fig. 8.10.
In the case of the velocity microphone the system is a doublet. There­
fore, the acoustical radiation resistance is proportional to the fourth power
of the frequency at the lower frequencies. The ultimate acoustical resistance
on one side is 42jA, where A = area of the ribbon. The acoustical resistance
frequency characteristic of a velocity microphone is shown in Fig. 8.28.
C. Noise Due to Thermal Agitation of the Atoms in the Vibrating System.­
Noise is created in the acoustical resistances in the vibrating system of a
microphone. In the dynamic pressure type microphone the controlling
element over a major portion of the frequency range is an acoustical resist­
ance. The effective sound pressure generated in this element may be
determined from equation 8.120 in the preceding section. This pressure
is, of course, generated in the acoustical resistance and may be considered
to be a generator in series with the acoustical resistance in the acoustical
network.
In some instances it is more convenient to employ a mechanical network.
In this case the rms thermal mechanical force, fM, in dynes, in the frequency
interval betweenfI and h may be obtained from the equation
jM
where f2M! df
=
=
JJ~2
f2M!df
=
JJ~2
4kTrM df
8.121
square of the thermal mechanical force in the interval df.
df = frequency interval, in cycles per second,
rM = mechanical resistance, in mechanical ohms,
k = Boltzmann's constant, 1.37 X 10-16 , and
T
=
absolute temperature, in degrees Kelvin.
D. Noise Due to Thermal Agitation of the Electrons in the Conductor.­
The thermal agitation of the electrons in the conductor of the electrical
system of a microphone generates a fluctuating voltage. 94 ,95 The voltage,
94
95
Johnson, J. B., Phys. Rev., Vol. 32, No. 1, p. 97,1928.
Nyquist, H., Phys. Rev., Vol. 32, No. 1, p. 110, 1928.
MICROPHONES
337
e, in abvolts, due to the thermal agitation of the electrons in a conductor
is given by
8.122
where
k
=
T
12 - /I =
rE =
X 10-16 ,
Boltzmann's constant, 1.37
absolute temperature, in degrees Kelvin,
width of the frequency band, in cycles per second, and
electrical resistance of the conductor, in abohms.
E. Noise Due to Barkhausen Effect in the Transformer.-In the magnetiza­
tion of a piece of ferromagnetic material by continuously varying magneto­
motive force the resultant flux does not vary in a continuous manner but
is made up of small steps. This phenomenon is termed the Barkhausen
effect. In a well-designed transformer the only source of Barkhausen noise
of any consequence is in the leakage reactance. Since the leakage reactance
is small the Barkhausen noise will be relatively small. Furthermore in
most high-grade transformer alloys the Barkhausen effect is also quite
small.
F. Noise in the Vacuum Tube.-There are a large number of sources of
noise in the vacuum tube. A few of these are shot effect, thermal noise
in the plate impedance, ionization, and hum. These noises are treated at
length in books 96 on vacuum tubes. The voltage generated in the plate
of a well-designed triode, with an amplification of 20, from all sources
except hum, is 2.8 X 10-5 volt. This is 1.4 X 10-6 volt at the grid terminals.
G. Noise due to Thermal Agitation of the Electrons in the Plate Resistor.­
The noise voltage generated in the plate resistor can be obtained from
equation 8.122 in Sec. 8.15D.
H. Example of Noise in a Sound Pickup System.-It is the purpose of
this section to give the actual magnitude of the noise in each element of
a sound pickup system. For the studio a very low level will be assumed
namely, 10 db. The microphone will be the velocity type with a sensitivity
of 600 microvolts per dyne per square centimeter at the 2S0-ohm terminals
(see Sec. 8.3B). The final step-up transformer raises the impedance to
50,000 ohms at the grid of the triode vacuum tube. All noise voltages
will be referred to the grid terminals of the vacuum tube. The frequency
range is 30 to 15,000 cycles.
1. Ambient noise in the studio, 5.0 X 10-6 volt.
2. Noise due to thermal agitation of the air molecules, 2.5 X 10-6 volt.
3. Noise due to thermal agitation of the atoms in the ribbon vibrating
system, negligible.
4. Noise due to thermal agitation of the electrons in the ribbon, 3.5 X 10-6
volt.
5. Noise due to the Barkhausen effect in the transformer, negligible.
6. Noise in the vacuum tube, 1.4 X 10-6 volt.
96 Terman, .. Radio Engineers Handbook," McGraw-Hill Book Company, New
York, N.Y., 1943.
338
ACOUSTICAL ENGINEERING
M
o
Q
R
T
y
Z
FIG. 8.84. Microphone shapes. A. Pressure carbon. B. Pressure carbon. C.
Pressure magnetic. D. Pressure dynamic. E. Pressure dynamic. F. Pressure
dynamic. G. Pressure inductor. H. Pressure dynamic. I. Pressure crystal or
dynamic.
J. Pressure crystal. K. Pressure dynamic. L. Pressure crystal.
M. Velocity ribbon. N. Velocity ribbon. O. Pressure dynamic. P . Pressure
dynamic. Q. Pressure condenser. R. Pressure condenser. S. Pressure dynamic.
T. Pressure ribbon. U . Velocity ribbon . V. Unidirectional ribbon. W. Uni­
directional dynamic or crystal. X. Unidirectional combination ribbon and dynamic.
Y. Unidirectional ribbon.
Z. Unidirectional dynamic. D. . Unidirectional ribbon.
MICROPHONES
339
The above data show that the noises from all sources are comparable
in magnitude. In a microphone of lower sensitivity the electrical noise
sources in the conductor, resistor, and vacuum tube would be the limiting
factors. For this reason it is very important to employ high-sensitivity
microphones in wide frequency range and high-quality reproduction of
sound.
8.16. Shapes of microphones.-Microphones may be classified in many
different ways. One classification involves the type of response, namely,
pressure, velocity, or combination of pressure and velocity. Another
classification involves the type of transducer used to convert acoustical
variations into the corresponding electrical variations, as for example,
carbon, magnetic, dynamic, electrostatic, crystal, ribbon, etc. The con­
figuration of the elements of a microphone is determined to a large extent
by the type of response and the transducer. The outside shape in turn is
largely determined by the configuration of the elements. A few typical
examples of microphone shapes are shown in Fig. 8.84. The actual number
of microphone shapes commercially available today is somewhere around
500. For this reason it is impossible to depict all of the different shapes.
However, most of the microphones in use today follow the general patterns
shown in Fig. 8.84. Some of the shapes shown are not necessarily the most
common. The drawings are not replicas but reasonably accurate sketches.
The pressure microphones with different types of transducers are as follows:
A and B, Carbon; C, Magnetic; D, E, F, H, I, K, 0, P, and S, Moving Coil,
dynamic; G, Inductor dynamic; T, Ribbon; I, J, and L, Crystal; Q and R,
Electrostatic. The velocity microphones with ribbon transducers are as
follows: M, N, and U. Unidirectional microphones with different trans­
ducers are as follows: Wand Z, Moving coil; W, Crystal; X, Combination
ribbon and moving coil; V, Y, and ~, Ribbon.
9
MISCELLANEOUS TRANSDUCERS
9.1. Introduction.-Interest in the science of sound reproduction has
been stimulated during the past three decades by the almost universal use
of the phonograph, radio, and the sound motion picture. The two most
important acoustical elements in electrical reproduction of sound are loud­
speakers and microphones. For this reason, considerable space has been
given in this book to complete discussion of the most common instruments.
There are innumerable electro acoustic, mechanoacoustic, and electro­
mechanoacoustic transducers in use today for all types of applications.
However, the major portion of the applications discussed in this text will
be confined to sound reproduction. In addition to loudspeakers and micro­
phones, the following transducers are in common use in various types of
sound reproduction: telephone receivers, phonograph recorders and pick­
ups, mechanical phonographs, magnetic tape or wire recorders and repro­
ducers, sound motion-picture recorders and reproducers, sound powered
phones, electrical musical instruments, and hearing aids. His the purpose
of this chapter to consider typical examples of these transducers.
9.2. Telephone Receivers.-A telephone receiver is an electroacoustic
transducer actuated by energy in the electrical system and supplying energy
to an acoustical system.
A. Magnetic Telephone Receiver.-The bipolar telephone receiver is a tele­
phone receiver in which the alternating force, due to the alternating current
in the electromagnet, operates directly upon a diaphragm armature of
steel. A cross-sectional view, electrical circuit, and mechanical network of
the vibrating system are shown in Fig. 9.1. The steel diaphragm is spaced
a small distance from the pole pieces which are wound with insulated wire.
A permanent magnet supplies the steady magnetic flux.
A schematic view and the magnetic network of the magnetic system is
shown in Fig. 9.2. The elements of magnetic circuits and networks have
been considered in Sec. 6.28.
The force,! in dynes, upon the diaphragm when an alternating current
flows in the coils is
1M = 4>2 = M2M + 2MMNimax sin wt + 27TN2i2max
4TTA
4TTRn2A
RnRAA
RA2A
9.1
27TN2i2max cos 2wt
RA2A
1
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J., 1943.
340
MISCELLANEOUS TRANSDUCERS
where
341
cp = total flux, in maxwells, Fig. 9.2,
A = effective area of one pole, in square centimeters,
N = number of turns per coil,
RD = reluctance of the permanent field circuit, in gilberts per
maxwell,
RA = reluctance of the alternating magnetic circuit, in gilberts per
maxwell,
MM = magnetomotive force of the magnet, in gilberts,
i max = maximum current in the coil, in abamperes,
w =
27Tj,
j
frequency, in cycles per second, and
=
t = time, in seconds.
The reluctance RD of the permanent magnetic field circuit and the reluc­
tance RA of the alternating magnetic field circuit can be obtained from the
magnetic network of Fig. 9.2.
The first and third term on the right-hand side of equation 9.1 represent
a steady force, the second term represents a force of the same frequency
and the last term represents a force of twice the frequency of the current
in the coil. Equation 9.1 shows the necessity for the polarizing field in
order to obtain high sensitivity and to reduce second harmonic distortion.
The diaphragm in the bipolar telephone receiver is a circular plate clamped
at the edge (see Sec. 3.5). The effective mass of the diaphragm, when it is
a clamped plate, is one-third the actual mass of the diaphragm. The
effective area of the diaphragm is one-third the total area of the diaphragm.
The first resonant frequency is usually placed at 1000 cycles. The effective
compliance of the diaphragm can be obtained from the effective mass
and the resonant frequency for the frequency region at and below the
first resonant frequency of the diaphragm. Referring to the mechanical
network it will be seen that the system is stiffness controlled in the region
below the resonant frequency. This means that, for a constant driving
force,1M, the force applied to the compliance, CM3, of the ear cavity will
be independent of the frequency and hence the sound pressure in the ear
cavity will be independent of the frequency.
The sound pressure delivered by a bipolar telephone receiver to a cavity
as a function of the frequency is shown in Fig. 9.1. In the range below
the resonant frequency the response is independent of the frequency. At
the first resonant frequency of the diaphragm the response is very high.
Above the resonant frequency the amplitude decreases rapidly with fre­
quency. The peak at 3000 cycles is the second resonant frequency of the
diaphragm.
The pressure response frequency characteristic labeled A, Fig. 9.1, was
obtained with no leak between the ear and the earcap. In all hard earcaps
a leak occurs between the ear and the telephone receiver and the acoustical
342
ACOUSTICAL ENGINEERING
impedance presented to the telephone receivers is considerably more complex
than that of an acoustical capacitance of a small cavity. In the case of
telephone receivers worn in the customary manner the acoustical impedance
has three components, namely, the resistive and inertive components due
to the leak between the earcap and the ear and the acoustical capacitance due
to the ear cavity. These factors will be considered in detail in the section
on the testing of telephone receivers (see Sec. lOA).
The pressure response frequency characteristic indicated as B in Fig. 9.1
was taken on an artificial ear which simulates the conditions encountered
~CHC~
TCMZTcM3
flO
ELEC.
MECHANICAL
CIRCUIT
30
i\
CD
a
'"z
A
,,
20
II)
o
.. 10
NETWORK
, ,
\
\
\ ./1\
,'B
\
II)
'"
'"
100
1000
f"REQUE:NCY
10000
CROSS -SECTIONAL VIEW
FIG. 9.1.
Cross-sectional view. mechanical network. electrical
circuit and response frequency characteristics of a bipolar telephone
receiver. In the mechanical network, JM = the mechanical driving
force. mo, rMO, and C MO = the mass, mechanical resistance, and
compliance of the diaphragm. CMl = the compliance due to the air
in the case. C M2 = the compliance of the air space between the
diaphragm and cover. ml = the mass of the air in the aperture in
the cover. CM3 = the compliance of the ear cavity. In the elec­
trical circuit, JfEM = the motional electrical impedance. L and rEI
= the damped inductance and electrical resistance of the coils.
rEO = the electrical resistance of the electrical generator.
e = the
voltage of the electrical generator. The graph shows the pressure
response frequency characteristics. A. Receiver feeding a closed
cavity. B. Receiver feeding an artificial ear.
in actual practice. The artificial ear (see Sec. 10.4B) introduces a leak
which corresponds to the leak between the ear and the earcap. It will
be seen that the effect of this leak is to reduce the response at the lower
frequencies. Those familiar with telephone receivers have noticed that
the low-frequency response is increased when the leak is reduced by pressing
the telephone receivers tightly against the ears.
Since the development of the bipolar telephone receiver by Alexander
Graham Bell the construction has remained essentially the same. Improve­
ments have been made in sensitivity and response by the use of better
materials. However, the clamped plate diaphragm characterized by
prominent resonant peaks was retained. Referring to Fig. 9.1, it will be
MISCELLANEOUS TRANSDUCERS
343
seen that the peaks due to the first and second resonance fall within the
response range. These resonances not only introduce frequency distortion,
but increase the intensity of reproduction of clicks due to the poor transient
response (see Sec. 6.15).
A bipolar telephone receiver 2 has been designed in which all the prominent
resonances within the response range have been eliminated and the response
frequency characteristic improved both from the standpoint of uniformity
as well as the frequency range. The new telephone receiver is of the bipolar
SCHEMATIC VIEW
9.2. Schematic view and magnetic circuit of
the magnetic system of a bipolar telephone receiver.
In the magnetic network, My = the magneto­
motive force developed by the permanent magnet.
M 1 and M 2 = the magnetomotive forces developed
by the current in the two coils. i = the current in
the coils. RM = the reluctance of the permanent
magnet. RI and R2 = the reluctances of the air
gaps between the pole pieces and the diaphragm.
Ra = the reluctance of the shunt air gap. '" = the
flux through the pole pieces and the diaphragm.
FIG.
permanent magnet type. The magnetic circuit consists of pole pieces of
45 per cent Permalloy, two straight bar magnets of Cobalt, and a Permandur
diaphragm (see Sec. 6.19). The use of these materials increases the
efficiency of the unit.
The mechanical network of the mechanical system is shown in Fig. 9.3.
The magnetic system and magnetic network of the telephone receiver of
Fig. 9.3 is essentially the same as the magnetic network of Fig. 9.2.
The mass of the diaphragm is represented by mo. The compliance and
mechanical resistance of the diaphragm are designated as CMO and TMO.
The back of the diaphragm is enclosed, forming the compliance, CM!,
due to the resulting cavity. This cavity is connected to the recess in the
receiver handle by a hole in the plate. A special silk covers this hole,
forming the mechanical resistance TMI and the mass mI. The volume due
to the recess in the receiver handle forms the compliance CM2. The holes
in the earcap form the mechanical resistance TMa and the mass ma. The
compliance CM 3 is due to the cavity between the earcap and the diaphragm.
S
Jones,
w.
C., Jour. A .I.E.E., Vol. 57, No. 10. p. 559, 1939.
344
ACOUSTICAL ENGINEERING
The response of this receiver was taken by measuring the pressure generated
in a plain cavity. This cavity is designated by the compliance CM4. The
holes in the grid covering the receiver proper are large enough to have no
reaction upon the response. A resilient screen of silk is mounted on the
back of this grill. The mass of this screen is very small and is lumped with
the diaphragm mass mo.
The electrical portion of the circuit consists of the winding electrical
resistance rEI and inductance Ll. The eddy current elements are designated
ELECTRICAL
NETWORK
MECHANICAL
NETWORK
30
'a" 20
--.....
lA
'"
<II
Z
o
a.1 0
<II
'"0:
CROSS - SECTIONAL
VI EW
~o
100
1000
10000
FREQUENCY
FIG. 9.3.
Cross-sectional view, mechanical network, electrical circuit and
response frequency characteristics of an improved bipolar telephone receiver.
In the mechanical network, 1M = the mechanical driving force. mo, rMO, and
CMo = the mass, mechanical resistance, and compliance of the diaphragm.
CMl = the compliance of the air in the cavity behind the diaphragm. mi and
1'MI = the mass and mechanical resistance of the vent in the cavity behind the
diaphragm. C M 2 = the compliance of the air in the cavity in the handle.
C M 3 = the compliance of the air space between the diaphragm and cap.
m3 and r M3 = the mass and mechanical resistance of the apertures in the cap.
C M4 = the compliance of the ear cavity. In the electrical network, ZEM =
the motional electrical impedance. LI and rEI = the damped inductance and
electrical resistance of the coils. L2 and 1'E2 = the inductance and electrical
resistance due to eddy currents. e = the voltage of the electrical generator.
The graph shows the pressure response frequen~y characteristic of the receiver
feeding a plain cavity. The dots represent the response computed from the
mechanical and electrical networks.
as rE2 and L2. The electrical impedance3 due to the mechanical system is
designated by the motional electrical impedance ZEM. The force 1M can
be obtained from equation 9.1.
The pressure response computed by means of the mechanical network is
shown by the dots on the graph of Fig. 9.3. The measured pressure response
is given by the curve on this graph. The agreement is very good and shows
that it is possible to predetermine the response and to evaluate the effect of
3
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J., 1943.
345
MISCELLANEOUS TRANSDUCERS
changes in the constants of the component parts. Comparing the response
of Figs. 9.1 and 9.3 it will be seen that large gains have been effected in
uniform response over the entire range and in sensitivity from 1500 to
3000 cycles.
The new magnetic telephone receiver4 shown in Figs. 9.4 and 9.5 differs
radically from any previous commercial telephone receiver. The novel
ELECTRICAL
NETWORK
CROSS - SECTIONAL VI EW
MECHANICAL NETWORK
@II I ~II~IIIIII
40
100
1000
FREQUENCY
10000
FIG. 9.4.
Cross-sectional view, mechanical network. electrical circuit. and the response
frequency characteristic of a ring armature telephone receiver. In the mechanical
network,iM = the mechanical driving force. mo. rMO. and C MO = the mass. mechanical
resistance and compliance of the diaphragm. ml and rMI = the mass and mechanical
resistance of the thin membrane. C M l = the compliance of the air space between the
diaphragm and the membrane. m2 and rM2 = the mass and mechanical resistance of the
holes in the ear cap. ma and rMa = the mass and mechanical resistance of the air gap
aperature. CMs = the compliance of the air volume between the earcap and the thin
membrane. CMs = the compliance of the air volume in the coil space. m4 ami rM4 =
the mass and mechanical resistance of the control mechanical resistance. C M 4 = the
compliance of the air volume in the handle. C MS = the compliance of the air cavity
between the earcap and the ear. C M6 = the compliance of the air cavity between the
diaphragm and back enclosure. m? and I'M? = the mass and mechanical resistance of
the small hole in the diaphragm. In the electrical network. ZEM = the motional
electrical impedance. Ll and I'EI = the damped inductance and electrical resistance of
the coil. Lz and rE2 = the inductance and electrical resistance due to eddy currents.
e = the voltage of the electrical generator. The graph shows the pressure response
frequency feeding a cavity.
features are a ring-type armature and a new magnetic and vibrating system.
The driving armature is a disk of permandur. See Sec. 6.28. The action
and the performance of the magnetic system may be obtained from the
sectional view of the magnetic system and the magnetic network shown in
Fig. 9.5. The action and performance of the vibrating system may be
obtained from the mechanical network shown in Fig. 9.4. The diaphragm
is a dome-shaped phenolic impregnated fabric material. A major portion
4
Mott and Miner, Bell Syst. Tech. Jour., VoL 30, No. 1, p. 110. 1951.
34-6
ACOUSTICAL ENGINEERING
of the control of the system is obtained from the mechanical resistancerM4
in the form of a cloth disk. See Sec. 5.1. The composite armature and
diaphragm design results in a lower mechanical impedance and an appreciable
increase in the ratio of effective area to effective mass of the diaphragm
as compared to a simple magnetic disk-type combination diaphragm and
armature shown in Figs. 9.1 and 9.3. The sensitivity is about 5 db 'higher
R~
SECTIONAL VIEW
"L
MO
MAGNETIC NETWORK
9.5. Schematic view and magnetic network of
the magnetic system of a ring-type telephone
receiver. In the magnetic network, Mo = the
magnetomotive force developed by the permanent
magnet. Ml and M2 = the magnetomotive forces
developed by the current in the coil and in the lower
and upper magnetic circuits. Rl = the reluctance
of the lower path in the magnetic material. R2 =
the reluctance of the upper path in the magnetic
material. Ra = reluctance of the armature. R4 =
the reluctance of the upper air gap. R5 = the
reluctance of the lower gap. </>1 and </>2 = the flux
through the upper and lower air gaps. </>3 = the flux
through the armature. i = the current through the
coil.
FIG.
than the receiver shown in Fig. 9.3. The frequency range is extended about
700 cycles. With the lower mechanical impedance, the effect of holding the
receiver off the ear does not produce as marked loss in intelligibility as in the
case of the disk diaphragm armature types because the response frequency
characteristic is not altered to any appreciable extent.
B. Crystal Telephone Receiver.-A crystal telephone receiver 5 consists
of a light diaphragm connected to a Rochelle salt crystal (Fig. 9.6). The
three corners of a "bender" crystal are fastened to the case. The fourth
corner is connected to the diaphragm.
The electrical impedance of a crystal is primarily a capacitive electrical
reactance. The electrical network of Fig. 9.6 shows that the low-frequency
response can be raised relative to the high-frequency response by connecting
a high electrical resistance in series with the telephone receivers. A relatively
high electrical resistance must be used because the electrical impedance
of the crystal is relatively high, being 80,000 ohms at 1000 cycles.
S
Williams, A. L., Jour. Soc. Mot. Pic. Eng., Vol. 32, No.5, p. 552, 1939.
MISCELLANEOUS TRANSDUCERS
347
The performance of the vibrating system may be obtained from the
mechanical network of Fig. 9.6.
A pressure response frequency characteristic with the telephone receiver
feeding a plain cavity is indicated by B, Fig. 9.3. The pressure response
frequency characteristic taken on an artificial ear is indicated by A, Fig. 9.6.
CROSS-SECTIONAL VIEW
FREQUENCY
FIG. 9.6. Cross-sectional view, mechanical network, electrical network, and
response frequency characteristics of a crystal telephone receiver. In the
mechanical network, 1M = the mechanical driving force. mo, rMO, and
CAlO = the mass, mechanical resistance, and compliance of the diaphragm.
mI, rAIl, and C MI = the mass, mechanical resistance, and compliance of the
crystal. C M2 = the compliance due to the air in the case. C M3 = the
compliance of the air space between diaphragm and cover. m2 and rM2 =
the mass and mechanical resistance of the holes in the cover. C M 4 = the com­
pliance of the ear cavity. In the electrical network, CEO and YEO = the
electrical capacitance and electrical resistance of the crystal. rEI = the
electrical resistance of the series resistor. e = the voltage of the electrical
generator. The graph shows the pressure response frequency characteristics.
A. Receiver feeding a closed cavity. B. Receiver feeding an artificial ear.
C. Dynamic Telephone Receiver.-A dynamic telephone receiver 6 consists
of a light diaphragm coupled to a voice coil and a suitable mechanical net­
work for controlling the response. A cross-sectional view of a typical
dynamic telephone receiver is shown in Fig. 9.7. The mechanical network
of the mechanical system is also shown in Fig. 9.7.
The electrical impedance,7 III abohms, due to the mechanical system is
given by
(Bl)2
9.2
ZEM=-­
ZM
where B = flux density in the air gap, in gausses,
1 = length of the conductor in the voice coil, in centimeters, and
ZM = total mechanical impedance at 1M, in mechanical ohms.
In dynamic telephone receivers the flux density is relatively low and
is small compared to rEI and may be neglected.
6
7
ZEM
Wente and Thuras, Jour. Acous. Soc. Amer., Vol. III, No. 1, p. 44, 1932.
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J., 1943.
348
ACOUSTICAL ENGINEERING
The force,fM, in dynes, is given by
1M
=
9.3
Bli
where i, the current in abamperes, is obtained from the electrical circuit.
In general, the force,IM, is practically a constant and may be considered
a constant in the mechanical network.
The pressure response frequency characteristic feeding a plain cavity is
indicated by A, Fig. 9.7. The response measured on an artificial ear
CM'
SLIT
CROSS - SeCTIONAL view
rjl ~;:1'1 'tl l l fll
50
100
1000
10000
FReQueNCY
FIG. 9.7. Cross-sectional view, mechanical network, electrical circuit, and
response frequency characteristics of a dynamic telephone receiver. In
the mechanical network, 1M = the mechanical driving force. mo = the
mass of the diaphragm. rMO and GMO = the mechanical resistance and
compliance of the suspension. GMI = the .compliance of the air space
behind the diaphragm. mi and rMI = the mass and mechanical resistance
of the slit. GM 2 = the compliance of the air space between the diaphragm
and cover. m2 .and rM2 = the mass and mechanical resistance of the holes
in the cover. GM3 = the compliance of the ear cavity. In the elect­
rical circuit, ZEM = the motional electrical impedance. L and rEI =
the damped inductance and electrical resistance 9f the voice coil. rEG =
the electrical resistance of the electrical generator. e = the voltage of the
electrical generator. The graph shows the pressure response frequency
characteristics. A. Receiver feeding a closed cavity. B. Receiver
feeding an artificial ear.
indicated by B, Fig. 9.7, shows that the response at the low frequencies is
reduced due to the leak.
D. Inductor Telephone Receiver.-An inductor telephone receiver 8 ,9 is
a telephone receiver in which a straight-line conductor, located in a magnetic
field, drives a "V" shaped diaphragm. An acoustical network is used to
compensate the response of the inductor-type telephone receiver shown in
Fig. 9.8. The acoustical network compensates for the leak between the
ear and the earcap. The effect of the leak between the ear and the earcap
upon the response of a telephone receiver has been outlined in the preceding
sections. Obviously, from a practical standpoint the performance of a
8
9
Olson and Massa, Jour. Acous. Soc. Amer., Vol. 6, No.4. p. 240, 1935.
Olson, H. F., Jour. Soc. Mot. Pic. Eng., Vol. 27, No.5, p. 537, 1936.
MISCELLANEOUS TRANSDUCERS
349
telephone receiver should be independent of the leak between the ear and
the earcap. In order to design the vibrating system of the telephone
receiver so that constant sound pressure will be delivered to the ear, the
nature of the acoustical impedance looking through the aperture of the
earcap must be considered as a part of the vibrating system. The acoustical
impedance characteristic, looking through the aperture of the earcap of
~C~AI~~~ ~
__
P
I
__- ,
rAE
"-v~·,
'
,
(~
:---L___ :1._;
: CAE
M_E____...s.__-'-__--I
ACOUSTICAL
'o"
CROSS -SECTIONAL VIEW
NETWORK
20~~---.--"-n~nr--'--r,,nnTn
B
~o~HIOO~~~~~~I~OO~O--~~~~~IOOOO
FREQUENCY
FIG. 9.8.
Cross-sectional view, acoustical network and response
frequency characteristic of an inductor telephone receiver. In the
acoustical network, M1 = the inertance of the diaphragm and con­
ductor. CAl and r A1 = the acoustical capacitance and acoustical
resistance of the suspension system. M 2 and r A2 = the inertance
and acoustical resistance of the bolt of silk. C A 2 and rA3 = the
acoustical capacitance and acoustical resistance of the cavity behind
the diaphragm and the bolt of silk. M4 = the inertance of the tube
connecting the cavity behind the diaphragm with the case cavity.
CAS = the acoustical capacitance of the case volume.
Ms and
r AS = the inertance and acoustical resistance of the hole in the case.
ME, rAE, and CAE = the inertance, acoustical resistance, and
acoustical capacitance of the ear. p = the driving pressure,
p = IM/S' 1M = the mechanical driving force. S = the area of
the diaphragm. In the electrical circuit, ZEM = the motional
electrical impedance. Land rE1 = the damped inductance and
electrical resistance of the conductor. rEG = the electrical resistance
of the electrical generator. e = the voltage of the electrical genera­
tor. The graph shows the pressure response frequency characteris­
tics. A. Receiver feeding an artificial ear. B. Receiver feeding a
plain cavity.
a telephone receiver, is shown in Fig. 10.36, Sec. lO.4B. These charac­
teristics show that the acoustical impedance is positive and increases with
frequency up to 400 cycles; between 300 and 500 cycles it is practically
resistive and above 400 cycles it is negative and decreases with frequency.
350
ACOUSTICAL ENGINEERING
A generalization of the requirements for maintaining constant sound
pressure in the ear cavity under these conditions is as follows: the velocity
of the diaphragm below 300 cycles must be inversely proportional to the
frequency, between 300 cycles and 500 cycles the velocity should be inde­
pendent of the frequency, and above 500 cycles the velocity should be pro­
portional to the frequency.
The acoustical network of a telephone receiver which delivers practically
uniform sound pressure to the ear cavity in the presence of a normal leak
is shown in Fig. 9.8. The acoustical network of the ear is shown dotted.
The "V" shaped diaphragm is driven by a straight conductor located in
the bottom of the "V."
The electrical circuit of the inductor telephone receiver is shown in
Fig. 9.8. The pressure p may be considered to be independent of the
frequency.
The pressure response frequency characteristic taken on an artificial ear
is indicated by A, Fig. 9.8. The constants were chosen to give the smooth­
est response between 60 and 7000 cycles. The pressure response frequency
characteristic with the receivers feeding a plain cavity is indicated by B,
Fig. 9.8. The small difference between the response with and without a
leak indicates the effectiveness of this type of vibrating system in minimizing
the effect of the leak between the ear and the earcap.
A dynamic telephone receiverlO employing an acoustical system similar
to the inductor telephone receiver described above has also been developed.
The acoustical network is similar to that of the inductor telephone receiver
shown in Fig. 9.8. The essential difference between the inductor and dy­
namic acoustically compensated telephone receivers resides in the driving
system, in the former a straight conductor is used to drive" V" diaphragm
while in the latter a circular voice coil is used to drive a dome-shaped
diaphragm.
9.3. Phonograph.-A phonograph is used to designate a system for
recording and reproducing sound from a record. Today, a phonograph
usually refers to a system in which a stylus (needle) follows the undulations
in the groove of a record and transforms these undulations into the cor­
responding acoustical or electrical variations. The record may take the
form of a cylinder or a flat disk. Today, the flat disk record is almost
universally used for entertainment while the cylindrical record is used for
dictographs. In the hill and dale or vertical-type record the undulations
are cut in a direction normal to the surface. In the lateral-type record the
undulations are cut in a direction parallel to the surface of the record. The
lateral-type records are used for home reproduction. Both vertical- and
lateral-type records are used for high quality reproductions as, for example,
in transcriptions for broadcasting. The complete system used in the
recording and processing of phonograph records will be considered in Sec.
13.16. It is the purpose of the sections which follow to consider phono­
10
Anderson, L. J., Jour. Soc. Mot. Pic. Eng., Vol. 37, No.3, p. 319, 1941.
MISCELLANEOUS TRANSDUCERS
351
graph recorders, mechanical phonograph, record player, phonograph pickup
records, and distortion in phonograph reproduction.
A. Recording Systems.-l. Recorders.-A phonograph recorder is an
equipment for transforming acoustical or electrical signals into motion of
approximately like form and inscribing such motion in an appropriate
medium by cutting or embossing. For the recording of disk phonograph
records, the electrical phonograph recorder replaced the mechanical recorder
about three decades ago. An electrical phonograph recorder is shown in
Fig. 9.9. The lacquer disk used in recording the master record is placed on
MICROSCOPE
FIG. 9.9.
Perspective view of a disk phonograph recorder.
the recording turntable. To insure against spurious motions, the turntable
is made very heavy. A suitable mechanical filter is placed between the
driving motor and the turntable so that uniform rotational motion of
the turntable will be obtained. The drive system is arranged so that all
the standard record speeds can be cut. In general, the recording turntable
is driven with a synchronous motor to insure uniform absolute speed of
rotation. The lead screw drives the cutter in a radial direction so that a
spiral groove is cut in the record. Lead screws of different pitches are used
ranging from 100 to 500 grooves per inch. In some recordings a variable
pitch is used. In this procedure the spacing between the grooves is made to
correspond to the amplitude-small spacing for small amplitudes and large
spacing for large amplitudes. Under these conditions the maximum amount
of information can be recorded on a record. The material which is removed
in the cutting process is in the form of a fine thread. The thread is pulled
into the open end of a pipe located near the cutting stylus and connected to
a vacuum system. A complete phonograph recording system is described
in Sec. 13.6.
2. Lateral Cutter.-In the lateral type of recording the undulations are
cut in a direction parallel to the surface of the record and perpendicular to
352
ACOUSTICAL ENGINEERING
the groove. Perspective and sectional views, the electrical circuit, and the
mechanical network of a lateral-type magnetic phonograph cutterll are
shown in Fig. 9.10.
A schematic view of the magnetic system and magnetic circuit of the
Z5,-,--,,-'--'-''--'-'--TT-'
~20~4--+~+--+~~-r~~~~
PERSPECTIVE
VIEW
SECTIONAL
VIEW
9.10. Perspective and sectional views, mechanical network and velocity
response frequency characteristic of a lateral-type phonograph recorder. In the
mechanical network, JM = the mechanical driving force. ml = the mass of the
armature. C Ml = the compliance of the restoring spring. m2, rM, and C M 2 =
the mass, mechanical resistance, and compliance of the damping element. M 3
and C M 3 = the mass of the stylus and holder. ZM = the mechanical impedance
of the load presented to the stylus. In the electrical circuit: ZEM = the motional
electrical impedance. L and rEI = the damped inductance and electrical
resistance of the coil. rEO = the electrical resistance of the generator. e = the
voltage of the electrical generator. The graph depicts the velocity response
frequency characteristic of the recorder.
FIG.
lateral-type phonograph cutter is shown in Fig. 9.11.
upon the armature is given by
r
_ (CPl
jM -
where CPl
CP2
=
=
+ CP2)2
81TA
The force/M, in dynes,
9.4
flux, in maxwells, through the reluctances Rl and R2 produced
by the permanent magnet, and
flux, in maxwells, through the reluctances Rl and R2 produced
by the current in the coil.
The reluctance of the armature may be neglected because it is small com­
pared to the reluctances of the air gaps R 1 , R 2, Ra, and R4. The flux CPl
11 Hasbrouck, H. J., Jour. Soc. Mot. Pic. Eng., Vol. 32, No.3, p. 246, 1939.
MISCELLANEOUS TRANSDUCERS
353
due to the magnetomotive force MD of the magnet can be obtained from the
magnetic network of Fig. 9.11. The magnetomotive force M A, in gilberts,
due to a current in the coil is given by
MA
where N
i
=
=
=
4-rrNi
9.5
number of turns in the coil, and
current in the coil, in abamperes.
The flux 4>2, in maxwells, due to the magnetomotive force M A developed by
the current in the coil can be computed from the magnetic circuit of Fig.
9.11. The force applied to the armature in a magnetic driving system is
MAGNET
ARMATURE
COIL
N
TURNS
_--++-.....:jL.....P....1 1 MI
PIVOT -t--==~-I.o
~
R2
R4
SCHEMATIC VIEW
MAGNETIC NETWORK
FIG. 9.11. Schematic view and magnetic network of a
lateral-type phonograph recorder. In the mangetic
network. MD = the magnetomotive force developed by
the permanent magnet. MA = the magnetomotive force
developed by the current in the coil. RM = the
reluctance of the path in magnetic material. R1 and R2
= the reluctances of the two variable air gaps. Ra and
R4 = the reluctances of the two fixed air gaps. i = the
current in the coil.
proportional to the current in the coil. The mechanical network is designed
so that, for constant applied force, the amplitude will be independent of the
frequency below approximately 800 cycles and the velocity will be inde­
pendent of the frequency above approximately 800 cycles.
A sectional view, the mechanical circuit, and the electrical system of a
feedback lateral-type phonograph cutter12 ,13 is shown in Fig. 9.12. The
vibrating system is of the dynamic type with two wire coils. The vibrating
system is designed so that there is a single degree of freedom over the operat­
ing frequency range. The response frequency characteristic of the vibrating
system shows that it is a system of one degree of freedom from 30 cycles to
16,000 cycles with the fundamental resonant frequency at 700 cycles. The
output of the sensing coil is fed to the input of the amplifier. The output
12
13
Davis. C. C., Jour. Audio Eng. Soc., Vol. 2. No.4. p. 228, 1954.
Morgan. A. R., Unpublished Report.
354
ACOUSTICAL ENGINEERING
of the amplifier is fed to a driving coil in an out-of-phase relationship. The
signal is fed to the input of the amplifier. With the feedback in operation
the velocity of the vibrating system is practically independent of the fre­
rM
m
f~CM
T
I
:-.,
MECHANICAL CIRCUIT
40
/~
CIRCUIT DIAGRAM
.. 30
a
~
... 20
(J)
z
a
~
...g; 10
V
''(
I\.
/
0::,
o
V
/
B
/
"
-10
100
20
10000
1000
FREQUENCY
SECTIONAL VI EW
9.12. Sectional view, mechanical circuit, electrical system, and velocity
response frequency characteristic of a feedback lateral-type phonograph cutter.
In the mechanical circuit, 1M = the mechanical driving force. m, rM, and C M =
the mass, mechanical resistance and compliance of the vibrating system. In the
graph, A = the velocity frequency response characteristic without feedback.
B = the velocity response frequency characteristic with feedback.
FIG.
'"'"'=TIt'"'""" tP
ELECTRICAL
SYSTEM
50
., 4 0
E
a
w 30
"
<I)
.
~ 20
<I)
w
0::
10
o ,.
/
~
"­
V
8
8 10 2
2
•
81032
FREQUENCY
9.13. Sectional view, mechanical circuit, electrical system, and
velocity response frequency characteristic of a feedback vertical-type
phonograph cutter. In the mechanical circuit, 1M = the mechanical
driving force. m, YM, and C M = the mass, mechanical resistance, and
compliance of the vibrating system. In the graph, A = the velocity
response frequency characteristic without feedback. B = the velocity
response frequency characteristic with feedback.
FIG.
355
MISCELLANEOUS TRANSDUCERS
quency over the frequency range from 30 to 16,000 cycles. The input to the
amplifier can be compensated to provide the desired recording charac­
teristic.
3. Vertical Cutter.-In the vertical type of recording the undulations are
cut in a direction perpendicular to the surface of the record. A sectional
view, the mechanical circuit, and electrical system of a feedback type of a
vertical-type phonograph cutter14 are shown in Fig. 9.13. The mechanical
system as depicted by the mechanical circuit is a system of one degree of
freedom. The response frequency characteristic of the system is designated
as A in Fig. 9.13. By feeding the output of the pickup coil in out-of-phase
relationship with the input to the amplifier, the velocity response frequency
characteristic with about 40 db of feedback will be as shown in Fig. 9.13B.
The use of a feedback in conjunction with a simple vibrating system yields
a uniform response frequency characteristic. The amplifier which drives
the system can be compensated to yield the appropriate recording response
frequency characteristic.
4. Recording Characteristics.-The velocity response frequency of one
type of standard frequency record used in obtaining the response frequency
characteristics of phonograph pickups and mechanical phonographs is
shown in Fig. 9.14. The reason being that the characteristic shown in
30
//
11120
o
,.
l­
/
i)
9WID
>
./
/
~
/'
100
1000
FREQUENCY
IN
CYCLES
PER
10000
SECOND
FIG. 9.14. Typical velocity response frequency characteristic of an old
type, standard frequency phonograph record.
Fig. 9.14 was used up to about two decades ago in recording phonograph
records. To prevent overcutting the groove, the recording is made so that
the amplitude is essentially independent of the frequency below approxi­
mately 800 cycles. The velocity under these conditions falls off 6 db per
octave with decrease of the frequency. Above approximately 800 cycles
the recording is made so that the velocity is independent of the frequency.
The amplitude in this frequency range falls off 6 db per octave with increase
of the frequency.
14
Vieth and Wiebusch, Jour. Soc. Mat. Pic. Eng., Vol. 30, No. 1, p. 96, 1938.
356
ACOUSTICAL ENGINEERING
In radio transcription recording, the orthacoustic15 type of recording
characteristic is employed. The orthacoustic velocity frequency charac­
teristic for constant voltage input to the microphone amplifier is shown in
Fig. 9.15. This characteristic is essentially a constant amplitude frequency
characteristic. In reproduction of the record, an inverse response frequency
0
2/
/:y
5
,0
en
o
,,' k(.:
5
>­
~
;::;::;
0
u
o
oJ
!oJ
>
_
"..
5
-I 0
-IS
.,­ ;..
"..
/~ ~
,,/
/~
.......; V
.. "'V
.... ..,. .'
~
-2<yo~
-;:.;".,'
~ 3
rooo
100
FREQUENCY
IN
CYCLES
10000
PER
20000
SECOND
FIG. 9.15.
Velocity frequency characteristics of various types of recording
characteristics used in commercial phonograph records. 1. RIAA Standard.
2. Orthacoustic Standard. 2 and 3 represent the upper and lower high-frequency
limits and 4 and 3 represent the upper and lower low-frequency limits of the
recording characteristics in use in commercial phonograph records.
characteristic is used to obtain a uniform over-all response frequency
characteristic. The use of this type of response frequency characteristic
reduces ground noise and distortion.
In recording of commercial phonograph records, high-frequency accentua­
tion is employed. The compensation used today varies over wide limits
among different record manufacturers from the lowest characteristic shown
in Fig. 9.15 to the orthacoustic characteristic of Fig. 9.15. The RIAA
characteristic16 is the proposed standard playback characteristic for lateral
disk recordings.
Standard frequency records exhibiting the characteristics of Fig. 9.15
have replaced the record characteristic of Fig. 9.14-. The use of frequency
records exhibiting the characteristics of Fig. 9.15 gives the performance of
the system without any further corrections. In the reproduction of com­
mercial phonograph records, an inverse response frequency characteristic
is employed to obtain a uniform over-all response frequency characteristic.
15 Recording and Reproducing Standards, Proc. Inst. Rad. Eng., Vol. 30, No.8,
p. 355, 1942.
16 Record Industry Association of America.
357
MISCELLANEOUS TRANSDUCERS
The use of high-frequency accentuation, as shown in Fig. 9.15, reduces
record ground noise and distortion.
5. Heated Stylus.17-In the production of phonograph records the original
recording is cut in a lacquer disk by means of the cutting stylus actuated by
the cutter. The walls of the resultant groove are not smooth. This lack
of smoothness is due to nonlinear effects which occur in all cutting processes.
It was discovered that by heating the cutting stylus the roughness of the
groove wall was reduced. As a matter of fact, an improvement in signal­
to-noise of 20 db was obtained by means of the heated stylus. This gain in
signal-to-noise is most apparent in the high-frequency range. Two types
of heating have been employed as shown in Fig. 9.16. In Fig. 9.16A, a fine
wire is wound around the stylus. An electrical current is sent through the
wire which heats the stylus. The mass of the fine wire is small compared to
the mass of the stylus and, therefore, does not effect the vibrating per­
formance. In Fig. 9.16B, the stylus is plated with a thin ring of copper.
COIL NOT IN CONTACT
WITH STYLUS
STYLUS
A
B
FIG. 9.16. Two systems for heating the cutting stylus of a phonograph re­
corder. A. Direct current system. B. Radio frequency system.
A coil fed from a radio-frequency oscillator induces a current in the copper
ring which heats the stylus. The heavy coil which surrounds the stylus is
not in contact with the stylus.
B. Reproducing Systems.-l. Record Player.-A phonograph record player
is an equipment for transforming the undulations in a groove in a medium
into the corresponding electrical or acoustical variations. In the early days
of the .phonograph the acoustical phonograph was used exclusively.
However, about three decades ago the electrical phonograph was developed.
Today, the electrical form of reproduction has almost completely displaced
the mechanical phonograph.
a. Electrical Record Player.-An electrical record player and changer is
shown in Fig. 9.17. The record is rotated by the reproducing turntable at
the same angular speed as that used in recording. The turntable is rotated
by means of an electric motor. The stylus or needle of the pickup follows
the wavy spiral groove and generates a voltage corresponding to the undula­
tions in the groove. Pickups for use in disk-record reproduction will be
described in later sections. The record player and changer shown in Fig.
9.17 will play three rotational speeds, namely 33t, 45 and 78 RPM. It Vl--ill
also play and change a stack of eight records. The small spindle is used for
17
Bachman, W. S., Audio Eng., Vol. 34, No.6, p. 11, 1950.
358
ACOUSTICAL ENGINEERING
the reproduction of 331 and 78 RPM records. The large spindle is used for
the reproduction of 45 RPM records.
Another type of record changer and player plays and changes a single type
of record. One of the most common is the 45 RPM record player and
changer. IS
A record player is the simplest type of disk-record reproducer. It is
manually operated. It ranges from the simplest of all disk-record players
RECORD
PICKUP ARM
".".....,..---++ PICKUP
g""?}3
45 RPM
SPINDLE
SPEED
SELECTOR
t:::======s;;;;======:::;;;;;=-~~~~~~t-START
STOP
LEVER
FIG. 9.17.
player.
Perspective view of a three-speed disk phonograph
to elaborate transcription types with very uniform rotational velocity and
high quality pickups.
b. Mechanical Phonograph. I9-A mechanical phonograph is a mechano­
acoustic transducer actuated by a phonograph record and by means of an
acoustical system radiates acoustical energy into a room or open air.
The record is rotated by a turntable at the same angular speed as that
used in recording. The turntable is rotated either by a spring motor or by
an electric motor. A cross-sectional view of the mechanical network of the
reproducing system of a mechanical phonograph is shown in Fig. 9.18. The
system consists of a diaphragm coupled to a needle which follows the wavy
spiral groove and generates a sound output which corresponds to the undula­
tions in the groove. To improve the radiation efficiency, the diaphragm is
coupled to a horn. The record mechanical impedance is usually large
compared to the mechanical impedance of the remainder of the system
save at the high frequencies. The record mechanical impedance is a function
of the type of material. Obviously, it is higher for the harder materials.
18
19
Carson, Burt, and Reiskind, RCA Review, Vol. 10, No.2, p. 173, 1949.
Maxfield and Harrison, Bell Syst. Tech. Jour., Vol. 5, No.3, p. 493, 1926.
359
MISCELLANEOUS TRANSDUCERS
The generator in the mechanical network of this system is of the constant
current type. That is, JM delivers constant velocity to the mechanical
network Under these conditions the velocity is independent of the
impedence of the load.
The response frequency characteristic of a mechanical phonograph of
the console type is shown in Fig. 9.18.
MECHANICAL
NETWORK
40
/1
III
030
w
'"Z20
~
'"~IO
900
CROSS -SECTIONAL VIEW
IV
.A
r­
~
1\/
, \A N
1000
10000
fREQUENCY
FIG. 9.18. Cross-sectional view, mechanical network and response fre­
quency characteristic of a mechanical phonograph. In the mechanical
network, ZMR = the mechanical impedance of the record. CMl, C.1l 2,
C M 3, C M4 , C M5 , C M6 , and C M7 = the compliances of the needle, the needle
holder, the needle holder arm pivot, the needle holder arm, the connector,
the spider, the diaphragm suspension, and the coupling chamber. ml,
m2, and m3 = the masses of the needle holder arm, the spider, and the
diaphragm. ZMH = the mechanical impedance at the throat of the horn.
JM = the force generated by a velocity generator having the characteristic
of Fig. 9.14. The graph shows the pressure response frequency charac­
teristic of a console-type mechanical phonograph using a record having a
characteristic of Fig. 9.14.
2. Phonograph Pickups.-A phonograph pickup is an electromechanical
transducer actuated by a phonograph record and delivering energy to an
electrical system, the electrical current having frequency components cor­
responding to those of the wave in the record. The systems for converting
the mechanical vibrations in to the corresponding electrical variations are
as follows: magnetic, variable resistance, condenser, electronic, dynamic,
and crystal. It is the purpose of this section to consider examples of some
of the most common phonograph pickups in use today.
a. Crystal Pickup.-A crystal pickup20 is a phonograph pickup which
depends for its operation on the piezoelectric effect. The crystal in use
today is Rochelle salt. A cross-sectional view of a typical crystal pickup
used in commercial phonographs employing a replaceable needle is shown in
Fig. 9.19. The needle, driven by the record, is coupled to the crystal. The
elements of the system and the mechanical network are shown in Fig. 9.19.
The displacement of the crystal can be determined from the mechanical
20
Williams, A. L., Jour. Soc. Mot. Pic. Eng., Vol. 32, No.5, p. 552, 1939.
360
ACOUSTICAL ENGINEERING
network of the mechanical system and the velocity of the generator obtained
from Fig. 9.15. The voltage output of the crystal is proportional to the
displacement. The internal electrical impedance of the crystal increases
with the decrease in frequency since the crystal is essentially an electrical
capacitance. The open circuit voltage characteristic renders the low­
frequency compensation problem exceedingly simple.
Referring to the mechanical network of Fig. 9.19 it will be seen that the
velocity in the record, ZMR, is a function of the magnitude of the mechanical
impedance of the pickup. As the mechanical impedance of the pickup
becomes larger the vibration velocity of the record will be correspondingly
greater. Vibration of the record produces radiation of sound into the air.
r
'rOP VIEWlCRYSTAL EXPOSED)
CM•
r..,
CM, C" 2 r M,
CEG
--$-'ZEL
1-1
EL ECTRICAL CIRCUIT
RESISTANCE
~~C"3
""3'1,,(:
r" 2
CROSS- SECTIONAL VI EW
MECHANICAL NETWORK
9.19. Cross-sectional view, mechanical network, and electrical network of a
crystal pickup. In the mechanical network, ZMR = the mechanical impedance of the
record. C MO = the compliance of the needle. nt1 = the mass of the needle holder.
GM1 = the compliance of the shaft. C M 2 and C M 3 = the compliances of the crystal
supports. YM1 and rM2 = the mechanical resistances of the crystal supports. rM3 and
GM 4 = the mechanical resistance and compliance of the front bearing. mo, rMO, and
GMf} = the mass, mechanical resistance, and compliance of the crystal.
m2 = the mass
of the pickup and tone arm. JM = the force generated by a velocity generator. In the
electrical circuit, eo = the open circuit voltage developed by the crystal. CEO = the
electrical capacitance of the crystal. ZEL = the electrical impedance of the load.
FIG.
Most of this radiation occurs at the high frequencies. The sound produced
in this manner is termed mechanical noise. It is undesirable because it
interferes with the sound from the loudspeaker and produces distortion.
To overcome this, a low-noise crystal pickup21 has been developed. The
essential elements and mechanical network of a low-noise crystal pickup
are shown in Fig. 9.20. A permanent sapphire stylus is used instead of
a replaceable needle. The mechanical impedance of the pickup in shunt
with the mechanical impedance of the record is very small. Therefore, the
motion or vibration of the record due to the pickUp is very small. The
mechanical noise of the low-noise pickup of Fig. 9.20 is about 20 db lower
than the replaceable needle pickup of Fig. 9.19. The open circuit voltage
output is proportional to the amplitude of the crystal. The electrical
generator may be considered to be the open circuit voltage in series with the
electrical capacitance.
21
Burt, A. D., Electyonics. Vol. 16, No. 1, p. 90, 1943.
MISCELLANEOUS TRANSDUCERS
TOP VIEW (CRYSTAL EXPOSED)
ELECTRICAL CIRCUIT
SECTIONAL VIEW
MECHANICAL NETWORK
361
m,~~·:~·::·' ;. $ =tt Jf~f~·~::.
FIG. 9.20. Cross-sectional views, mechanical network and electrical circuit of a low
noise crystal pickup. In the mechanical network, ZMR = the mechanical impedance
of the record. ml = the mass of the stylus and holder. CMl = the compliance of
the stylus arm. m2 = the mass of the vertical member. C M 2 = the compliance of
the chuck. m4, rM2, and C M 4 = the mass, mechanical resistance and compliance of
the crystal. CM 5 and rM3 = the compliance and mechanical resistance of the chuck
bearing. C M3 and rMl = the compliance and mechanical resistance of the crystal
support. ma = the mass of the pickup and tone arm. 1M = the force generated by
a velocity generator. In the electrical circuit, eo = the open circuit voltage de­
veloped by the crystal. CEO = the electrical capacitance of the crystal. ZEL = the
electrical impedance of the load.
A crystal phonograph pickup similar to the crystal phonograph pickup
of Fig. 9.20 but with a reduction in mass of the vibrating system is shown in
Fig. 9.21. With the design of Fig. 9.21 it is possible to obtain uniform
response up to 15,000 cycles. On the other hand, if greater output over a
more restricted frequency range is desired, the design of Fig. 9.20 may be
used. The performance of the system may be deduced from the mechanical
network of Fig. 9.21. The open circuit voltage output is proportional to
CEG
r~
ELECTRICAL CIRCUIT
TOP VIEW (CRYSTAL EXPOSED)
MECHANICAL NETWORK
FIG. 9.21. Cross-sectional views, mechanical network, and electrical circuit of a low
noise, wide range crystal pickup. In the mechanical network, ZMR = the mechanical
impedance of the record . ml = the mass of the stylus and holder. C M1 = the com­
pliance of the stylus arm. m2 = the mass of the vertical member. CM2 = the
compliance of the chuck. m4, rM2, and CM4 = the mass, mechanical resistance, and
compliance of the crystal. C M3 and r,l(l = the compliance and mechanical resistance of
the crystal support. ma = the mass of the pickup and tone arm. rM5 and C M6 = the
mechanical resistance and compliance of the front bearing. C M 5 and rM3 = compliance
and mechanical resistance of rear bearing. 1M = the force generated by a velocity
generator. In the electrical circuit, eo = the open circuit voltage developed by the
crystal. CEO = the electrical capacitance of the crystal. ZEL = the electrical im­
pedance of the load.
362
ACOUSTICAL ENGINEERING
the amplitude of the crystal. The electrical generator may be considered
to be the open circuit voltage in series with the electrical capacitance.
b. Ceramic Turnover Phonograph Pickup.-A turnover phonograph
pickup22 employing a ceramic (barium titanate) transducer is shown in
Fig. 9.22. The stylus arm contains two stylii located at the end of the stylus
TURN OVER
LEVER
ELECTRICAL CIRCUIT
PERSPECTIVE VIEW
m,
m6 C"6 r M6
VIBRATING SYSTEM
MECHANICAL NETWORK
FIG. 9.22.
Perspective views, mechanical network and electrical circuit of a turnover
ceramic phonograph pickup. In the mechanical network, ZMR = the mechanical im­
pedance of the record. ml = the mass of the stylii and holders. m2 = the mass of
the connecting arm. C M2 = the compliance of the turnover arm. rMa and C M 3 = the
mechanical resistance and compliance of the turnover lever arm. C M4 = the com­
pliance of the connecting arm. rM5 and C MS = the mechanical resistance and com­
pliance of the ceramic support. m6. rM6, and C M6 = the mass, mechanical resistance,
and compliance of the ceramic transducer. m7 = the mass of the pickup and tone arm.
1M = the force generated by a velocity generator. In the electrical circuit, eo = the
open circuit voltage developed by the ceramic transducer. CEO = the electrical capaci­
tance of the ceramic transducer. ZEL = the electrical impedance of the load.
arm and angularly displaced by 180°. A stylus with a .003" radius at the
tip is used to reproduce 78 RPM coarse groove records and a stylus with a
.001" radius at the tip used to reproduce 45 RPM and 33! RPM fine groove
records. By means of the turnover lever either stylus may be presented
to the record. The stylus arm is coupled to the ceramic transducer by means
of a cradle lever arm. The performance of the system may be deduced from
the mechanical network of the system. The open circuit voltage output is
proportional to the amplitude of the ceramic transducer. The internal
electrical element of the ceramic transducer is an electrical capacitance.
The electrical generator may be considered to be open circuit voltage in
series with an electrical capacitance.
22
Koren, Pearson, Klingener, and Sabol, Jour. Acous. Soc. Amer., Vol. 26, No. 1,
p. 15, 1954.
r
MISCELLANEOUS TRANSDUCERS
363
A turnover pickup23 employing two separate stylus arms is shown in
Fig. 9.23. One stylus arm is fitted with a stylus having a tip radius of .003
for the reproduction of 78 RP:!'.1 coarse groove records and the other stylus
H
PERSPECTIVE VIEW
STYLUS ASSEMBLY
r--eG~
cEGT
~
VIBRATING SYSTEM
ELECTRICAL CIRCUIT
L I :L 2
Ls:L4
MECHANICAL NETWORK
FIG. 9.23.
Perspective views, mechanical network and electrical
circuit of a two stylii ceramic phonograph pickup. In the mechani­
cal circuit, ZMR = the mechanical impedance of the record. ml =
the mass of the stylus. YMI and C MI = the mechanical resistance
and compliance of the stylus arm. m2, YM2, and C M2 = the mass,
mechanical resistance and compliance of the coupling arm. ma,
rM3, and C M 3 = the mass, mechanical resistance and compliance of
the ceramic transducer. m4 = the mass of the pickup and tone arm.
The levers LI, L2, La, and L4 are represented as transformers with
turn ratios corresponding to the ratios of the lever arms. 1M = the
force generated of a velocity generator. In the electrical circuit,
eG = the open circuit voltage developed by the ceramic transducer.
CEG = the electrical capacitance of the ceramic transducer. ZEL =
the electrical impedance of the load.
arm is fitted with a stylus having a tip radius of .001" for the reproduction
of 45 RPM and 33t RPM fine groove records. The lever located at the
front of the cartridge is turned 180 0 to change from one stylus arm to the
other. The stylus arm under operation rests in the cradle of the lever arm
connected to the ceramic transducer. The use of the vibrating system
shown in Fig. 9.23 reduces the mechanical impedance at the stylus. The
lever system provides the proper stylus for the particular application. The
performance of the system may be deduced from the mechanical network.
23
Bauer, Gunter, and Steeler, Jour. A udio Eng. Soc., Vol. 2, No.4, p. 239, 1954.
ACOUSTICAL ENGINEERING
364
The internal electrical element is an electrical capacitance. The electrical
generator may be considered to be the open circuit voltage in series with the
electrical capacitance.
c. Magnetic Pickup.-A magnetic pickup 24,25 is a phonograph pickup
whose electrical output is generated in a coil or conductor in a magnetic
field or circuit. A magnetic pickup of early design is shown in Fig. 9.24.
The motion of the needle is transferred to the armature. The steady flux is
furnished by a permanent magnet. The armature is of the balanced type
so that in its central position there is no flux through the armature. When
the armature is deflected, a flux flows through the armature which induces
a voltage in the coil.
A schematic view of the magnetic system and magnetic network of a
magnetic pickup is shown in Fig. 9.25. The system of Fig. 9.25 applies
to Fig. 9.24 assuming the variable air gaps 1 and 3 of Fig. 9.25 are the
reluctances Rl and R2 of Fig. 9.25. If the armature is deflected a distance
Llx, there will be a change in the reluctances Rl and R2 and there will be a
flow of flux through the armature. The reluctance of the armature is small
compared to the reluctances of the air gaps Rl and R2 and may therefore be
neglected. Furthermore, since the combined reluctance of Rl and R2 is
large compared to the reluctance R M , the flux through the armature due
to a deflection ~x is given by
9.6
Since Rl = R2
becomes
=
Ra
=
R4
=
R in a symmetrical system, equation 9.6
Llcp
=
MMLlR
R2
9.7
The reluctance R is given by
9.8
where a = spacing between the armature and pole pieces, in centimeters, and
A = area of the pole piece, in square centimeters.
The incremental change in reluctance is given by
~R=~x
A
9.9
where ~x = change in distance a, in centimeters.
From equations 9.7, 9.8, and 9.9,
~cp
=
AMMLlx
a2
24
25
Kellogg, E. W., Jour. A.I.E.E., Vol. 46, No. 10, p. 1041, 1927.
Hasbrouck, H. J., Proc. I.R.E., Vol. 27. No.3, p. 184, 1939.
9.10
365
MISCELLANEOUS TRANSDUCERS
rEG
~
MAGNET
r
C M2 MI
ARMATURE
ELECTRICAL CIRCUIT
m2
S:;~--;-ICOIL
m,
rMI
~.$ f=TJ:::
SCHEMATIC VIEW
SIDE VIEW
LG
MECHANICAL NETWORK
FIG. 9.24. Front and side views, mechanical network, and electrical circuit of a magnetic
pickup. In the mechanical network, ZMR = the mechanical impedance of the record.
GMO = the compliance of the needle. ml = the mass of the needle holder and armature.
GMl = the compliance of the needle holder pivot. C M2 and rMl = the compliance and
mechanical resistance of the damping material. m2 = the mass of the pickup and tone
arm. 1M = the force generated by a velocity generator. In the electrical circuit,
eo = the open circuit voltage developed in the coil. La and rEO = the inductance and
electrical resistance of the coil. ZEL = the el,ectrical impedance of the load.
STYLUS
R2
~
I
R,
R,
ARMATURE
e
cp
SPACER
MAGNET
MM
R2
~
COIL
N
TURNS
RM
MM
RM
SCHEMATIC VIEW
MAGNETIC CIRCUIT
FIG. 9.25. Schematic view and magnetic circuit of
the magnetic system of a magnetic phonograph
pickup. In the mechanical circuit, MM = the
magnetomotive force developed by the permanent
magnet. RM = the reluctance of the path in the
magnetic material. Rl and R2 = the reluctances
of the variable air gaps. Ra and R4 = the reluct­
ances of the fixed air gaps. <p = the flux in the
armature. e = the voltage induced in the coil.
The generated voltage e, in abvolts, in the coil is given by
de/>
9.11
e=N­
dt
where N = number of turns in the coil.
From equations 9.10 and 9.11, the generated voltage in the coil is given by
NMMA.
9.12
e=~x
where .i
=
velocity of the armature at the pole pieces, in centimeters per
second.
366
ACOUSTICAL ENGINEERING
Equation 9.12 shows that the generated voltage will be independent of the
frequency if the velocity of the armature is independent of the frequency.
The mechanical network of the mechanical system is shown in Fig. 9.24.
Damping, represented by the compliance CM2 and the mechanical resist­
ance rMI, is furnished by a suitable material such as viscoloid.
A more recent design of magnetic pickup 26 is shown in Fig. 9.26. The
horizontal stylus arm also serves as the armature. The pole pieces are
located at the stylus. This design makes it possible to obtain a relatively
ARMATURE
DAMPING BLOCK
--
-------~l
I
I
I
I
I
I
I
I
I
t__ ~~=_~~
_____ ________
..1
ELECTRICAL CIRCUIT
BOTTOM VIEW
SECTIONAL VIEW
MECHANICAL NETWORK
FIG. 9.26. Bottom and sectional views, mechanical network, and electrical
circuit of a magnetic pickup. In the mechanical network, 3MB = the mechanical
impedance of the pickup. ml = the mass of the stylus and stylus holder. YMI
and CMl = the mechanical resistance and compliance of the armature. m2 = the
mass of the pickup and tone arm. J.v = the force generated by a velocity
generator. In the electrical circuit, eo = the open circuit voltage developed in
the coil. LG and rEG = the inductance and electrical resistance of the coil.
ZEL = the electrical impedance of the load.
low mechanical impedance. The steady flux is supplied by a small perma­
nent magnet. The performance of the relatively simple vibrating system
may be obtained from the mechanical network. The open circuit voltage
output is proportional to the velocity of the armature. The electrical
generator may be considered to be the open circuit voltage in series with the
electrical impedance. Since the electrical impedance is primarily inductive,
the electrical impedance is proportional to the frequency.
Another more recent design of magnetic pickup 27 is shown in Fig. 9.27.
The vertical stylus arm also serves as the armature. The coil surrounds the
armature. The armature is of the balanced type with air gaps between
the armature and the pole pieces at the two ends of the armature. When
26
27
Bachman, W. S., Elec. Ind., Vol. 4, No.7, p. 86,1945.
Stanton, W.O., Jour. Audio Eng. Soc., Vol. 3, No.2, p. 70, 1955.
367
MISCELLANEOUS TRANSDUCERS
the armature is in its central position, there is no flux in the armature. When
the armature is deflected, a flux flows in the armature which induces a
voltage in the coil. The performance of the relatively simple vibrating
system may be obtained from the mechanical network. The open circuit
voltage output is proportional to the velocity of the armature. The electrical
generator may be considered to be the open circuit voltage III senes with
rEG
[j['IT. - COIL
STYLUS:;.u
SECTIONAL VIEW
LG
C)
ELECTRICAL CIRCUIT
TOP VIEW
PERSPECTIVE VIEWS
MECHANICAL NETWORK
FIG. 9.27. Top and bottom perspective views, mechanical network, and the
electrical circuit of a magnetic pickup. In the mechanical circuit, ZMR = the
mechanical impedance of the record. ml = the mass of the stylus, stylus holder,
and armature. rMl and C Ml = the mechanical resistance and compliance of the
armature. rM2 and C M2 = the mechanical resistance and compliance of the
armature longitudinal support wire. rMa and CMa = the mechanical resistance
and compliance of the armature base support. ma = the mass of the pickup and
tone arm. 1M = the force generated by the velocity generator. In the electrical
circuit, ea = the open circuit voltage developed in the coil. La and rEG = the
inductance and electrical resistance of the coil. ZEL = the electrical impedance
of the load.
the electrical impedance. Since the electrical impedance is primarily
inductive, the electrical impedance is proportional to the frequency.
d. Dynamic Pickup.-A dynamic pickup is a phonograph pickup in which
the output results from the motion of a conductor in a magnetic field.
Fig. 9.28 shows a cross-sectional view and mechanical network of a dynamic
pickup 28 for the reproduction of hill-and-dale type records. The principal
mechanical impedance is due to the mass of the needle and coil. The
output of the coil is proportional to the velocity. Therefore, the response
characteristic is similar to that of the magnetic pickup. The coil is
practically a constant electrical resistance over the audio-frequency range.
A dynamic pickup 29 employing a stylus arm attached to a coil located
in a magnetic field for the reproduction of lateral phonograph records is
shown in Fig. 9.29. The performance of the system may be obtained from
28
29
Frederick, H. A., Jour. Soc. Mot. Pic. Eng., Vol. 18, No.2, p. 141. 1932.
Lindenberg, T. Jr., Electronics, Vol. 18, No.6, p. 108, 1945.
ACOUSTICAL ENGINEERING
368
the mechanical network of Fig. 9.29. Since the system may be made very
small and light, it is possible to reproduce the entire audio-frequency range.
The open circuit voltage is proportional to the velocity of the coil. The
electrical generator may be considered to be the open circuit voltage in
MAGNET
VOICE COIL
SECTIONAL VIEW
MECHANICAL NETWORK
ELECTRICAL CIRCUIT
FIG. 9.28. Cross-sectional view, mechanical network, and electrical circuit of a vertical
dynamic pickup. In the mechanical network, ZMR = the mechanical impedance of
the record. ml = the mass of the stylus and voice coil. GMl and YMl = the compliance
and mechanical resistance of the suspension system. m2 = the mass of the pickup and
tone arm. 1M = the force generated by a velocity generator. In the electrical circuit,
eo = the open circuit voltage developed in the voice-coil.
LG and YEO = the inductance
and electrical resistance of the voice-coil. ZEL = the electrical impedance of the load.
PERSPECTIVE VIEW
ELECTRICAL CIRCUIT
BOTTOM VIEW
MECHANICAL NETWORK
FIG. 9.29.
Perspective and bottom views, mechanical network and
electrical circuit of a lateral dynamic pickup. In the mechanical circuit,
ZMR = the mechanical impedance of the record.
ml = the mass of the
stylus and stylus holder. YMl and GMl = the mechanical resistance and
compliance of the stylus arm. m2 = the mass of the coil. YM2 and
GM2 = the mechanical resistance and compliance of the coil supports.
YM3 and GMa = the mechanical resistance and compliance of the longi­
tudinal coil support. ma = the mass of the pickup and tone arm. 1M
= the force generated by velocity generator. In the electrical circuit,
eo = the open circuit voltage developed in the coil.
La and rEO = the
inductance and electrical resistance of the coil. ZEa = the electrical
impedance of the load.
series with the electrical impedance of the coil. The coil is practically a
constant electrical resistance over the audio-frequency range.
Another form of dynamic pickup, shown in Fig. 9.30, is capable of repro­
ducing both lateral- and vertical-type phonograph records by merely chang­
ing the transformer connections. The vibrating system consists of two
r
369
MISCELLANEOUS TRANSDUCERS
parallel ribbons located in a magnetic field. When the stylus is actuated
by a lateral-type phonograph record the ribbons rotate about the center
axis. When the stylus is actuated by vertical-type phonograph record the
two ribbons move together in a direction normal to the plane of the record.
The direction of the currents in the two ribbons differs for the two types of
motion. Each ribbon is connected to a separate transformer. In this
way the outputs of the two ribbons can be brought into phase for either
lateral- or vertical-cut records by merely changing the transformer con­
nection. The open circuit voltage is proportional to the velocity of the
VERTICAL
CONNECTION
LATERAL
CONNECTION
{::]J
rEG
TRANSFORMERS
LG
ELECTRICAL CIRCUIT
$ TIC
M,
rM
f.------r--;M
M
MECHANICAL NETWORK
PERSPECTIVE VIEW
FIG. 9.30. Perspective view, electrical connection arrangement, mechan­
ical network, and electrical circuit of a combination vertical or lateral
dynamic pickup. In the mechanical network, ZMR = the mechanical
impedance of the record. mv rM, and eM = the mass, mechanical
resistance, and compliance of the stylus and conductors. m2 = the mass
of the pickup and tone arm. f M = the force generated by a velocity
generator. In the electrical circuit, eo = the open circuit voltage developed
by the conductors. Lo and rEO = the inductance and electrical resistance
of the conductors. ZEL = the electrical impedance of the load. Note:
Either vertical- or lateral-type phonograph records can be reproduced by
merely changing the transformer output connections.
ribbons. The electrical generator may be considered to be the open circuit
voltage in series with the electrical impedance of the ribbons. This pickup
is designed for wide-range reproduction of transcription phonograph records.
e. Frequency Modulation Pickup.-A frequency modulation pickup is a
phonograph pickup in which the frequency of a high-frequency oscillator
is varied by altering one of the elements in the oscillating circuit. By use
of a discriminator the modulated high-frequency output is transformed to
the vibration frequency of the stylus.
A perspective view, electrical diagram, mechanical network, and response
frequency characteristic of a frequency modulation pickup30 are shown in
Fig. 9.31. A stretched ribbon is mounted in a plane parallel to an insulated
plate and spaced by a small air gap. The stylus supporting wire is anchored
30
Beers and Sinnett, Proc. Inst. Rad. Eng., Vol. 31, No.4, p. 138, 1943.
370
ACOUSTICAL ENGINEERING
at its upper end. It is attached to the ribbon at approximately the mid­
point of its length and the free end is bent in a plane parallel to the record
groove. A sapphire stylus is attached to the end of the wire. It is evident
r[Iim,l~:,
MECHANICAL NETWORK
PERSPECTIVE VI EW
9.31. Perspective view, electrical system, and mechanical
network of a frequency modulation pickup. In the mechanical
network, ZMR = the mechanical impedance of the record. ml = the
mass of a stylus and holder. CMl = the compliance of the stylus
arm. m2 = the mass of the vertical member. C M2 and YMl = the
compliance and mechanical resistance of the ribbon. ma = the
mass of the pickup and tone arm. JM = the force generated by a
velocity generator.
FIG.
that a lateral displacement of the stylus will produce a change in the spacing
between the ribbon and insulated black plate and thus produce a change in
electrical capacitance. The electrical capacitance formed by the ribbon and
insulated back plate is made a part of the oscillating circuit of a 30-megacycle
ELECTRICAL CIRCUIT
ELECTRICAL SYSTEM
DIAPHRAGM
TONE ARM
m4
SECTIONAL VIEW
MECHANICAL NETWORK
FIG. 9.32.
Sectional view, electrical system, mechanical network, and electrical circuit
of an electronic pickup. In the mechanical network, ZMR = the mechanical impedance
of the record. m l = the mass of the stylus. YM1 and C Ml = the mechanical resistance
and compliance of the stylus arm. YM4 and C M4 = the mechanical resistance and
compliance of the damping member. m2' YM2' and C M2 = the mass, mechanical
resistance, and compliance of the anode lever. ma, YMa, and C Ma = the mass, mechan­
ical resistance, and compliance of the diaphragm. m4 = the mass of the pickup and
tone arm. JM = the force generated by a velocity generator. In the electrical circuit,
eG = the open circuit voltage developed in the tube.
YEG = the internal electrical
resistance of the tube. ZEL = the electrical impedance of the load.
r
371
MISCELLANEOUS TRANSDUCERS
oscillator. The change in capacity due to the motion of the stylus produces
a change in the frequency of the oscillator. The output of the oscillator is
impressed upon a discriminator and detector. The output of the detector
corresponds to amplitude of the stylus.
f. Electronic Pickup.3I-An electronic pickup is a phonograph pickup
in which the output is generated by the motion of an electrode in a vacuum
tube. A cross-sectional view, electrical circuit, mechanical network, and
the response frequency characteristic of an electronic pickup are shown in
Fig. 9.32. The voltage is generated by the change in distance between
the cathode and anode. The anode is the movable element. :Motion of
the anode is transferred through the envelope of the tube by means of a
thin metal diaphragm. A permanent sapphire stylus is used in this pickup.
The voltage output is proportional to the amplitude.
g. Variable Resistance Pickup.-A variable resistance phonograph pick­
up is a pickup in which the voltage is generated in a current polarized
variable electrical resistance element. The electrical resistance of the
element is varied by compressions and rarefactions of the element. A
schematic view of a variable resistance pickup 31& is shown in Fig. 9.33.
m,
MECHANICAL
SCHEMATIC
NETWORK
VIEW
FIG. 9.33. Perspective view and mechanical network of a variable resistance
pickup. In the mechanical network, ZMR = the mechanical impedance of the
record. m l = the mass of the stylus. CMl = the compliance of the stylus
arm. m2 = the mass at the base of the stylus arm. C M2 and rM = the com­
pliance and mechanical resistance of the damping material for the base
support. ma = the mass of the tone arm. 1M = the force generated by the
velocity generator.
A variable electrical resistance element is cemented on each side of the
stylus arm. Bending of the stylus arm produces rarefactions on one side
and compressions on the other side. The compressions and rarefactions
produce a corresponding decrease and increase in electrical resistance of
the variable resistance element. Since the element is polarized by a current,
the change in electrical resistance produces a corresponding change in
voltage. The electrical schematic diagram shows the polarizing battery
and transformer system. The electrical system is similar to the double­
button carbon microphone. The voltage output is proportional to the
31
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 19, No.2, p. 307, 1947.
Bachman, W. S., Elec. Eng., Vol. 65, No.3, p. 159, 1946.
ala
372
ACOUSTICAL ENGINEERING
amplitude. The base of the stylus arm is embedded in damping material.
The performance of the vibrating system may be obtained from a considera­
tion of the mechanical network.
h. Feedback Pickup.32-Feedback may be used in electrical and electro­
mechanical systems to change such factors as the transmission and distortion
characteristics, the terminal impedances and the noise levels of the systems.
STEEL
ARMATURE
MAGNET
STYLUS
PERSPECTIVE VIEW
MECHANICAL NETWORK
tTl4
CMI
40
ml
\ I
I
I")Q 35
'1'
.,30
CM2 rM2
u
z
.. 25
.,0
CM4
!20
BOTTOM
CERAMIC
UNIT'\.
.: 15
I
~ 10 V
u
:t
COIL
ELECTRICAL DIAGRAM
:oJ 5
::E
--
~ .......
B
100
r--.I.
1000
FREQUENCY
~
10000
FIG. 9.34.
Perspective and bottom views, mechanical network, electrical diagram, and
mechanical impedance frequency characteristics of a feedback pickup. In the mechan­
ical network, ZMR = the mechanical impedance of the record. ml = the mass of the
stylus and stylus holder. C Ml = the compliance of the stylus arm. m2 = the mass
of the front portion of the armature and ceramic transducer. f'M2 and C M2 = the
mechanical resistance and compliance of the damping block under the stylus arm. f'Ma
and C Ma = the mechanical resistance and compliance of the damping blocks on the
ceramic transducer and magnetic armature. ma = the mass of the rear portion of the
ceramic transducer and magnetic armature. CM4, = the compliance of the ceramic
transducer and magnetic armature. m4, = the mass of the pickup and tone arm.
rM5 and C M5 = the mechanical resistance and compliance of the support for the ceramic
transducer and armature. IMl and/M2 = the forces generated by the velocity generator
and the magnetic driving system. In the graph: A. mechanical impedance characteristic
without feedback. B. mechanical impedance characteristic with feedback.
Feedback has been applied to cutters, calibrators, and other electroacoustic
devices. However, the application of feedback to phonograph pickups is a
recent development. The problem in the phonograph pickup is to reduce
the mechanical impedance at the stylus of the pickup so that the load
presented to the record will be reduced. The feedback phonograph pickup
shown in Fig. 9.34 employs two electromechanical transducers in the feed­
32 Halter, J. B., Unpublished Report.
MISCELLANEOUS TRANSDUCERS
373
back loop. The system for driving the stylus is an electromagnetic trans­
ducer in which the stylus is attached to the armature. The sensing and
reproducing system is a ceramic transducer consisting of two barium titanate
strips attached to the two sides of the steel armature. The electrical diagram
of the feedback phonograph pickup is shown in Fig. 9.34. The performance
of the system may be deduced from the mechanical network of Fig. 9.34.
In the ceramic transducer, the open circuit voltage e, in volts, is given by
9.13
where x
=
KB
=
amplitude of the transducers, and
constant of the system involving the material and construction
of the transducer.
In the electromagnetic transducer the force 1M, in dynes, produced by a
current i, in abamperes, in the coil is given by
9.14
where KJ = constant involving the parameters of the electromagnetic
transducer.
The problem is to adjust the amplitude and phase of the system so that a
maximum displacement will be produced in CM4 for a minimum force 1M! at
the stylus. The performance of the system with and without feedback is
shown in Fig. 9.34. It will be seen that a tremendous reduction in the force
1M! is obtained with feedback.
i. Compliance 01 Pickups.-A consideration of the pickups described in the
preceding sections shows that the lateral compliance at the stylus is an
important element. The compliance determines the force applied to the
record in the low-frequency range where the pickup system is stiffness
controlled. The force is the ratio of the displacement to the compliance.
See Sec. 4.6. As an indication of the magnitude of the compliance of pick­
ups, the compliance of a pickup for fine-groove records is of the order of
1 X to- 6 centimeters per dyne.
j. Tone Arm Resonance.-Tone arm resonance occurs in the low-frequency
range when the mechanical reactance due to the effective mass of the pickup
and tone arm is equal to the mechanical reactance of the compliance of the
pickup. The response at the resonant frequency is determined by the
damping in the pickup and tone arm pivot. In one design of tone arm 33
additional mechanical resistance has been added by a viscous damped tone
arm. The tone arm resonant frequency determines the low-frequency range
because the response falls off rapidly below the tone arm resonance.
3. Distortion in Record Reproduction.-The recording and reproducing
of a phonograph record is a complicated process and there are many sources
of nonlinear distortion. The record does not present an infinite mechanical
33
Bachman, W. S., Trans. I.R.E., Prof. Group Audio, March, 1951.
374
ACOUSTICAL ENGINEERING
impedance to the needle. As a consequence, the vibrating system of the
pickup is shunted by the effective mechanical impedance of the record at the
needle. Nonlinear distortion will be introduced if the record is a variable
element.. Another source of distortion is due to a deviation in tracking
commonly termed tracking error. Still another form of distortion is known
as tracing distortion, due to the finite size of the stylus. These and other
types of distortion will be discussed in this section.
A source of nonlinear distortion due to a deviation in tracking,34,35 is
commonly termed tracking error. The angle between the vertical plane
containing the vibration axis of the pickup and the vertical plane con­
taining the tangent to the record is a measure of the tracking error. If the
vibration axis of the pickup passes
through the tone arm pivot, the tracking
error can be zero for only one point on
the record. The tracking error can be
reduced if the vibration axis of the
pickup is set at an appropriate angle
with respect to the line connecting the
stylus point and the tone arm pivot
together with a suitable overhang dis­
tance between the stylus and the record
axis. Fig. 9.35.
The amount of overhang d, in inches,
is given by
Rt 2
d=
FIG. 9.35.
Geometry of a tone arm
system for reducing the tracking error.
1
R
9.15
L [- (1 + ~)2 + R t ]
4
Ro
Ro
where L ='length of the tone arm from
the pivot to the stylus, in
inches,
Ro = radius of the start groove of
the recording, and
R t = radius of the end groove of
the recording.
The angle a in degrees between the vertical plane containing vibration
axis of the pickup and the line joining the stylus and the tone arm pivot is
given by
9.16
84
35
Olney, Benj., Electronics, Vol. 10, No. 1, p. 19, 1937.
Bauer, B. B., Electronics, Vol. 18, No.3, p. 110, 1945.
r
MISCELLANEOUS TRANSDUCERS
375
With the application of equations 9.15 and 9.16 in the design of a tone arm,
the tracking error can be reduced to ±5 per cent.
A form of distortion in lateral-disk record reproduction known as tracing
distortion 36 ,37,38 is a function of the diameter of the stylus, the lateral
velocity, and the groove velocity. The distortion is due to the fact that
there is not a one-to-one correspondence between the shapes of the cutting
and reproducing stylii. The shape of the groove produced by the cutting
stylus of a lateral-disk recorder is shown in Fig. 9.36. The groove narrows
as the cutting stylus approaches the center position because the cutting
stylus is moving at an angle with respect to the motion of the record. A
sectional view of a groove with the reproducing stylus in contact with the
TOP VIEW
rmym
A-~
mvm
B-~
SECTiONAL VIEWS
J.
rmyrnC-~
FIG. 9.36.
The shape of the groove cut by the stylus of a lateral
phonograph recorder in a lacquer disk.
walls of the groove is shown in Fig. 9.37. Referring to Figs. 9.36 and 9.37,
it will be seen that, when the reproducing stylus moves in this groove, it
will rise as the groove narrows. The frequency of the rise is twice the
frequency of the modulation. The narrowing of the groove is termed
the pinch effect. The two sides of the groove are symmetrical; therefore, the
stylus must execute symmetrical motion about the center line which means
that there should be no even harmonics. However, odd harmonics are
produced. The equations for the magnitude of the harmonics have been
developed.
With regard to nonlinear distortion, the two-frequency method of distor­
tion testing has been found to agree quite well with subjective testing of
nonlinear distortion. 39 The nonlinear distortion, in per cent, which occurs
when two frequencies are combined is given by
DiToro. M. J.. Jour. Soc. Mot. Pic. Eng.• Vol. 29. No.5. p. 493.1938.
Pierce and Hunt, Jour. Acous. Soc. Amer.• Vol. 10. No. 1, p. 14. 1938.
38 Sepmeyer, L. W., Jour. Acous. Soc. Amer.• Vol. 13. No.3. p. 276.1942.
39 Roys. H. E .• RCA Review, Vol. 10. No.2. p. 254. 1949.
36
37
376
ACOUSTICAL ENGINEERING
9.17
where
peak lateral velocity of the lower frequency ft, in centimeters
per second,
Uz = peak lateral velocity of the upper frequency /Z, in centimeters
per second,
r = radius of the reproducing stylus, in centimeters,
ft = lower frequency, in cycles per second,
h = upper frequency, in cycles per second, and
S1 = groove velocity, in centimeters per second.
Ul
=
FIG. 9.37.
Sectional view of a stylus in a groove. The dimen­
sions for coarse groove, fine groove, and ultra fine groove are as
follows:
Dimension
Coarse
Fine
Ultra Fine
A
·006
·0027
·0010
B
·0008
·0004
·0004
C
·0019
·0007
·00017
D
·0027
·00025
·001
E
·0023
·00027
·00015
F
45°
45°
45°
G
90°
90°
90°
Subjective tests have shown that when the intermodulation distortion
is less than 10 per cent the distortion is practically imperceptible. This then
establishes a value for the terminal velocity for a certain stylus diameter
and the peak lateral velocity.
MISCELLANEOUS TRANSDUCERS
377
The record does not present an infinite mechanical impedance to stylus.
As a result, the vibrating system of the pickup is shunted by the mechanical
impedance of the record, as depicted in the mechanical networks of pickups
in this chapter. Nonlinear distortion 4o ,41 will occur if the record is a
variable element. If the force which the stylus presents to the record is of
such magnitude that it exceeds the yield point of the record material, the
mechanical impedance of the record will not be a constant. The result is
production of nonlinear distortion. Furthermore, if the force exceeds the
yield point by a considerable amount, the record may be permanently
damaged.
As the needle or stylus is worn by the groove the shape of the point
changes from a spherical surface to a wedge shape. The wedge-shaped
stylus 42 introduces nonlinear distortion and a loss in the high-frequency
response.
A consideration of the load and needle forces at the stylus tip shows that
there is force which is proportional to the tracking angle. This force is
usually directed toward the center of the record and is applied to the inner
boundary of the record groove. This force is known as the side thrust and
it is responsible for the unequal wear43 on the two sides of the stylus.
Another source of distortion is due to the lack of correspondence between
the linear groove speed in the recording and ultimate reproduction. This
type of distortion is termed" wows. "44 This may be due to a nonuniform
speed of the record turntable during recording or reproduction, misplace­
ment of the center hole or configuration distortion during the processing.
In general, the major source of "wows" is due to nonuniform speed of the
reproducing turntable.
The record surface noise,45 in the absence of any signal, is one of the factors
which limits the volume range and the frequency range of shellac phono­
graph records. The amount of surface noise for a given record is propor­
tional to the frequency band width. In order to reduce the surface noise
to a tolerable value in shellac records it is usually necessary to limit the
high-frequency range in reproduction. A method of decreasing the effective
surface noise consists of increasing the amplitude of the high-frequency
response in recording and introducing complementary equalization in repro­
duction. The volume range of a phonograph record, in general, does not
permit recording the full range of a symphony orchestra without some
compression. To offset this compression complementary expansion may
be introduced in the reproduction.
4. Record Noise.-When the stylus of a phonograph pickup is actuated
by the groove of a phonograph record a force is developed between the stylus
Begun and Lynch, Jour. Acous. Soc. Amer., Vol. 13, No.3, p. 284, 1942.
Max, A. M., Jour. Audio Eng. Soc., Vol. 3, No.2, p. 66, 1955.
42 Bauer, B. B., Jour. Acous. Soc. Amer., Vol. 16, No.4, p. 246, 1945.
43 Bauer, B. B., Trans. I.R.E., Prof. Group Audio, PGA-6, p. 11, 1952.
44 Comerci, Frank A., Jour. Soc. Mot. Pic. Tel. Eng., Vol. 64, No.3, p. 117, 1955.
45 Reid, J. D., Jour. Acous. Soc. Amer., Vol. 13, No.3, p. 274,1942.
40
41
378
ACOUSTICAL ENGINEERING
and the walls of the record. The force, in dynes, developed by the inter­
action of the pickup stylus and the record is given by
9.18
where
w
= 27rj,
j
x
=
=
ZMR =
ZMP =
frequency, in cycles per second,
amplitude of the groove, in centimeters,
mechanical impedance of the record, in mechanical ohms, and
mechanical impedance of the pickup at the stylus, in mechani­
cal ohms.
Equation 9.18 illustrates the importance of a pickup with a small mechani­
cal impedance. If the pickup mechanical impedance is comparable to
the mechanical impedance of the record, a considerable part of the amplitude
of the record groove will take place in motion ot the record. This motion
or vibration of the record produces sound which is radiated into the air.
The radiated sound corresponds somewhat to the sound recorded on the
record, but it is very much distorted due to the way in which it is produced
and is, therefore, disagreeable. Furthermore, there is interference between
this sound and the sound radiated from the loudspeaker. The force which
drives the stylus is a function of the record mechanical impedance, if the
mechanical impedance of the pickup at the stylus is relatively large. This
may produce distortion in the reproduced sound because the mechanical
impedance of the record varies over wide limits from the outside to the inside
groove and is a function of the mounting of the record supporting means.
A pickup with a high mechanical impedance also produces excessive record
wear. Equation 9.18 together with the above discussion shows that
record noise and wear and distortion can be reduced by making the
mechanical impedance of the pickup small compared to the mechanical
impedance of the record. The measurement of record noise is described in
Sec.lO.SD.
C. Selection oj Rotational Speed and Record Diameter. 46-The playing
time, the diameters of the start and end grooves of the recording, the rota­
tional speed of the grooves per inch, and the terminal linear velocity are all
factors involved in the determination of a record. These relations are inter­
connected by the following equations:
p=D02~DtN
9.19
and
9.20
46
Reiskind, H. I., Trans. I.R.E., Prof. Group Audio, PGA-S, February, 1952.
379
MISCELLANEOUS TRANSDUCERS
where P = playing time, in minutes,
Do = diameter of the start groove of the recording, in inches,
D t = diameter of the end groove of the recording, in inches,
R = rotational speed, in revolutions per minute,
N = grooves, per inch, and
St = terminal linear velocity.
Equations 9.19 and 9.20 show that there are many parameters involved
in the selection of the rotational speed and the playing time. One of the
most important of these is nonlinear distortion, discussed in the preceding
section. The other factors such as playing time, record diameter, grooves
per inch, rotational speed, etc. are determined by the particular application.
12 INCH-78 RPM
TINCH - TaRPM
IOINCH-78RPM
7 INCH -
16~
RPM
liNCH - 33.j.RPM
9.38. Typical dimensions of the most common commercial-type disk phonograph
records. The dimensions are the diameter of the outside of the record, the diameter of
the outside and inside record groove, the diameter of the label, and the diameter of the
center hole.
FIG.
D. Commercial Disk Phonograph Records. 47 ,48,49-Commercial phonograph
records are made in four speeds, namely, 78, 45, 331. and 161 revolutions
per minute. The 78 RPM records are made in three diameters, termed
12 inches, 10 inches, and 7 inches. The normal maximum playing times
are 5, 3l and 2t minutes, respectively. The 331 RPM records are made in
thre~ diameters, termed 12 inches, 10 inches, and 7 inches. The nominal
47
48
49
Goldmark, Snepvangers, and Bachman, Proc. I.R.E., Vol. 37, No.8, p. 923, 1949.
Carson, Burt, and Reiskind, RCA Review, Vol. 10, No.2, p. 173, 1949.
Goldmark, Peter C., Audio, Vol. 39, No. 12, p. 15, 1955.
380
ACOUSTICAL ENGINEERING
maximum playing times are 25, 17, and 8 minutes, respectively. The 45
RPM records are made in a diameter of 7 inches. The nominal maximum
playing time is 8 minutes. The 161 RPM records are made in a diameter
of 7 inches. The nominal maximum playing time of the records with the
large center hole is 30 minutes. The nominal maximum playing time for
the small-hole records is 45 minutes for music and 60 minutes for speech.
The diameter of the outside, the diameters of the first and last program
grooves, the label diameters and the diameter of the center hole of the
different records are shown in Fig. 9.38. It may be mentioned in passing
that the specifications of Fig. 9.38 are given as representative and do not
include all the variations.
The dimensions of the coarse groove, the fine groove, and the ultra-fine
grooves and the dimensions of the corresponding stylii are shown in Fig.
9.37. The coarse groove is used in 78 RPM records. The fine groove is
used in the 45 and 33! RPM records. The ultra-fine groove is used in the
16i RPM records.
The maximum nominal grooves per inch for the different size grooves are
as follows: coarse groove, 125; fine groove, 275; and ultra-fine groove, 550.
The maximum amplitudes, in inches, in the frequency range 200 to 2000
cycles for the different size grooves are as follows: coarse groove, .004-.005
inch; fine groove, .0015-.002 inch; and ultra-fine groove, .0007-.001 inch.
9.4. Vibration Pickup.-Measurement and study of vibration have
become an important factor in the elimination of noise in machinery, vehicles,
and household appliances. Depending upon the requirements, it may be
desirable to measure amplitude, velocity, or acceleration.
Direct measurement of acceleration, velocity, or displacement of vibra­
tion requires the establishment of a stationary body to serve as a reference
frame against which these functions may be determined. Any type of
transducer may be used to convert the motion into the corresponding
electrical current. It is the purpose of this section to describe a piezo­
electric inertia-type vibration pickup.
The structure of a typical inertia-type piezoelectric vibration pickup 50
is shown in Fig. 9.39. The crystal is a Rochelle salt bimorph type. With
the crystal held at the three corners the output voltage is proportional to
the force acting on the free corner... The crystal is enclosed in a rigid metal
case. When the case is driven by a vibration normal to the plane of the
crystal element, a force is developed at the unsupported section of the
crystal due to inertia reaction. The mechanical network of the vibrating
system is shown in Fig. 9.39. The mechanical resistance is small and does
not influence the mechanical network save near the resonant frequency
which occurs at about 1500 cycles. The velocity, in centimeters per second,
is given by
9.21
50
Bauer, B. B., Jour. Acous. Soc. A mer., Vol. 11, No.3, p. 303, 1940.
381
MISCELLANEOUS TRANSDUCERS
where 1M
=
m
=
eM =
driving force developed at the free edge of the crystal, in
dynes,
effective mass of the crystal, in grams, and
effective compliance of the crystal, in centimeters per dyne.
For frequencies well below the resonant frequency the velocity is given by
X,-....,
!M
9.22
Jwm
The acceleration is given by
'.
1M
JXw=
-
9.23
m
The displacement is given by
x
9.24
JW
Below the resonant frequency the force is proportional to and in phase
with the acceleration. The voltage output of the unit then corresponds
X=
ELECTRICAL
SECTIONAL VIEW
SUPPOR~S
CRYSTAL
LEADS
PERSPECTIVE
VIEW
SYSTEM
10
[flu
..
flO
r.
MECHANICAL
NETWORK
III
o
...J
m
w
>
0
~ -10
A-#P'
V D'
10
100
,REQUENCY
1000
FIG. 9.39.
Sectional view, perspective view of the crystal mounting arrangement,
mechanical network, electrical connection, and response frequency characteristic
of a vibration pickup. In the mechanical network, m = the mass of the crystal.
YM and eM = the mechanical resistance and compliance of the crystal and
supports. f M = driving force. In the electrical system, with the switch lever
on D, V, and A the response corresponds to displacement, velocity, and accelera­
tion, respectively. The graph depicts the voltage response frequency charac­
teristics for A, acceleration, V, velocity, D, displacement.
to the acceleration. The output of the acceleration-type pickup may be
integrated once or twice by means of an electrical network as shown in
Fig. 9.39 to obtain velocity and displacement.
The response frequency characteristics of the vibration pickup and the
electrical system are shown in Fig. 9.39. It has been found that, above
1000 cycles, the performance of the pickup is influenced by manner of
coupling to the vibration machine.
.
Magnetic and dynamic vibration pickups have also been developed.
382
ACOUSTICAL ENGINEERING
In these devices two different types are used, in one the armature or voice
coil is free and the field structure is coupled to the vibrating system under
test, in the other the armature or voice coil is driven by the vibrating system
under test. The electrical compensation in these devices differs from the
crystal type because the voltage output is proportional to the velocity.
A ceramic vibration pickup 51 similar to the crystal vibration pickup has
been developed. The vibrating system and the electrical characteristics
are similar to those of the crystal vibration pickup. The essential difference
being that the ceramic element is mounted on the four corners and the
acceleration acts upon the center.
An accelerometer 52 has been developed employing a mechano-electronic
transducer similar to that employed in the electronic microphone of Sec.
8.2F and the electronic pickup of Sec. 9.3B2f. A small weight is attached
to the anode rod. The stiffness and restoring force are supplied by the
diaphragm. The system exhibits a high order of sensitivity.
9.5. Sound-Powered Phones.-A sound-powered phone system is a
point-to-point telephone communicating system employing no batteries,
amplifiers or any other means of external power. The sequence of events
in a sound-powered telephone system is as follows: The human voice pro­
duces a sound wave which actuates the microphone at the transmitting end.
The microphone converts the acoustical energy into the corresponding
electrical energy. This energy is carried by wires to the receiving end.
At the receiving end the electrical variations are transformed into the cor­
responding sound vibrations by the receiver.
A sound-powered telephone is shown in Fig. 9.40. Cross-sectional views,
mechanical and electrical networks of the microphone and receiver are shown
in Fig. 9.40. In order to obtain a tolerable sound level at the receiver the
over-all efficiency of the system must be quite high. This high efficiency
is accomplished by the use of multi-resonant .elements which reduce the
mechanical impedance of the vibrating system. The transmission frequency
band is made relatively narrow so that a low value of mechanical impedance
can be obtained with a simple vibrating system. The response frequency
characteristics of the microphone, the receiver, and the combination of the
microphone and receiver are shown in Fig. 9.41. In the combination system
it will be seen that there is a gain in sound pressure over the useful trans­
mission frequency range which means that the sound pressure in the ear
cavity is greater than that at the microphone. The transmission of sound
without pressure loss requires a very efficient system.
9.6. Electrical Megaphone.-The electrical megaphone 53 consists of
the combination of a microphone, an amplifier, and a horn loudspeaker
(Fig. 9.42). The microphone and horn loudspeaker form a single unit.
In use, the operator speaks into the microphone. The voice is reinforced
by the amplifier and loudspeaker. The resulting power output is many
51
52
53
Carlson, E. V., Trans. I.R.E., Prof. Group on Audio, PGA-10, p. 2, 1952.
Lewis, Robert C., Jour. Acous. Soc. Amer., Vol. 22, No.3, p. 357, 1950.
Sanial, A. J., Communications, Vol. 25, No.7, p. 33, 1945.
r
383
MISCELLANEOUS TRANSDUCERS
rEI
L
~
REC.
REC. ELEC. CIR.
HANDSET
MICROPHONE
SECTIONAL
VIEW
RECEIVER
MECH.
NETWORK
SECTIONAL
VIEW
9.40. The sound powered handset, sectional views of the microphone and receiver,
mechanical networks of the microphone and receiver, and electrical circuit of the
receiver. In the microphone mechanical network, ZME = the mechanical impedance
due to the electrical circuit. m}, r.lf} , and C M } = the mass, mechanical resistance, and
compliance of the diaphragm and armature. C M2 = the compliance of the air chamber
in front of the diaphragm. m2 = the mass of the air in the aperture in the diaphragm
cover plate. CMa = the compliance of the mouthpiece cavity. m3 and rM3 = the
mass and mechanical resistance of the air load upon the mouthpiece. 1M = the driving
force. 5 = the area of the diaphragm, and p = the sound pressure. In the receiver
mechanical network, mv rM}, and elf} = the mass, mechanical resistance, and com­
pliance of the diaphragm and armature. C M2 = the compliance of the cavity in front
of the diaphragm. m2 = the mass of the air in the aperture of the diaphragm cover
plate. 2MB = mechanical impedance of the external load upon the receiver. In the
receiver electrical circuit, ZEM = the electrical motional impedance. Land rEI = the
damped inductance and electrical resistance of the receiver. ZEa = the electrical
impedance of the microphone. e = the developed voltage output of the microphone.
FIG.
A
ttl
B
MICROPHONE
C
RECEIVER
o
II
..
I­
::l
020
o
o
f\
I­
::l
I0
\A
30
0
500
::>
r'\
/
1000
r--f\/
I­
V
>
HANDSETS
I~t\
aJ
Z
- 30
"'«j"
TWO
10
40
\
\
2000 3000
FREQUENCY
a.
I­
520
V
!oJ
a:
::>
~ 10
.
V
!oJ
a:
o
500
1000
\
\
2000 3000
FREQUENCY
\
\
\
Q
!;(-20
a:
II
-30
500
/
1000
\
2000 3000
FREQUENCY
9.41. Response frequency characteristics of a sound powered telephone.
A. Voltage response frequency characteristic of the microphone. B. Voltage
response frequency characteristic of the receiver. C. Over-all pressure ratio
response frequency characteristic of two handsets, one used as a transmitter and
the other as a receiver.
FIG.
384
ACOUSTICAL ENGINEERING
times that of the unaided voice or the voice and an acoustical megaphone.
The only theoretical limitation to the amount of reinforcing is the produc­
tion of continuous oscillations due to regenerative feedback from the loud­
speaker to the microphone. The directional pattern of horns shows that
the rear radiation is quite small compared to that directly in front when
the dimensions of the mouth of the horn are comparable to the wavelength
(see Sec. 2.19). By placing the microphone at the rear of the horn and
HORN
LOUD SPEAKER
MECHANISM
~~=L
__
"::~
L.::===:::;;;;;;1'-===L-=~~~
FIG. 9.42. Sectional view depicting the elements and electrical
connections of an electrical megaphone.
attenuating the low-frequency range the amount of sound picked up by
the microphone is small. This makes it possible to obtain a relatively
large output before- oscillations begin. The microphone employed for the
electrical megaphone is of the close-talking type. The loudspeaker is a
conventional, light-weight horn loudspeaker. Two types of amplifiers
have been used-namely, a portable battery operated amplifier and a semi­
portable a-c line operated amplifier.
9.7. Magnetic Tape Sound Recording and Reproducing System.
54,55,56,57,58,59,60,61,62,63_Magnetic recording and reproducing were invented
Carlson and Carpenter, U.S. Patent, 1,640,881, 1927.
Wetzel, W. W., Audio Eng., Vol. 31, No. 12, p. 12,1947.
56 Camras, Marvin, Jour. Soc. Mot. Pic. Eng., Vol. 48, No.!, p. 14, 1948.
57 Begun, "Magnetic Recording," Murray Hill Books, New York, N.Y., 1949.
58 Frayne and Wolfe, "Elements of Sound Recording," John Wiley and Sons, New
York, N.Y., 1949.
59 Gratian, J. W., Jour. Acous. Soc. Amer., Vol. 21, No.2, p. 74, 1949.
60 Kornei, 0., Jour. Audio Eng. Soc., Vol. 1, No. 1, p. 225,1953.
61 Joryz, Alfred, Bibliography of Magnetic Recording, Jour. Audio Eng. Soc., Vol. 2,
No.3, p. 183, 1954.
62 Camras, Marvin, Convention Record I.R.E., Part 3, Audio, p. 16, 1953.
63 Selsted and Snyder, Trans. I.R.E., Prof. Group on Audio, Vol. AU-2, No. S, p. 137,
1954.
54
55
MISCELLANEOUS TRANSDUCERS
385
more than a half century ago by Poulsen. Since that time there has been a
periodic revival of magnetic recording and reproducing. During the past
decade the swing has been to magnetic
tape with the result that today it is
universally employed for all types of
magnetic reproduction. Magnetic tape
consists of a plastic base with a coating
of magnetic oxide as shown in Fig. 9.43.
PLASTIC
BASE
The base material in use today varies in
SECTIONAL VIEW
thickness from .001 inch to .0015 inch.
The magnetic coating varies in thick­
FiG. 9.43. Sectional view of magnetic
ness from .0002 inch to .0006 inch.
recording tape.
A typical BH curve of the magnetic
coating is shown in Fig. 9.44. The retentivity is of the order of 700 gausses
and the coercive force is of the order of 300 gilberts.
The recording and reproducing process is depicted in Fig. 9.45. The
passage of the tape past the reB- H
cording head leaves a series of
magnetized sections which corres­
pond to the signal which was
applied to the head when the tape
was in contact with the head at
each of these sections. In the
reproduction process, the tape is
moved past the head with the
result that a change in magnetic
H flux is produced in the head as
a magnetized section passes the
head. This change in flux in­
duces a voltage in the coil which
corresponds to the voltage of the
original applied signal.
A schematic view and the mag­
netic network of a magnetic re­
cording head and magnetic tape
are shown in Fig. 9.46. The cur­
rents il and iI' produce a flux cp in
FIG. 9.44. Typical B-H characteristic of the
the tape. Due to the retentivity
iron oxide coating on magnetic tape.
of the coating material of the tape,
a magnetized section is produced.
The action and performance of the recording process can be deduced from
the magnetic network of Fig. 9.46.
A schematic view and magnetic network of a magnetic reproducing head
and magnetic tape is shown in Fig. 9.47. When the magnetic tape is moved
past the head, the magnetomotive force of the magnetized reactions pro­
duces the magnetic flux CP5 and CP5' in the two coils. When this flux changes,
~'"
386
ACOUSTICAL ENGINEERING
MAGNETIC
COATING
PLASTIC
BASE
N~\
COIL
-I""'1I-+-~
e
N
RECORDED
SIGNALS
9.45. Schematic diagram depicting the
magnetic tape recording and reproducing process.
FIG.
RI
'"
Rz
,...M
,
R'
R2
R.
R.
''';y
R'
•
R•
R.
R4
•
R'
Rs
tjd'
R7
SCHEMATIC V I EW
MAGNETIC NETWORK
FIG. 9.46. Schematic view and magnetic net­
work of the head and tape of a magnetic tape
recording system. In the magnetic circuit,
M and M' = the magnetomotive forces
developed by the currents in the two coils.
R5 and R5' = the reluctances of the magnetic
material of the heads. Ra and R7 = the
reluctances of the top and bottom air gaps.
R4 and Rs = the reluctance of the magnetic
leakage path. R2 and R 2' = the reluctances
of the air gaps between the head and the
tape. RI = the reluctance of the magnetic
tape. rP = the flux through the magnetic
tape. i and i' = the currents in the two coils.
q;
•
e
"'5
"'.
"",R.
'!1.
SCHEMATIC VIEW
MAGNETIC NETWORK
FIG. 9.47.
Schematic view and magnetic net­
work of the head and tape of a magnetic tap!
reproducing system. In the magnetic circuit
M = the magnetomotive force stored in th!
tape. R5 and R5' = the reluctances of th!
magnetic material of the heads. Ra and R~
= the reluctances of the top and bottom ail
gaps. R4 and R6 = the reluctances of tht
magnetic leakage paths. R2 and R 2' = th!
reluctances of the air gaps between the head
and the tape. RI = the reluctance of tht
magnetic tape. rP5 and rP5' = the flux in the
coils. e and e' = the induced voltages in
the coils.
as the magnetomotive force changes as different magnetized sections of
the tape pass over the head, a voltage el and el' is induced in the coils.
These voltages correspond to the currents il and iI' applied to the recording
head.
In the recording and reproducing process there is a loss due to the finite
gap length. In the recording process the gap width is of little importance
because the recording process takes place from the trailing edge of the gap.
It is this edge rather than the gap that is of importance in the recording
head. In the reproducing process it is the length of the gap that determines
MISCELLANEOUS TRANSDUCERS
387
the magnetomotive force. The loss due to the finite length of the gap in
the reproducing head is given by
R
where R
=
~
Slll­
. 7TdJ
[
- 20 log
',/
9.25
loss in decibels,
d = length of the gap, in centimeters, and
" =
wavelength of the signal along the tape, in centimeters.
The gap loss response frequency characteristic is shown in Fig. 9.48.
It
5
o
-......
III
o
" r--­
.
.
z
- -10
(/)
~
:J; -15
\
\
/'""\
\
II:
-20
r'\
n11\
-25
II
-30
01
.02
.03 .04.05
d/ }..
.2
.3.4.5
I
2
3
4
5
EFFECTIVE GAP LENGTH ... WAVELENGTH
FIG. 9.48.
The response loss due to the reproduction with a finite gap as
a function of the ratio of the gap length to the wavelength.
will be seen that the response is zero when the gap length equals a wavelength
and multiples of the wavelength.
The output of the reproducing head is
d~
e = N dt
9.26
where e = voltage output, in abvolts,
N = number of turns in the coil,
~ = flux in the coil, in gausses, and
t = time.
If the amplitude of ~ is a constant, the voltage will increase at the rate of
6 db per octave. However, this characteristic must be multiplied by the
388
ACOUSTICAL ENGINEERING
gap loss. The open circuit voltage response of a magnetic reproducing
system is given by
RM
20 log (sin
=
"7)
9.27
The open circuit voltage response frequency characteristic of a magnetic
reproducing system is shown in Fig. 9.49.
35
30
25
/
CD
o
;!; 20
....
Ul
/
z
a'
....
'"a:
15
/
10
5
o
v
'\, ('
~
/
V
V
.01
.02.03.04.05
.1
d/). EFFECTIVE
.2.3
.4.5
GAP LENGTH -
I
2
3
4
5
WAVELENGTH
FIG. 9.49. The response of a magnetic tape reproducing system as a
function of the ratio of the gap length to the wavelength.
The magnetic material used in the coating of the plastic tape is of necessity
nonlinear because it must pos!?ess retentivity to retain the magnetic signal
applied to the tape in recording. The characteristic which depicts the
magnetomotive force or magnetizing force H produced by the recording head
in the magnetic tape and the residual induction Br after the magnetic tape
leaves the head is depicted by the characteristic 1, 2, 0, 3, 4- of Fig. 9.50.
The nonlinear portion in the vicinity of 0 of Fig. 9.50 will produce distortion.
Various means have been developed for reducing the effects of this non­
linear characteristic. The system which is universally used in sound repro­
duction today is the alternating current bias. The high-frequency signal
50 to 150 kilocycles is added to the audio signal in the recording head, there
being no modulation of one signal by the other. The action of the addition
of the high-frequency bias in reducing the effect of the nonlinear charac­
teristic is shown in Fig. 9.50.
The system used in recording and reproducing from magnetic tape is
shown in Fig. 9.51. The tape transport mechanism consists of the takeup
and payoff reels and the capstan drive. Three heads are used, namely,
erasing, recording, and reproducing heads. The reproducing head is used,
in recording, to monitor the recording. In reproducing the erasing and
MISCELLANEOUS TRANSDUCERS
389
recording heads are immobilized. In recording, any previous signal is
removed by the erase head which is accomplished by feeding a high-fre­
quency signal of high intensity to the erase head.
When the recording process is carried out with constant current in the
Br
H
3/
0/
2/
{
S
1/
FIG. 9.50. The recording on magnetic tape WIth a high-frequency
bias. The characteristic 1, 2, 0, 3,4 is the residual induction Br
produced by the magnetizing force H produced by the recording
head. The high-frequency bias and audio signal are applied to
the tape. The resultant characteristic in reproduction is the
characteristic S- F.
DRIVE
CApSTAN
TAKE UP
REEL
FIG. 9.51.
system.
PAY OFF
REEL
The elements of a magnetic tape recording and reproducing
390
ACOUSTICAL ENGINEERING
head and the reproducing process is carried out with an amplifier in which
the response is independent of the frequency, the overall response will be
given by the characteristic of Fig. 9.49. Therefore, suitable compensation
must be provided in order to obtain an over-all uniform response frequency
characteristic. The upper frequency limit is determined by the first dip,
where the gap is equal to the wavelength. In recording, suitable high­
frequency accentuation is applied in the range above dj).. = .3 so that in
reproduction no additional compensation will be required in this frequency
range. In reproduction, the response is accentuated 6 db per octave with
decrease in frequency in the frequency range below dj).. = .3. In this way
a uniform recording-reproducing characteristic is obtained. The accentua­
tion of high-frequency response in recording increases the signal-to-noise
ratio. There is no overload problem in this frequency range due to accentua­
tion of response in recording, because the amplitude of speech and music is
lower in the high-frequency range compared to the mid- and low-frequency
ranges. In the frequency range below dj).. = .3 the amplitude of flux if> will
be constant for constant current in the recording head. Therefore, in order
to obviate overloading of the tape in recording, the low-frequency compensa­
tion must be supplied in the reproducing amplifier.
The standard tape speeds are as follows: 30, 15, 7~, 31, and 1~ inches per
second. The higher speeds are used for high quality recording. The lower
speeds are used for speech reproduction. The standard tape width is ! inch.
Both single- and double-track recordings are used on the i-inch tape.
The upper frequency limit of reproduction will depend upon the air gap
of the head and the tape speed. With heads in use today, and at tape
speeds of 30 and 15 inches per second an upper frequency limit of 20,000
cycles can be easily achieved. At a tape speed of 71 inches per second an
upper frequency limit of 15,000 cycles per second is possible in a well­
designed system. The upper frequency limit employing tape speeds of
31 and 1i inches is correspondingly lower.
9.8. Magnetic Tape Conversion Systems.-A. Frequency Conversion
System.-The fact that magnetic tape can be operated over a tremendous
range of speeds from a fraction of an inch per second to a thousand inches
per second provides a means for frequency conversion. The frequency
conversion may be either to a lower or a higher frequency. The use of a
magnetic tape system for providing frequency conversion is shown in
Fig. 9.52. The tape is recorded with a linear speed V and a maximum
frequency bandwidth of j. The system over which the signal is to be
transmitted is limited to a maximum frequency band of less than j. The
tape is reproduced at a slower speed Vi so that the maximum frequency
corresponds to the capabilities of the transmission system. The output
of the transmission system is fed to a recorder operating at a tape speed
of Vi and the signal is recorded. The record produced in this manner is
reproduced at a tape speed V. In this way the original signal is recovered.
The time of transmission is increased by the ratio of
~.
391
MISCELLANEOUS TRANSDUCERS
TAPE
HEAO
CAPSTAN
REEL
FIG. 9.52.
The elements and processes in a magnetic
tape frequency conversion system.
The preceding example illustrated a reduction in the frequency band­
width. The system may also be used to step up the frequency. For
example, if the transmission takes place on a transmission system with a
wider bandwidth, the frequency may be increased and the time of trans­
mission reduced.
B. Frequency Compression System.-The redundancy in speech is com­
paratively large. Therefore, in the transmission of speech another method
of reducing bandwidth, termed frequency compression, may be used. The
use of magnetic tape for a frequency compression system is shown in Fig.
9.53. The original signal is recorded at a tape speed V. Then this tape is
reproduced by means of a system of rotating heads which move in the same
direction as the tape at peripheral velocity of VI. The frequency is reduced
V - Vl
by the factor
V
However, sections of the tape must be left out.
The amount of the signal on the original tape that is used is V
~ VI.
The
392
ACOUSTICAL ENGINEERING
HEAD
CAPSTAN
ROTATING
HEADS
LOUDSPEAKER
9.53. The elements and processes in a magnetic
tape frequency compression system..
FIG.
output of the rotating heads is fed to a recorder operating with a tape speed
of V - VI. In reproduction, the system of rotating heads rotated in a
direction opposite to the motion of the tape with a peripheral velocity of
VI. The original frequencies are restored.
The advantage of the system resides in the fact that the amount of tape
required for the storage system is reduced and the frequency band required
for transmission is reduced.
In an alternative reproducing system the tape is reproduced at a velocity V
as shown in Fig. 9.54. The time required for the reproduction is now
V - VI
reduced by a factor
V
C. Time Compression System.-A time compression system similar to the
frequency compression system may be used for reducing the time of trans­
mISSIOn.
The use of a magnetic tape for time compression is shown in Fig. 9.55.
The original signal is recorded at a tape speed V. Then this tape is repro­
MISCELLANEOUS TRANSDUCERS
393
LOUDSPEAKER
9.54. The elements and processes in a magnetic
tape frequency compression system.
FIG.
duced at a tape speed V + VI by means of a system of rotating heads which
move in the same direction as the tape at a peripheral velocity VI. The
frequency output is the same as the original frequency. However. sections
of the original signal are left out. The amount of the signal on the original
tape that is used is V
~ Vl.
The output of the rotating heads is fed to
a recorder operating with a tape speed of V. The tape is reproduced at a
speed V on a conventional machine. The frequency is the same as the
original. Therefore. since sections of the signal are left out, the time
required for reproduction is V :
v-;: times that of the original.
9.9. Sound Motion Picture Recording and Reproducing System. 64
-A sound motion picture recording system consists of a means for producing
a modulated light beam and means for moving a light sensitive film relative
64 Kellogg, E. W., History of Motion Pictures, Jour. Soc. Mot. Pic. Tel. Eng., Vol. 64.
No.6, p. 291,1955; Vol. 64, No.7, p. 356, 1955; Vol. 64, No.8, p. 422,1955.
394
ACOUSTICAL ENGINEERING
MICROPHONE
LOUDSPEAKER
9.55. The elements and processes in a magnetic
tape time compression system.
FIG.
to the beam for recording signals derived from sound signals. A sound
motion picture reproducing system is a combination of a light source, optical
system, and photoelectric cell and a mechanism for moving a film carrying
an optical sound record by means of which the recorded variations may be
converted into electrical signals of approximately like form. The elements
of a complete motion picture recording and reproducing system are described
in Sec. 13.7. It is the purpose of this section to describe the film and sound
track, the modulators, and film transport used in the recording of sound on
photographic film and the optical-electronic reproducer and film transport
used in reproduction of sound from photographic film.
A. Film and Sound Track.-In the recording of sound motion pictures the
picture and sound are recorded on separate photographic films. See Sec.
13.7. Therefore, the camera and sound recorder must be synchronized.
This is accomplished by the use of an interlock system between the camera
and sound recorder and the use of perforated film in the form of sprocket
395
MISCELLANEOUS TRANSDUCERS
holes along the two edges of the film for both the camera and sound recorder.
Fig. 9.56.
The sound track on 35-millimeter film occupies a space about .1 inch
wide just inside the sprocket holes as shown in Fig. 9.56. There are two
types of sound track in general use today-namely, variable area and
variable density. The type of sound track shown in Fig. 9.56 is termed
bilateral variable area. There are also other types as, for example, unilateral,
duplex, class A pushpull, and class B pushpull variable area sound tracks.
SOUND
TRACK
0
SOUND
TRACK
ICYV'
0
0
0
SPROCKET
HOLE:S
PICTURE:
ARE:A
0
0
VARIABLE
AREA
0
0
0
tj 0
o
o
0
o
o
PICTURE:
AREA
Oc--_ _ _ _--'
VARIABLE
DENSITY
FIG. 9.56. The position of the picture and sound track in 35-millimeter sound
motion-picture film. Two types of sound track are shown-namely, variable area
and variable density.
In addition to the conventional variable density sound track shown in
Fig. 9.56, there are other types as, for example, squeeze, class A pushpull,
and class B pushpull variable density sound tracks.
B. Recording System.-l. Variable area. 65,66,67 ,68,69_In the variable
area system the transmitted light amplitude is a function of the amount of
unexposed area in the positive print. This type of sound track is produced
by means of a mirror galvanometer which varies the width of the light slit
under which the film passes. The elements of a variable area recording
system are shown in Fig. 9.57. The triangular aperture is uniformly
illuminated by means of a lamp and lens system. The image of the tri­
angular aperture is reflected by the galvanometer mirror focused on the
mechanical slit. The mechanical slit in turn is focused on the film. The
galvanometer mirror swings about an axis parallel to the plane of the paper.
The triangular light image on the mechanical slit moves up and down on
the mechanical slit. The result is that the width of the exposed portion
65 Albin, F., Clark, L. E., Hill, A. P., Hilhard, J. K., Kimball, Harry, Lambert,
Kenneth, and Miller, Wesley, "Motion Picture Sound Engineering," D. Van Nostrand
Company, Princeton, N.J., 1938.
66 Hardy, A. C., Trans. Soc. Mot. Pic. Eng., Vol. 11, No. 31, p. 475, 1927.
67 Dimmick, G. L., Jour. Soc. Mot. Pic. Eng., Vol. 15, No.4, p. 428, 1930.
68 Kellogg, E. W., Jour. Soc. Mot. Pic. Eng., Vol. 25, No.3, p. 203, 1935.
69 Sachtleben, L. T., Jour. Soc. Mot. Pic. Eng., Vol. 25, No.2, p. 175, 1935.
396
ACOUSTICAL ENGINEERING
of the negative sound track corresponds to the rotational vibrations of the
galvanometer. In the positive record the width of the unexposed portion
corresponds to the signal.
The amount of ground noise produced is proportional to the exposed
portion of the positive sound track. For this reason it is desirable to make
the unexposed portion 70 ,71,72 of the record just wide enough to accommodate
the modulation. This is accomplished by applying a bias signal to the
galvanometer. In the absence of a signal a very narrow exposed portion
MECHANICAL
GALVANOMETER
MIRROR
¥:-;;;.-
S~I:'J
r
.-*..-(Hl'i
LENS
$...~
I
,;rI'17"'0
OUTLINE or
TRIANGULAR
LIGHT
BEAM ON
NEGATIVE
MEC~ti~ICAL
SOUND
FILM
POSITIVE
TRIANGULAR
LAMP
APERTURE
RECORDING SYSTEM
MECHANICAL
SLIT
rLn}-m2
ARMATURE~ r..c.. z
MAGNET
OAMPER
o
n
T"
SOUND
MECHANICAL
NETWORK
1111
VIEW
SECtiONAL
TRACK
L
ELECTRICAL
CIRCUIT
200
PERSPECTIVE
TRACK
-j-.-jt
1000
FREQUENCY
I
10000
VIEW
FIG. 9.57. The elements of a variable area sound motion-picture film recording
system. The negative and positive sound tracks. Perspective and sectional views,
mechanical network, and electrical circuit of the galvanometer. In the mechanical
network, JM = the mechanical driving force. ml and C MI = the mass and com­
pliance of the armature. m2, rM, and C,1f2 = the mass, mechanical resistance, and
compliance of the damper. In the electrical circuit, ZEM = the electrical motional
impedance. L and rEI = the damped inductance and electrical resistance. YEG
= the electrical resistance of the generator. e = the voltage of the generator. The
graph depicts the amplitude response frequency characteristic of the galvanometer.
Dotted and solid lines depict the amplitude response for the galvanometer alone and
with an electrical capacitance in shunt with the galvanometer, respectively.
is produced on the negative record which means a correspondingly narrow
unexposed portion on the positive record. When a signal appears, the
triangular spot on the mechanical slit moves down just enough to accom­
modate the signal. The initial bias is accomplished within a millisecond.
However, the return to normal bias after a large signal followed by a small
70
71
72
Kreuzer, B., Jour. Soc. Mot. Pic. Eng., Vol. 16, No.6, p. 671, 1931.
Dimmick, G. L., Jour. Soc. Mot. Pic. Eng., Vol. 29, No.3, p. 258, 1937.
Kellogg, E. W., Jour. Soc. Mot. Pic. Eng., Vol. 36, No.2, p. 137, 1941.
MISCELLANEOUS TRANSDUCERS
397
signal is about 1 second. Faster return action produces thumping in the
reproduced record.
A film sound reproducing system is an amplitude system, that is, the
voltage output is proportional to the amplitude on the film. Thereiore,
in order to obtain a uniform response frequency characteristic, neglecting
the frequency discrimination due to finite recording and reproducing slits,
the amplitude of the galvanometer should be independent of the frequency.
Perspective and sectional views, the electrical circuit, and the mechanical
network of a film recording galvanometer are shown in Fig. 9.57. The
controlling element in the vibrating system in the low-frequency range is
the compliance CMl. Under this condition the ratio of the amplitude to
the applied force is independent of the frequency. A damper, m2, rM,
CM 2, reduces the amplitude in the region of the resonant frequency of
ml with CMl. The amplitude response frequency characteristic is shown
in Fig. 9.57. It will be seen that the rotational amplitude is uniform with
respect to frequency to about 10,000 cycles.
2. Variable Density.73-In the variable density system the transmitted
light amplitude is an inverse function of the amount of exposure in the
positive print. This type of sound track is produced by means of a light
valve which varies the amount of light which falls upon the moving film.
The elements of a variable density recording system are shown in Fig.
9.58. The ribbons of the light valve are illuminated by means of a lamp
and lens system. The image of the illuminated slit produced by the ribbons
of the light valve is focused on the film. The amount of exposure on the
negative film varies with the aperture at the ribbons. In the positive
record the amount of exposure is an inverse function of the input to the
light valve. Ground noise reduction 74 ,75 can also be obtained with a light
valve. In the absence of a signal the light valve is biased so that the aperture
between the ribbons is almost closed. When a signal appears the ribbons
open just enough to accommodate the signal. The action is similar to that
in the variable area system. The elements of a light valve and the electrical
and mechanical circuits are shown in Fig. 9.58. Below the resonant fre­
quency the controlling element in the mechanical circuit is the compliance
eM. Therefore, in this frequency range the ratio of the applied force and
the amplitude is independent of the frequency. At the resonant frequency
of the ribbons the response is accentuated. The amplitude response fre­
quency characteristic of a light valve is shown in Fig. 9.58.
3. Recording Film Transport. 76 ,77-The film transport mechanism used
in recording sound on film consists of a positive drive of the perforated film
and a constant speed drive of the film where the modulated light beam strikes
the film. A film transport mechanism of this type is shown in Fig. 9.59.
MacKensie, D., Trans. S.M.P.E., Vol. 12, No. 35, p. 730, 1928.
Silent and Frayne, Jour. Soc. Mot. Pic. Eng., Vol. 18, No.5, p. SSt, 1932.
75 Scoville and Bell, Jour. Soc. Mot. Pic. Eng., Vol. 38, No.2, p. 125, 1942.
76 Kellogg, E. W., Jour. Soc. Mot. Pic. Eng., Vol. 15, No.5, p. 653, 1930.
77 Collins, M. E., Jour. Soc. Mot. Pic. Eng., Vol. 48, No.2, p. 148, 1947.
73
74
398
ACOUSTICAL ENGINEERING
NEGATIVE
SOUND
TRACK
POSITIVE
SOUND
TRACK
LAMP-(j)
rEI
E:S
LENS
llt~::~
~
ELECTRICAL
CIRCUIT
m
r"
~
i"
C
30
m
.A
rM co2 5
C M <>
..,20
~
BRIDGE
15
.....V
0­
PEG
RIBBON
ASSEMBLY
_-,==,--",/,---FlLM
MT
MECHANICAL
CIRCUIT
VI
LENS
ASSEMBLY
::: 10
J
"
a:
o
200
RECORDING
L
SYSTEM
1000
FREQUENCY
10000
FIG. 9.58. The elements of a variable density sound motion-picture film
recording system. The negative and positive sound tracks. Sectional
and ribbon assembly views, mechanical circuit, and electrical circuit of the
light valve. In the mechanical circuit, 1M = the mechanical driving force.
m, YM, and eM = the mass, mechanical resistance, and compliance of the
ribbons. In the electrical circuit, ZEM = the motional electrical impedance.
Land YEL = the damped inductance and electrical resistance of the ribbons.
YEG = the electrical resistance of the generator.
e = the voltage of the
generator. The graph depicts the amplitude response frequency charac­
teristic of the light valve.
TO
PAYOFF
REEL
TO
TAKEUP
REEL
I
RECORDING DRUM
FIG. 9.59. Schematic view of the photographic film
transport mechanism of a motion-picture film sound
recorder.
r
399
MISCELLANEOUS TRANSDUCERS
Positive drive of the film is obtained by the sprocket drive. The sprocket
drive is interlocked with the camera drive so that synchronism of the picture
and sound will be obtained. See Sec. 13.7. When the film passes over the
sprocket drive, variations in the motion of the film at the sprocket hole
frequency are produced. These variations in the film speed musLbe removed
at the recording point to eliminate spurious frequency modulation of the
image on the film. Uniform speed at the recording point is provided by
the filter between the sprocket drive and recording point consisting of the
inertia of the recording drum and the compliance of the film between the
recording drum and the sprocket drive. The recording drum is driven by a
magnetic system from the motor which drives the sprocket which provides
a slight amount of drive to the film. This form of drive provides isolation
from variations in the rotational speed of the motor drive. The combination
of the isolating filter and magnetic drive provides a system with very uniform
motion at the surface of the drum. The image of the modulator is focused
on the film while it is in contact with the drum.
C. Reproducing System.-l. Optical-Electronic Reproducer.-The elements
of a motion picture film sound reproducing system are shown in Fig. 9.60.
The light source, in the form of an incandescent lamp, is focused upon a
mechanical slit by means of a condensing lens. The mechanical slit in turn
is focused on the negative film. The height of the image on the film is
i 1IIIIIImi
OUTPUT
II
'~~~''''II~ :~I
PHOTO-
SLIT
20
TRANSFORMER
REPRODUCING
SYSTEM
2
4
810~ 2
4
8 103 2
4
'104 2
FREQUENCY
9.60. The elements of a motion picture film sound reproducing system and
the voltage response frequency characteristic with a constant amplitude film.
FIG.
usually about .00075 inch. Under these conditions the amount of light
which impinges upon the photocell is proportional to the unexposed portion
of the sound track in variable area recording or to the inverse function of
the density in variable density recording. When the film is in motion the
light undulations which fall upon the photocell correspond to the voltage
variations applied to the recording galvanometer. The voltage output of the
photocell is proportional to the amount of light which falls upon the cathode.
The voltage output response frequency characteristic of a typical motion
picture film sound reproducing system using a constant amplitude film is
shown in Fig. 9.60. The falling off in response at the high-frequency portion
of the range is due to the finite dimensions of the slits in the recording and
reproducing systems. This reduction in response can be overcome by
compensations in the recording and reproducing systems.
400
ACOuSTICAL ENGINEERING
2. Reproducing Film Transport. 78 -The film transport used in reproducing
sound on photographic film consists of a positive drive of the perforated
film and a constant speed drive where the light passes through the film to
the photoelectric cell. A film transport mechanism of this type is shown in
Fig. 9.61. Positive drive of the film is obtained by means of the two sprocket
drives. The sprocket drives are geared with the positive picture drive so
that a constant loop of film is maintained between the sound head and the
picture head. The positive drive also insures that the film speed in repro­
duction will be same as that in
fROM
recording. Since there is a loose
PICTURE
PROJECTOR
loop of film between the picture
head and sound head, variations
ROLLER
in the picture drive will not be
imparted to the sound head.
After the film enters the sound
head it passes over a drum. The
light beam of the reproducing
system passes through the film
to the photocell located inside the
drum while the film is on the
drum. Since the drum is rotated
at a constant speed, the film will
move past the light beam at a
constant speed. The drum is
FIG. 9.61.
Schematic view of the photo­
driven by the first sprocket drive.
graphic film transport mechanism of a
The second sprocket isolates the
motion picture film sound reproducer.
takeup reel from the reproducing
system. Uniform speed at the
reproducing point is provided by the filter between the sprocket drive
and the reproducing point consisting of the inertia of the drum and
the compliance of the film between the drum and sprocket drive. To
provide damping and stabilization, the drum drives a flywheel through a
fluid coupling. This is termed a rotary stabilizer. The combination of
the filter and the rotary stabilizer provides a system with very uniform
motion at the surface of the drum.
9.10. Motion Picture Magnetic Tape Sound Recording and Repro­
ducing System. 79-During the past few years the original recording of
the sound directly upon motion picture film has to a large extent been
replaced by recording on magnetic tape. The magnetic system for recording
on magnetic tape is the same as that described in Sec. 9.7. However, in
order to maintain synchronism between the picture recorded by the camera
and the sound recorded on the magnetic tape, perforated film or tape is used
in the sound recorder.
Loomis and Reynolds, Jour. Soc. Mot. Pic. Eng., Vol. 25, No.5, p. 449, 1935.
Frayne and Wolfe, "Elements of Sound Recording," John Wiley and Sons, New
York, N.Y., 1949.
78
79
r
401
MISCELLANEOUS TRANSDUCERS
During the past three years, wide screen motion picture systems with
stereophonic sound have been introduced on a wide scale. The information
is rerecorded on the magnetic strips cemented to the positive picture release
print. The magnetic system for reproducing the magnetic strips is the same
as that described in Sec. 9.7. It is the purpose of this section to describe
the magnetic tape, the tape transport used in recording and the tape trans­
port used in reproducing.
A. Magnetic Tape.-The camera and sound recorder must be interlocked
as in the case of the recording on photographic film. Therefore, a magnetic
coating on a perforated plastic base is used as the recording medium. The
dimensions are the same as those of 35-millimeter photographic film.
Fig. 9.62A. One reason for the use of magnetic tape of these dimensions is
SPROCKET
HOLE
SPROCKET
HOLE
\
-...J
0
~
0
~
........
0
0
0
0
0
0
0
0
0
0
0
0
0
0
In
lr->
A
MAGNETIC
COATING
B
PICTURE
AREA
0
0
~:AGNETIC
TRACK/
.030 INCH WIDE
MAGNETIC TRACKS
.050 INCH WIDE
9.62. A. Magnetic tape for original sound recording in sound motion
pictures. B. Magnetic strips on a motion-picture positive film.
FIG.
that if desired the existing photographic recorder may be used by the addi­
tion of magnetic heads. Special magnetic recorders have also been de­
veloped, but these also use magnetic tape of the same dimensions. Another
reason for the use of the wide tape is that several tracks representing several
channels may be recorded on the tape shown in Fig. 9.62A because the total
width of the magnetic coating is approximately an inch. For example, a
soloist may be recorded on one track and the sections of the orchestra may be
recorded on other tracks. The three or more channels of stereophonic sound
may be recorded on the magnetic tape. See Sec. 13.7.
The information recorded on the magnetic tape is rerecorded on the
photographic film by means of the systems described in Sec. 9.8.
During the past three years wide screen motion picture systems 80 with
stereophonic sound have been introduced on a wide scale. The elements
80
Sponable, Brazz, and Grignon, Jour. Soc. Mot. Pic. Tel. Eng., Vol. 63, No. 1, p. I,
1954.
402
ACOUSTICAL ENGINEERING
of the system are described in Sec. 13.7. The three or four channels are
recorded on the tape shown in Fig. 9.62A. The information is rerecorded
on the magnetic strips cemented to the positive picture release print as
shown in Fig. 9.62B.
B. Recording Tape Transport.8l.82-The tape transport used in recording
sound on magnetic tape for sound motion pictures consists of a positive
sprocket drive of the perforated film and a constant speed drive where the
magnetic recording head is in contact with the tape. A magnetic tape
transport mechanism of this type is shown in Fig. 9.63. Positive drive of
the tape is obtained by means of the sprocket drive. The sprocket drive is
interlocked with the camera drive so that synchronism of the picture and
TO
PAYOFF
REEL
RECORD
HEAD
MAGNETIC
TAPE
TA~~UP
MONITOR
HEAD
REEL
FIG. 9.63. Schematic view of the magnetic tape transport
mechanism of a magnetic sound recorder for sound motion
pictures.
sound will be obtained. See Sec. 13.7. When the tape passes over the
sprocket drive, variations in the motion of the film at the sprocket hole
frequency are produced. These variations in the film speed must be removed
at the recording point to eliminate spurious frequency modulation of the
magnetic recording on the magnetic tape. Uniform speed at the recording
point is provided by the filter between the sprocket drive and the recording
point consisting of the inertia elements of the two rollers and the two drums
and the compliance of the tape between the sprocket drive and the rollers
and the drums. The drums are equipped with flywheels to provide additional
inertia. Damping of the inertia and compliance system is provided by the
mechanical resistance of the dash pot. The spring system maintains a tight
loop for the magnetic tape. This design of mechanical system provides
uniform motion at the magnetic heads. Two magnetic heads are provided,
one for recording and the other for monitoring.
The film recorder of Fig. 9.59 may also be used for recording one track of
magnetic tape by placing a magnetic recording head inside the drum in
81
82
Hittle, C. E., Jour . Soc. Mot. Pic. Eng., Vol. 58, No.4. p. 323. 1952.
Davis and Manley. Jour. Soc. Mot. Pic. Tel. Eng.• Vol. 62. No.3. p. 208. 1954.
MISCELLANEOUS TRANSDUCERS
403
contact with overhanging portion of the tape on the drum. The magnetic
head is shown dotted in Fig. 9.59.
C. Reproducing Tape Transport. 83-The information on the tape in the
magnetic system of sound reproduction is rerecorded on the magnetic strips
of the positive film. Fig. 9.62B. The transport mechanism used in repro­
ducing sound recorded on magnetic strips on positive film consists of a
positive drive of the perforated film and a constant speed drive at the point
of contact of the magnetic heads.
FROM
Fig.9.64 Positive drive of the film is
PROJECTOR
obtained by means of two sprocket
drives. These two sprocket drives
are geared to the positive picture
drive so that constant loop of film is
maintained between the sound head
and the picture head. The positive
drive also insures that the film
speed in reproduction will be the
same as in recording. The inertia
drum coupled to a flywheel provides
the inertia element of the filter
system. The film loop tension rollers
TO
TAKEUP
provides the compliance elements in
REEL
the filter system. Damping of the FIG. 9.64. Schematic view of the film
loop tension rollers provides the transport mechanism of a magnetic re­
mechanical resistance element. The producer for sound motion pictures.
filter system removes the variations
introduced by the sprocket drive and provides uniform motion of the film at
the magnetic heads.
.
9.11. Volume Limiters, Compressors, and Expanders. 84 ,85,86,87­
A volume compressor is a system that reduces the amplification of an
amplifier when the signal being amplified is large and increases the amplifi­
cation when the signal is small. Compressors are used to reduce the volume
range in sound motion picture and phonograph recording, sound broad­
casting, public address, and sound reinforcing systems, etc.
A volume expander is a system that increases the amplification of an
amplifier when the signal is large and decreases the amplification when the
signal is small. In reproduction, a volume expander is used to counteract
the effect of the compressor in recording.
Volume compressors and expanders are amplifiers in which the amplifi­
cation varies as a function of the general level of the signal. The elements
of a compressor, limiter, or expander are shown in Fig. 9.6SA. The input
Phyffe and Hittle, Jour. Soc. Mot. Pic. Eng., Vol. 62, No.3, p. 215, 1954.
Sinnett, C. M., Electronics, Vol. 8, No. 11, p. 14, 1935.
85 Norman, N. C., Bell Labs. Record, Vol. 13, No.4, p. 98, 1934.
86 Mathes and Wright, Bell Syst. Tech. Jour., Vol. 13, No.3, p. 315, 1934.
87 Steinberg, J. C., Jour. Acous. Soc. Amer., Vol. 13, No.2, p. 107, 1941.
83
84
404
ACOUSTICAL ENGINEERING
signal is amplified and rectified. The rectified signal is applied to a resist­
ance condenser network. The d-c voltage across the condenser is used to
vary the bias and, as a consequence, the amplification of a push pull amplifier
employing tubes with variable transconductance. The constants of the
system can be adjusted to obtain limitation, compression or expansion.
In the limiter characteristic, shown in Fig. 9.6SA, the relation between
the output and input is linear up to a certain level, beyond this point the
output remains constant regardless of the input. The limiter type is useful
for protection against a sudden overload, as, for example, in the input to a
broadcast transmitter.
In the case of the compressor characteristic, shown in Fig. 9.6SA, there is
a gradual reduction in the gain with increase of the input. A reduction
i[2Jil2]
INPUT IN DB
LIMITER
INPUT IN DB
COMPRESSOR
A
il21
INPUT IN DB
EXPANDER
B
FIG. 9.65.
Input vs. output characteristics of limiters. compressors. and expanders.
A. The elements of a limiter or a compressor. B. The elements of an expander.
in the volume range in radio and phonograph reproduction makes it possible
to reproduce the wide range of orchestra music in the home without excessive
top levels. It also improves the signal-to-noise ratio. It improves the
intelligibility of speech and enhances music reproduction when the ambient
noise is high, as, for example, in sound motion theater reproduction.
In the case of the expander characteristic, shown in Fig.9.65B, there is
a gradual increase in gain with increase in output. The combination of a
compressor and expander may be used to improve the signal-to-noise ratio
in sound reproduction.
The attack time for a gain reduction of 10 db, in compressors and limiters,
is of the order of a millisecond. The retreat to normal is of the order of 1
second.
9.12. Synthetic Reverberation.-The reverberation time of studios may
be changed and controlled within certain limits by varying the absorption.
The amount of control that may be obtained by varying the amount of
absorption by means of hard panels which cover the absorbing material or
r
MISCELLANEOUS TRANSDUCERS
405
other similar systems is limited. Furthermore, the reproducing conditions
may also require additional reverberation. Where the reverberation time
of reproduced sound is far below the optimum value, the reproduction may
be enhanced by artificially adding reverberation.
Artificial reverberation may be added to a sound signal by means of the
loudspeaker, reverberant chamber, and microphone combination shown in
Fig. 9.66A. The reverberant chamber consists of an enclosure with highly
reflecting, nonparallel walls, ceiling, and fioor. The cubical content varies
from 1000 to 10,000 cubic feet. If a reduction in the reverberation time is
desired, fiats of absorbing material may be brought into the chamber. A
modification of the single-chamber system is the addition of a second
chamber coupled to the first by means of a door. The use of two rooms
makes it possible to obtain a wide variety of reverberant effects by varying
TRANSDUCERS WITH PROGRESSIVE
DELAY AND ATTENUATION
FIG. 9.66.
Schematic arrangement of systems for introducing synthetic
reverberation in reproduced sound. A. Loudspeaker, microphone, and
reverberant chamber combination. B. A system consisting of transducers
with progressive delay and attenuation.
both the reverberation time of the chambers and the coupling between the
chambers. The loudspeakers, microphones, and amplifiers used for these
systems should be of the highest quality. Mixers are provided so that any
ratio of the original sound to reverberant sound may be obtained.
Reverberation, in the chamber described above, consists of the multiple
reflection of a large number of pencils of sound. Each pencil of sound
suffers a decrease in intensity with each reflection. These conditions can be
simulated by the system shown in Fig. 9.66B. The amplified sound signal
is passed through a number of transducers with progressive delay and attenu­
ation. These transducers may be a series of pipes with loudspeakers and
microphones terminating the ends. The transducer may be a recorder and
a series of pickups on a phonograph record or magnetic tape 88 or phosphor
wheel. 89 The reverberation time may be varied by varying the progressive
88
89
Wolf, S. K.. Proc. I.R.E., Vol. 27. No.7, p. 365. 1939.
Goldmark and Hendricks. Proc. I.R.E .• Vol. 27. No. 12. p. 747. 1939.
406
ACOUSTICAL ENGINEERING
attenuation. Mixers are provided so that any ratio of the original sound
to the reverberant sound may be obtained.
9.13. Hearing Aids.-Test made upon representative cross-sections
of the people in this country show a very large percentage to be hard of
hearing. Practically all of these people may obtain satisfaction from the
use of a hearing aid. A hearing aid is a complete reproducing system which
increases the sound pressure over that normally received by the ear.
The first and simplest hearing aid 90 consisted of a carbon microphone, a
battery, an attenuator, and a telephone receiver (Fig. 9.67A). This hearing
aid gave satisfactory service where the hearing loss was about 20 db.
The hearing aid shown in Fig. 9.67B consisted of a carbon microphone, a
carbon amplifier, an attenuator, a battery, and a telephone receiver. This
hearing aid gave satisfactory service where the hearing loss was about 40 db.
CARBON
I.UCROPHONE ATTENUATOR
01.,
BATTERY
A
I
CARBON
MICROPHONE
ATTENUATOR
CARBON \
AMPLIFIER,I
""~m~
B
MICROPHONE
,------,
AMPLIFIER
WITH VOLUME AND
TONE CONTROL
RECEIVER
c
FIG. 9.67.
Hearing aids. A. Simple carbon microphone hearing aid. B. Carbon
microphone hearing aid with a mechanical carbon amplifier. C. Schematic diagram
of a vacuum tube or transistor hearing aid.
The quality of the carbon type hearing aids was usually very poor, due to
the frequency and the amplitude distortion produced by the carbon micro­
phone and amplifier.
During the past two decades and prior to the advent of the transistor,
hearing aids 91 ,92,93,94,95,96,97,98 employing vacuum tube amplifiers almost
completely replaced the carbon types. This has been due to the develop­
ment of small low-current drain vacuum tubes and small high-efficiency
batteries. The quality is far superior to that of the carbon type. Further­
more, suitable compensation circuits may be introduced to complement the
ear characteristics. The schematic arrangement of the components of a
vacuum tube hearing aid is shown in Fig. 9.67C. The microphone used in
hearing aids today is a diaphragm crystal or ceramic type similar to that
described in Sec. 8.2C2 and depicted in Fig. 8.9.
TufInell, W. L., Bell Labs. Record, Vol. 18, No. 1. p . 8, 1939.
Ramanow, F. F., Jour. Acous. Soc. Amer., Vol. 13, No.3, p. 295, 1942.
92 Sabine, P. E., Jour. Acous. Soc. Amer., Vol. 16, No. 1. p. 38, 1944.
93 Carlisle and Mundel, Jour. Acous. Soc. Amer., Vol. 16, No. 1, p. 45,1944.
94 Grossman and Molloy, Jour. Acous. Soc. Amer., Vol. 16, No. 1. p. 52, 1944.
95 Hanson, W. W., Jour. Acous. Soc. Amer., Vol. 16, No. 1, p. 60, 1944.
98 LeBel, C. j., Jour. Acous. Soc. Amer., Vol. 16, No. 1. p. 63,1944.
97 Watson, N. A., Jour. Acous. Soc. Amer., Vol. 16, No.3, p. 194, 1945.
98 Strommen, E., Jour. Acous. Soc. Amer., Vol. 15, No.4, p. 211, 1944.
90
91
r
407
MISCELLANEOUS TRANSDUCERS
Within the past three years, hearing aid amplifiers employing transistors 99
have been developed and commercialized on a wide scale. The advantage of
transistors over vacuum tubes is that only one low-voltage battery is required
as contrasted to separate filament and plate batteries for vacuum tube
systems. The total power required from the battery in a transistor amplifier
for hearing aids is a small fraction of the battery power required for a vacuum
tube amplifier for hearing aids. As a result the cost of operating a transistor
hearing aid is very small compared to a vacuum tube hearing aid. Further­
more, the weight and size of transistor amplifiers with the battery are very
much less than those of a corresponding vacuum tube amplifier and battery
c::::=s
rEG
L
ELECTRICAL
EAR
rEI
'''~
C M•
C M2
CROSS-SECTIONAL
NETWORK
25
<tI
020
DIAPHRAGM
m,rMI
MECHANICAL
CIRCUIT
.......
~ 15
COIL
MAGNET
eM'
z
:r
'"
~
10
5
o
100
1000
fREQUENCY
10000
VIEW
FIG. 9.68. Cross-sectional view, mechanical network, electrical circuit,
and response frequency characteristic of an insert-type telephone receiver.
In the mechanical network, iM = the mechanical driving force. mo, rMO,
and CMO = the mass, mechanical resistance, and compliance of the
diaphragm. C M1 = the compliance due to the air in the case. C Y2 = the
compliance of the air space between the diaphragm and the cover. m1 and
1'M1 = the mass and mechanical resistance of the tube.
C Ma = the com­
pliance of the ear cavity. In the electrical circuit, ZEM = the electrical
motional impedance. Land rE1 = the damped inductance and electrical
resistance of the coil. rEO = the electrical resistance of the coils. e = the
voltage of the electrical generator. The graph depicts the pressure
response frequency characteristic.
complement. Since the input electrical impedance of a transistor is of the
order of a few hundred ohms, magnetic microphones are used in transistor­
type hearing aids.
Two types of receivers are used-namely, the air conduction type and
the bone conduction type.
A cross-sectional view, the electrical circuit, the mechanical network,
and response frequency characteristic of the air conduction insert type
telephone receiver is shown in Fig. 9.68. A molded plug fits the ear cavity
and holds the receiver in place. Under these conditions the leak at the ear
99 Webster, Stanley K., Trans. I.R.E., Prof. Group Audio, Vol. AU-2, No.2, p. 65,
1954.
408
ACOUSTICAL ENGINEERING
is very small. Therefore, good response is obtained at the low frequencies.
The action of the system is essentially the same as that of a bipolar telephone
receiver considered in Sec. 9.2A and need not be repeated here.
In certain types of deafness, the middle ear, which consists of a series of
bones that conduct sound to the inner ear, is damaged while the inner ear
which consists of nerves, is normal (see Sec. 12.2). Under these conditions,
sound may be transmitted through the bones of the head to the inner ear
by means of a bone conduction receiver.lOO,lOl Usually the face of the bone
conduction receiver is placed against the mastoid bone behind the ear. A
cross-sectional view, the electrical circuit, mechanical network, and response
frequency characteristic of a bone conduction receiver is shown in Fig. 9.69.
~c~
~
ELECTRICAL
m,
~---1§-r.-"2~~
;M
MECHANICAL
CIRCUIT
NE TWORK
30
'o"
20
~
10
'\
Z
lr
0
'"
~ -10
SECTIONAL
VIEW
-20
V
100
-
./
Vr-...
\
1\
1000
fREQUENCY
10000
FIG. 9.69.
Cross-sectional view, mechanical network, electrical circuit,
and response frequency characteristic of a bone conduction receiver.
In the mechanical network, mo = the mass of the coil and magnetic
structure. ml = the mass of the armature.
C MI and rMl = the com­
pliance and the mechanical resistance connecting the coil and magnetic
structure to the case. CM2 and l' At2 = the compliance and mechanical
resistance connecting the armature to the case. m2 = the mass of the case.
ZME = the mechanical impedance of the mastoid bone. fM = the mechan­
ical driving force. In the electrical circuit, ZEM = the electrical motional
impedance. L and rEI = the damped inductance and electrical resistance.
rEG = the electrical resistance of the generator.
e = the voltage of the
electrical generator. The graph depicts the force developed on an artificial
mastoid.
By means of the multiple resonant system it is possible to deliver a large
force to the relatively high mechanical impedance, ZME, of the mastoid
bone. The response frequency characteristic is quite good considering the
difficult conditions under which the bone conduction receiver operates.
A modification of the insert hearing aid telephone receiver is shown in
Fig. 9.70. A small plastic tube connects an ear insert earpiece and the
telephone receiver. There is some attenuation and frequency discrimina­
100
101
1953.
Hawley, M. S., Bell Labs. Record, Vol. 18, No. 1. p. 12, 1939.
Hector, Pearson, Dean, and Carlisle, Jour. Acous. Soc. Amer., Vol. 25, No.6, p. 1195,
r
i
MISCELLANEOUS TRANSDUCERS
409
tion of sound transmitted by the small plastic tube. See Sec. 5.32. How­
ever, this loss can be overcome by compensation in the amplifier. The
advantage of the system of Fig. 9.70 is that it is somewhat less obtrusive
than the conventional system in that the earpiece is smaller and the telephone
receiver can be placed under the shirt or dress.
fi
=:;,,, "'"
RECEIVER
FIG. 9.70. Insert-type hear­
ing aid telephone receiver
with tube transmission line.
9.14. Sirens.l 02-The simplest siren consists of a revolving disk per­
forated with a ring of equally spaced holes which interrupt a jet of air from
a tube placed close to one side of the disk. The fundamental frequency of
the successive puffs of air issuing through the holes is equal to the product
of the number of holes and the revolutions per second of the disk. The
wave form, of course, depends upon the shape of the holes in the disk and
the shape of the projection of the air tube upon the disk. The pressure of
the air supply in large sirens is usually very high, of the order of 100 pounds
per square inch. In the smaller sirens the air pressure is supplied by a single­
stage centrifugal pump and the supply pressure is of the order of a pound
per square inch. Small sirens are used by police cars, ambulances, and fire
engines for signalling the approach of these vehicles. Large power sirens
are used on firehouses, lighthouses, and lightships.
A high-power siren103 has been developed in which the blower is driven
by a 95 horsepower automobile engine. The air stream represents about
38 kilowatts. The flow of air is interrupted by a rotary valve at a rate of
440 cycles per second and then passes into a horn. The use of a horn pro­
vides a certain amount of directionality and contributes to the high effi­
ciency of the siren. The sound output from the horn is about 25 kilowatts
in the fundamental.
9.15. Seismic Detectors.10L-The variation of the velocity of sound in
the various strata comprising the earth's crust forms the basis of geophysical
Wood, "A Textbook of Sound," The Macmillan Company, New York, N.Y.• 1930.
Jones, Clark. Jour. Acous. Soc. Amer., Vol. 18. No.2, p. 371. 1946.
104 Silverman, Daniel.. Jour. A.I.E.E., Vol. 58, No. 11. 455,1939.
102
103
410
ACOUSTICAL ENGINEERING
investigations in prospecting for oil. The detonation of a charge of dynamite
creates an acoustic wave which is reflected from the various strata of the
earth's surface. These reflected waves are picked up by microphones
connected to recording oscillographs and located in strategic positions on
the earth's surface. From the geometrical configuration of the apparatus, the
oscillograph record, and the velocity of sound in various types of strata, the
conformation of the various strata may be determined. Oil pools are located
in curved strata termed by geologists as anticlines. The presence of anti­
clines may be determined from seismic measurements.
Magnetic, carbon, crystal, condenser, and dynamic microphones have
been used for detectors. The large amplitude frequency components of
seismic waves are usually confined to the lower frequencies. Therefore,
the response of the microphone is confined to the range below 100 cycles.
For these applications a magnetic microphone has been found to be very
satisfactory. The armature is usually made massive and the stiffness small
in order to obtain high sensitivity in the low-frequency range. The micro­
phone is placed directly upon the earth's surface. The microphone proper
then vibrates with the earth's surface. The massive armature opposes
any change from its position of rest. As a consequence, there is relative
motion between the armature and the microphone proper which results in
the production of a voltage corresponding to the vibrations of the earth's
surface. By suitable orientation, the microphone can be made responsive
to only vertical vibrations. As a consequence, the wave transmitted directly
through the earth is not reproduced.
9.16. Stethoscopes.105,106,107-The ordinary acoustical stethoscope is
one of the most useful instruments which the physician uses in mediate
auscultation. By means of the stethoscope the physician is able to study
sounds produced within the heart, lungs, stomach, intestines, or other
portions of the body, and to determine whether normal or abnormal conditions
exist as indicated by normal or abnormal sounds. The most important
sounds are normal heart sounds, heart murmurs, breathing sounds, respira­
tory rales or rattles, and peristaltic squeaks or groans. Obviously, it is the
structure of the sound, which involves the intensity, the fundamental fre­
quency, the harmonic components, the duration, and the growth and decay,
that makes it possible to diagnose normal or abnormal conditions by ausculta­
tion.
Since diagnosis is based on the structure of the sounds, it is very important
that the stethoscope should not distort the sound by discrimination against
certain frequency bands or by attenuation of the sound. The sounds of the
body range from about 40 cycles to 4000 cycles. Fig. 9.71 shows the fre­
quency bands of some of the most common sounds. The fundamentals are
shown as dark areas and the harmonics or overtones as cross-hatched areas.
The fundamental of the systolic sound ranges from 40 to 80 cycles. There
105
106
107
Rappaport and Sprague, Amer. Heart Jour., Vol. 21, p. 257, 1941.
Frederick and Dodge, Bell Syst. Tech. Jour., Vol. 3, No.4, p. 531. 1924.
Singer, C., Electronics, Vol. 38, No.6, p. 66,1939.
MISCELLANEOUS TRANSDUCERS
411
are some lower components but from the standpoint of ear characteristic
these are very weak (see Sec. 12.6). The overtones are scattered over the
remainder of the frequency band up to 4000 cycles and above. Above
4000 cycles most of the sounds in the body are so weak that they are masked
by the ambient random noises generated in the body. The fundamental
diastolic sounds range from 60 to 100 cycles. The overtones are scattered
over the remainder of the frequency band up to 4000 cycles. The funda­
mental sounds of systolic and diastolic murmurs range from 300 to 800 cycles.
MUSICAL SCALE
M'gOLE
Cz
SYSTOLIC
DIASTOLIC
STOLIC
C'
SOUNDS
SOUNDS
MURMURS
PRESTOLIC
MURMLIRS
PERISTOLIC SOUNDS
RESPI RATORY SOUNDS
RESPIRATORY SOUNDSlRANDOMj
40
100
200
400
1000
2000
4000
FREQUENCY IN CYCLES PER SECOND
FIG. 9.71. The frequency ranges of the sounds
generated in the body. The frequency ranges
of the fundamental frequencies are shown as
solid lines. The frequency ranges of the
harmonics and overtones are shown as cross­
hatched lines.
The overtones in certain cases can be observed up to 2000 or 3000 cycles.
Prestolic murmurs usually range from 60 to 200 cycles. The overtones
range up to about 1000 cycles. Above this frequency the overtones are
masked by the body sounds. The fundamentals of peristaltic sounds have
a tremendous range in both frequency and intensity. Fundamentals up
to 2000 cycles are quite common. The overtones in the case of very intense
sounds extend beyond 4000 cycles. The fundamental frequency of respira­
tory squeaks, rales, crackles, and groans ranges from 60 cycles to 1000 cycles.
Respiratory sounds such as wheezes and the rushing of air are of a random
nature and do not possess a true fundamental. The components of these
sounds are scattered over the entire audible spectrum.
From Fig. 9.71 it is quite evident that, in order to obtain the maximum
intelligence from the stethoscope all frequencies over the range from 40 to
4000 cycles should be transmitted without attenuation or discrimination.
412
ACOUSTICAL ENGINEERING
Most acoustical and mechanical vibrating systems introduce distortion in
the form of discrimination against certain frequency bands. Extreme
distortion may alter the sound beyond recognition.
The two most common stethoscopes in use today are the open bell and
diaphragm types shown in Fig. 9.72A and B. The response frequency
characteristic of the open bell type is smoother and covers a wider frequency
range than the diaphragm type. However, the tuned diaphragm type
BINAURAL
TAPERED
TUBES
c
9.72. Sectional views of stethoscopes.
type. C. Wide range selective type.
FIG.
A. Diaphragm type.
B. Open bell
delivers greater output in the frequency range from 250 to 1500 cycles. The
open bell has better low-frequency response but the general output level in
the mid-frequency range is lower than the diaphragm type.
There are two reasons for the use of a diaphragm instead of an open
bell-namely, to exclude or attenuate external noises, and to eliminate
leakage between the body and the stethoscope. The open bell stethoscope
actually amplifies air-borne noises in the manner of the ear trumpet. If
the effective slit between the body and the bell of the open bell stethoscope
is just a small fraction of a thousandth of an inch, the low-frequency response
is attenuated due to this leakage. If the bell is pressed against the body so
this leak is effectively eliminated, the body stiffness represented in the
acoustic impedance of the body is increased with a resultant attenuation of
low frequencies.
MISCELLANEOUS TRANSDUCERS
413
In the existing diaphragm type stethoscopes the investigators have
found that it is necessary to use a resonant diaphragm in order to obtain
good output They have placed these resonances in the mid-frequency
range where the ear is quite sensitive. As a consequence the stiffness of
the diaphragm is quite high and the result is very high attenuation of the
low-frequency response.
A wide-range acoustical stethoscope108 ,109 is shown in Fig. 9.72C. The
chest piece of radical design consists of a light polythene diaphragm sup­
ported by a multipyramid resilient back plate. This structure provides
an efficient coupling means to the high acoustical impedance of the body.
The adequate resilience of the chest piece insures uniform response to low
tones. The light-weight diaphragm coupled directly to the body makes it
possible to obtain output beyond 4000 cycles. The acoustical impedance
of the chest piece is matched to the acoustical impedance of the tube or
line at the input end. The relatively high acoustical impedance at the
input end of the line is matched to the relatively low acoustical impedance
of the ear by the use of a tapered tube or line. The sensitivities in the low­
and high-frequency ranges are much greater than those of existing stetho­
scopes due to the matching of acoustical impedances. The high-frequency
response is maintained to 4000 cycles while most existing stethoscopes cut
off at 1500 cycles. There are certain instances in which the entire frequency
range is not desired. This is particularly true when the particular sounds in
question are confined to the low-, high-, or mid-frequency range. For
example, in listening to high-frequency prestolic murmurs, peristaltic, and
respiratory sounds, it may be desirable to eliminate the low frequencies.
In other instances, it may be desirable to attenuate the high-frequency
range. Therefore, to increase the usefulness of the stethoscope, an acoustical
filter has been added in which it is possible to attenuate either the low- or
high-frequency ranges, or both. The acoustical filter provides a system in
which frequency discrimination may be introduced at will, and thereby
increases the usefulness of the stethoscope by classification of the charac­
teristic sounds in the body into frequency bands.
The electrical stethoscope consists of the combination of a microphone,
amplifier, and telephone receivers. In one type the pickup device consists
of a bell-shaped horn, coupled to the microphone diaphragm. The coupling
system is similar to that of Fig. 8.1. Condenser, magnetic, and crystal
type transducers have been used in the microphone for these applications.
The amplifier is equipped with low- and high-frequency tone controls for
attenuating the response in either or both the high- and low-frequency
ranges. The addition of a recording system similar to the electrocardio­
graph may be used to obtain an oscillographic record depicting the sounds in
the body. Since the output of the electrical stethoscope is greater than that
of the acoustical stethoscope, noises generated by the clothing, movement
of the headpiece, etc., cause considerably more interference than in the
108
109
Olson, H. F., Electronics, Vol. 16, No.8, p. 184. 1943.
Olson, H. F., U. S. Patents 2,363,686 and 2,389,868.
414
ACOUSTICAL ENGINEERING
acoustical stethoscope. This is due to the fact that most of these noises in
the acoustical stethoscope fall below the threshold of hearing.
9.17. Ear Defenders. llO-Ear defender is a term used to designate a
device which introduces attenuation of sound between a point outside the
head and the eardrum. There are two types-namely, the cushion type
and the insert type. The cushion type is similar to a pair of headphones
with soft cushion ear pads. The cushion type is heavy, cumbersome, and
uncomfortable and for that reason it has not been used to any appreciable
extent. The insert type is some form of plug which is pushed into the ear
~
PLASTIC
WA~PL:~~IC
A
.-;" •• ..f..
~',~
·~;":i~r.~~:;~~?}
B
SECTIONAL
ACOUSTICAL
VIEWS
NETWORK
FIG. 9.73. A. Sectional view of an ear defender in the ear canal and the
acoustical network of the system. In the acoustical network, M 1 = the
inertance due to the mass of the ear defender. CAl and r Al = the effective
acoustical capacitance and acoustical resistance of the ear defender with
respect to the wall of the ear canal. M 2 and r A2 = the inertance and
acoustical resistance of the leak between the ear defender and the wall of
the ear canal. C A2 = the acoustical capacitance of the entrapped volume
of the ear canal. PI = the sound pressure outside the ear. P2 = the
sound pressure in the ear canal. The separate sectional views show two
different designs of ear defenders with one and two sealing flanges.
respectively. B. Wax filled ear defender.
canal. One form, which was used extensively a number of years ago,
consisted of a wad of cotton. The attenuation of a wad of cotton decreases
with decrease of the frequency. The attenuation below 500 cycles is quite
small. In order to obtain high attenuation at the low frequencies, the seal
between the defender and the ear canal must be practically airtight, because
a very minute hole will reduce the attenuation to a negligible amount. This
fact can be deduced from a consideration of the acoustic network of the ear
defender of Fig. 9.73A. A successful insert type of ear defender must be
made of suitable material combined with a shape which will provide
adequate attenuation, comfort, easy insertion, and positive retention.
110
Watson and Knudsen. Jour. Acous Soc. Amer.• Vol. 15, No.3. p. 153. 1944.
MISCELLANEOUS TRANSDUCERS
415
Ear defenders have been developed which satisfy the above requirements.
The most successful ear defenders have been made of synthetic rubbers or
soft plastics, because these materials remain resilient over long periods of
time and are resistant to ear wax. The shape which appears to be most
successful is a skirt closed at the top and equipped with one or more thin
flounces which rest against the ear canal and thereby provide the seal
(Fig.9.73A.). A tab, fastened at the bottom of the skirt, is used for inserting
or removing the defender. A good ear defender will introduce an attenua­
tion of between 30 to 35 db over the frequency range from 60 to 8000 cycles.
Another design ll1 of the ear defender employs a plastic case with soft
elastic walls and a viscous core of malleable wax. Fig. 9.73B. This design
is more comfortable because the body heat softens the wax with the result
that the defender corresponds to the ear canal in which it is placed without
distortion of the ear canal. Such distortion leads to discomfort.
9.18. Electronic Sound and Vibration Reducers and Absorbers.­
Existing systems for the absorption of sound and the control of vibrations
are all of the passive type. Recently, active systems have been developed
for the control of sound, reverberation, and vibration. These systems are
in the form of combinations of electronic elements. It is the purpose of this
section to describe electronic sound and vibration reducers and absorbers.
A. Free-Field Zone-Type Sound Reducer.1 12-The free-field, zone-type
sound reducer consists of a microphone, amplifier, and loudspeaker con­
nected in inverse fashion so as to reduce the sound pressure of any incident
sound wave in the vicinity of the microphone-loudspeaker combination.
A sectional view, schematic electrical diagram, and acoustical circuit of an
electronic sound reducer are shown in Fig. 9.74. The system is connected
and equalized for response and phase with respect to frequency, so that the
sound pressure is reduced at the microphone. The driving pressure P2,
of Fig. 9.74, is given by
9.28
where B =
1=
i =
S =
flux density in the air gap of the loudspeaker, in gausses,
length of the conductor of the voice coil, in centimeters,
current in the coil, in abamperes, and
area of the cone, in square centimeters.
The amplitude and phase relations of the sound pressures PI and P2 are
selected so as to make the sound pressure pa as small as possible. Under
these conditions the operation of the system is a sound pressure reducer.
The amount of sound pressure reduction is a function of the distance between
the microphone and loudspeaker, the wavelength of the sound wave, the
phase relation in the electronic system, and the distance from the
microphone-loudspeaker combination. Typical sound reduction frequency
111 Zwilocki, J., Jour. Acous. Soc. Amer., Vol. 27, No.3, p. 460,1955.
1120lson and May, Jour. Acous. Soc. Amer., Vol. 25, No.6, p. 1130, 1953.
416
ACOUSTICAL ENGINEERING
ACOUSTICAL
NETWORK
SECTIONAL VIEW
FIG. 9.74. Sectional view. schematic electrical diagram. and acoustical network of an
electronic sound absorber. PI = the sound pressure in free space. MI = the inertance
of the air load. rAI = the acoustical resistance of the air load. M2 = the inertance
of the cone and voice coil of the loudspeaker. CAl = the acoustical capacitance of
the suspension system of the cone. r A2 = the acoustical resistance of the cloth over the
apertures in the back plate. CA2
the acoustical capacitance of the volume of the
cabinet. r A3 = the acoustical resistance of the sound absorbing material in the cabinet.
P2 = the driving sound pressure in the loudspeaker. Pa = the sound pressure at the
microphone.
characteristics for various distances from the reducer are shown in Fig. 9.75.
These characteristics show that the electronic sound reducer may be used to
reduce undesired sounds over a limited zone of operation.
One application for the electronic noise reducer is in the form of a noise
reducer in airplanes and automobiles where the noise level is relatively high
in the low-frequency range. With the practical use of conventional sound
absorbing materials in automobiles and airplanes, the reduction in noise
level in this frequency region is relatively small. For these applications, the
noise reducer may be installed on the back of the seat. There are also
many other applications for the zone-type noise reducer. Other applica..
tions include the reduction in noise from machines, ducts, etc.
R
= 24"
/------ ..........
/
>'
"­
R = 10"
//
/
I
(
I
/
I
I
\
\
\
\
\
,
I
................
R
\
\
\
\
\
~LSI
/ 'I
\
...,-IM I
/
"
"'-
4"
(-~,
I
\
'\
--- ... "
'!;
"­
I
_",/
I
~
ILl
II:
:::>
~-5~--4-----~~-+--~~~L-~~
ILl
II:
~
/
'
.........
--
/ /
,/'
SCHEMATIC VIEW
FREQUENCY IN CYCLES PER SECOND
FIG. 9.75. Schematic view of an electronic sound reducer and sound
pressure reduction frequency characteristics for distances of 4. 10. and
24 inches from the microphone-loudspeaker combination.
r
MISCELLANEOUS TRANSDUCERS
417
B. Free-Field Electronic Sound Absorber.1 13- The free-field electronic sound
absorber consists of a microphone, amplifier, loudspeaker, and acoustical
resistance. Fig. 9.76. The microphone, amplifier, and loudspeaker are
connected in an inverse fashion so as to provide a low acoustical impedance
termination for the acoustical resistance. The principal application for the
electronic sound absorber is in the low-frequency region where it is difficult
to obtain high absorption due to the practical difficulty of providing a low
acoustical impedance termination for passive or inactive acoustical materials.
Since the application for the electronic sound absorber is in the low-frequency
ACOUSTICAL NETWORK
SECTIONAL VIEW
FIG. 9.76.
Sectional view, schematic electrical diagram and acoustical network of an
electronic sound absorber. PI = the sound pressure in free space. Ml = the inertance
of the air load. rAI = the acoustical resistance of the air load. M2 = the inertance
of the cone and voice coil of the loudspeaker. r AS = the acoustical resistance of the
screen covering the microphone and cone. Ms = the inertance of the screen. CAl
= the acoustical capacitance of the suspension system of the cone. r A2 = the acoustical
resistance of cloth over the apertures in the back plate. C A2 = the acoustical capacitance
of the volume of the cabinet. P2 = the driving sound pressure in the loudspeaker.
Pa = the sound pressure at the microphone. YAa = the acoustical resistance of the
sound absorbing material in the cabinet.
region, the system is operated as a diffraction absorber. The absorbing
efficiency of a diffraction sound absorber with the appropriate acoustical
resistance may be several hundred per cent. Therefore, for absorption of
sound in the low-frequency range, the spacing between the electronic sound
absorbers can be relatively large. See Sec. 9.2E and Fig. 11.8.
e. Headphone-Type Noise Reducer.-An application of the point-type
sound and voice reducer is for the telephone receiver as shown in Fig. 9.77.
The acoustic shielding of the earcap provides some reduction in the sound
from the outside. The electronic sound reducer provides additional reduc­
tion in the sound pressure in the small enclosed cavity of the headphone.
The telephone receiver consists of two separate diaphragm and voice coil
assemblies. The useful signal is applied to the voice coil attached to the
inner diaphragm and provides the driving force for reproducing the useful
information. The other voice coil is connected to the output of the ampli­
fier. The sensing microphone is connected to the input of the amplifier.
This feedback system provides the noise reduction. The action is similar
to the sound reducer of Sec. 9.18A.
113
Olson, H. F., Jour. Acous. Soc. Amer., Vol. 25, No.6, p. 1130, 1953.
+18
ACOUSTICAL ENGINEERING
RECEIVER
DRIVER
IAPHRAGM
SECTIONAL VIEW OF RECEIVER
SCHEMATIC VIEW OF SYSTEM
FIG. 9.77. Electronic noise reducing headset. The schematic
view shows the elements of the system. The sectional view
shows the elements of the headphone.
D. Electronic Vibration Reducer.-Reduction in the transmission of sound
through structures of solid materials is usually accomplished by the addition
of mass or by a compliant isolating system. The latter means is usually
preferred because the addition of mass is costly and for most applications
impractical. The idea is to insert an element which has a low mechanical
impedance and thereby provide a shunt for the vibrations. An electronic
system may he used to provide the low mechanical impedance and thereby
LEVER
SYSTEM
r
ln,
VOICE COIL
MAGNET -1'4--001
SENSOR
x,
MECHANICAL NETWORK
SCHEMATIC VIEW
FIG. 9.78. Sectional view, schematic electrical diagram, and mechanical network of an
electronic vibration reducer. Xl = the input vibrational velocity. IM1 = the input
driving force. m 1 = the mass of the lever system and voice coil. 1M2 = the driving
force generated in the voice coil. eM = the compliance of the lever system and the
centering spider. i2 = the vibrational velocity of the voice coil. m2 = the mass of
the magnetic structure. ZMS = the mechanical impedance of the support. 1M3 = the
driving force at the support. i3 = the vibrational velocity of the support.
control and isolate the vibrations. In most problems involving the control
of vibrations, the amplitudes are relatively small and the mechanical im­
pedance relatively large. Under these conditions, piezoelectric transducers
may be used. For example, an electronic vibration reducer may consist
of a piezoelectric driver and sensor with a suitable amplifier.
A dynamic system may also be used as a vibration reducer. Since the
amplitude of the vibrations produced by most machines is small compared
MISCELLANEOUS TRANSDUCERS
419
to the amplitude which may be obtained from a dynamic unit, a transformer
in the form of a lever may be used between the dynamic unit and the machine.
A schematic view of a dynamic vibration reducer is shown in Fig. 9.78. The
sensor is a piezoelectric transducer. The amplifier is similar to the amplifier
used for the electronic noise reducer. The performance of the system may
be obtained from the mechanical network. The mechanical impedances in
the mechanical network are all referred to the input and output terminal
impedances. Cognizance must be taken of the lever system, which is in
effect a mechanical transformer, in referring the force 1M2 and the masses
and compliances to the input and terminal mechanical impedances. The
problem in isolating the machine from the supports is to adjust the phase
and magnitude of the force 1M2 so that the resultant force 1M3 developed in
the support will be a minimum. The velocity X3 in the support may be
expressed as follows:
.
X3
=
1MIZM2 -
where ZMl
ZM2
=
=
ZM3 =
1M2ZMI
-----''---,---"''------,---ZMIZM2
+ ZMIZM3 + ZM2ZM3
9.29
mechanical impedance of the machine,
mechanical impedance of the driver and sensor, and
mechanical impedance of the support.
It may be that a part of the mechanical impedance of the driver and
sensor may be included in the mechanical impedance of the machine and/or
the mechanical impedance of the support.
From a consideration of equation 9.29, it will be seen that the magnitude
of the velocity X3 can be reduced by the application of the force 1M2 in the
proper magnitude and phase. For example, X3 = 0 when
1MIZM2 = 1M2ZMI
9.30
Under these conditions,1M3 is also zero. That is to say, no vibrations are
produced in the support. The machine is perfectly isolated from the
support.
Another problem is to reduce the vibration of the machine without regard
to the vibration transmitted to the support. The velocity Xl of the machine
may be expressed as follows:
.
Xl =
1MI
(ZM2
+ ZM3)
-
1M2ZM3
"---'------'----''---ZMIZM2
+ ZMIZM3 + ZM2ZM3
9.31
From a consideration of equation 9.31, it will be seen that the magnitude
of the velocity Xl can be reduced by the application of the force 1M2 in the
proper magnitude and phase. For example, the velocity of the machine
Xl will be zero if
9.32
If Xl = 0, there will be no motion of the machine. There will, however,
under these conditions be a larger velocity X3.
There are many applications for an electronic vibration reducer which
decreases the coupling between an offending vibration producer and a
420
ACOUSTICAL ENGINEERING
terminal location in which vibrations are undesirable. Most applications
for an electronic vibration reducer will involve the isolation of the vibrations
produced by machine from the foundation of the machine.
9.19. Noise Reduction Circuits.-Noise is one of the most disagreeable
forms of distortion that occurs in sound reproducing systems. Therefore,
any means which reduces or mitigates noise is extremely useful and important.
There are many ways of increasing the signal-to-noise ratio thereby
reducing the deleterious effects of noise. A few of the systems that have
been used may be listed: (1) a system in which the high-frequency response
is attenuated, (2) a system with suitable precompensation and postcompen­
sation so that the high-frequency response is accentuated in recording or
transmitting and attenuated in reproducing or receiving, (3) a system using
two channels--one channel is used to carry the signal and the other channel
to control the amplitude of the signal in reproduction, (4) a system in which
the high-frequency cutoff of the reproducing system is automatically made a
function of the general level of the signal, and (5) a system employing a
nonlinear element arranged so that signals below a certain threshold will be
attenuated.
Attenuating the high-frequency response is the most common method
for the reduction of noise.
The use of two channels, in which one is used as a volume control, has been
applied in some special cases but, in general, is impractical because two
channels are not available in conventional reproducing systems.
Systems in which the high-frequency response is accentuated in recording
or transmitting and attenuated in reproducing or receiving are used in
phonograph and sound motion-picture reproduction as well as frequency­
modulation radio broadcasting. This procedure is quite effective, but in
some systems it is also necessary to reduce the frequency transmission band in
order to obtain a substantial reduction in noise.
A block diagram and response frequency characteristics of a system1l4 in
which the high-frequency response is made a function of the high-frequency
level is shown in Fig. 9.79. This system is based on the fact that the masking
of the noise by the signal is a function of the level of the signal. Therefore,
in order to maintain masking of the noise by the signal for different signal
levels, the frequency range of reproduction is made a function of the signal
level. This is accomplished by means of an electronically controlled low­
pass filter. The response frequency characteristics of the system are shown
for different levels in Fig. 9.79. In the system shown in Fig. 9.79, high­
frequency cutoff is employed. If there is noise in the low-frequency range,
an electronically controlled high-pass filter may be used. If there is noise
in both the high- and low-frequency ranges, an electronically controlled
band filter may be used.
A block diagram, the amplitude characteristic, and the response charac­
teristics of a threshold-type noise reducer1l5 is shown in Fig. 9.80. Band-pass
114
115
Scott, H. H., Electronics, Vol. 20, No. 12, p. 96, 1947.
Olson, H. F., Electronics, Vol. 20, No. 12, p. 119, 1947.
421
MISCELLANEOUS TRANSDUCERS
5
I
~ 0
"\['\;\
...rn~ -5
INPUT
1,\\
\
z
oQ.
gJ -10
u:
-15
100
200
400
1000 2000 4000 10000
FREQUENCY
FIG. 9.79.
Schematic diagram and response frequency characteristics of a noise­
reducing system employing an electronically controlled low-pass filter. The response
frequency characteristics labeled 1, 2. 3. and 4 depict the response characteristics with
decreasing signal levels.
filters which pass frequencies over a range of an octave are used at the input
and output of the nonlinear element. The amplitude characteristic of the
nonlinear element is illustrated in Fig. 9.80. As will be described later,
this amplitude characteristic can be obtained with a properly biased diode
vacuum tube or crystal rectifier. By using this method the system will
exhibit high attenuation to signals of small amplitudes.
w... '" • L •
w\w...'w, 1",
NOISE INPUT
........
.. • ,~ \I
NOISE OUTPUT
/\/\/\
VV\J
SINE WAVE INPUT
f\ C\ !\
v V V
OUTPUT FROM NONLINEAR ELEMENT
/\/\/\
V\JV
SINE WAVE OUTPUT
FIG. 9.80. Schematic diagram. input VB. output charac­
teristics and the response to noise and a sine wave of a
threshold noise reducer.
The response of the noise-reduction system to noise and a sine wave is
depicted in Fig. 9.80. If the amplitude of the noise is kept below the
response range of the noise-reduction system, the noise will not be reproduced.
The response of the noise-reducing system to a sine wave signal is also shown.
The output of the nonlinear element contains the fundamental, harmonics,
and subharmonics of the fundamental. However, since the pass band of
the input and output band-pass filters is an octave, the harmonics and
subharmonics will not be transmitted by the system. The output wave,
422
ACOUSTICAL ENGINEERING
then, is a sine wave of the same frequency as the input sine wave. If two
sine waves of different frequencies are impressed upon the system, the two
frequencies must lie within the pass band octave in order to be admitted by
the input band-pass filter. The output of the nonlinear element contains
harmonics and subharmonics of the two fundamental frequencies, but these
are rejected by the output band-pass filter. The output of the nonlinear
element also contains the sum of the two frequencies and the difference of
the two frequencies . Since the input is confined to an octave, the band-pass
output filter will reject the sum and difference frequencies.
A system with an upper cutoff of 12,000 cycles and three channels of
noise reduction is depicted in the block diagram of Fig. 9.81. This system
INPUT
5
l":
0
\
lr
"'
~-IO
-1540
1. ;. "'!1\ \
I\/\ \
1!
l 1\ ~ '. \
~
...~ -5
100
200 400
1000 2000 4000
10KC
FREQUENCY IN CYCLES PER SECOND
20KC
FIG. 9.81. Schematic diagram and response frequency
characteristics of a three-channel threshold noise reducer.
uses the nonlinear elements of Fig. 9.80. Each nonlinear system is equipped
with a separate bias control so that the noise can be reduced in each band
without discrimination against the useful signal. The response-frequency
characteristic of the separate channels and the over-all response is shown in
Fig. 9.81. Conventional band-pass filters are used to confine the response
to octave bands. An amplifier overcomes the loss in the filters and nonlinear
elements. Noise reduction of up to 20 db can be obtained in each of the
channels.
10
MEASUREMENTS
10.1. Introduction.-The rapid progress made in acoustics during the
past three decades has resulted in a corresponding advance in acoustical
measurements. l In applied acoustics, as in any applied science, theoretical
analysis and analytical developments are substantiated by experimental
verifications. In view of the importance of acoustical measurements, it
seems logical to devote a portion of this book to this phase of acoustics.
It is the purpose of this chapter to consider the testing of microphones,
loudspeakers and telephone receivers together with fundamental acoustical
measurements.
10.2. Calibration of Microphones.2,3,~A number of different measure­
ments are required to determine the performance of a microphone. The
most important characteristics which depict the performance of a micro­
phone are as follows:
1.
2.
3.
4.
5.
6.
Response frequency characteristic
Directional characteristic
Nonlinear distortion characteristic
Phase distortion characteristic
Transient response characteristic
Electrical impedance characteristic
In addition to the above characteristics are such factors as the effect of
temperature, humidity, and changes in atmospheric pressure upon the per­
formance of the microphone. Carbon microphones exhibit characteristics
peculiar to granular contacts such as carbon noise, packing, and breathing.
A. Response Frequency Characteristic.-l. Pressure Response.-The pres­
sure response frequency characteristic of a microphone is the ratio e/p as a
1 Beranek, L. L., "Acoustic Measurements," John Wiley and Sons, New York, N.Y.,
1949.
2 American Standards Association Sectional Committee z-24, Report on, Calibration
of Microphones, Jour. Acous. Soc. Amer., Vol. 7, No.4, p. 330, 1936. Also American
Standards Association, z-24.4. 1938.
3 Standards on Electroacoustics of the Institute of Radio Engineers, 1933.
4 American Standards Association, "Pressure Calibration of Laboratory Standard
Pressure Microphones." z-24.4- 1949; "Laboratory Standard Pressure Microphones,"
z-24.8-1949; "Free Field Secondary Calibration of Microphones," z-24.11-19S4.
423
424
ACOUSTICAL ENGINEERING
function of the frequency where e is the open-circuit voltage generated by
the microphone, in volts, and p is the sound pressure, in dynes per square
centimeter, upon the diaphragm of the microphone. The ratio e/p is usually
expressed in decibels with respect to some arbitrary reference level. The
pressure upon the diaphragm may be generated by a piston-phone, thermo­
phone, or an electrostatic actuator.
a. Pistonphone. 5 •6 •7 •8-A schematic arrangement of a pistonphone for
use in calibrating a pressure type microphone having a diagram of high
acoustical impedance is shown in Fig. 10.lA. The small piston is driven
by a crank. The pressure generated at the diaphragm, assuming all of
the walls of the enclosure to be rigid, is
p=
where p
Vo
A
r
=
=
=
=
po =
y =
a
=
rAPoY{l
V
+ (y -l)A w + !..[(y -1)A w ]2}-1/2
aVo
2
10.1
aVo
peak pressure. in dynes per square centimeter,
volume of the enclosure, in cubic centir.leters,
area of the piston, in square centimeters.
radius of the crank, in centimeters,
atmospheric pressure, in dynes per square centimeter.
ratio of specific heats (1.4 for air),
Jw;~p =
3.9v'lfor air. 20° c..
Aw
= area of metallic walls, in square centimeters,
K = thermal conductivity of the enclosed gas (6.2 X 10-6 for air),
p = density of the gas. in grams per cubic centimeters,
C p = specific heat of the gas at constant pressure (.24 for air).
w = 21rj, and
j = frequency, in cycles per second.
This method is very useful for calibrating a microphone at the low fre­
quencies. The upper frequency limit is governed by the permissible speed
of the mechanical system which-is approximately 200 cycles.
Under test the output of the microphone is fed to an amplifier and output
meter. For a particular value of generated pressure the output is noted.
Then, the pistonphone is disconnected and a voltage of the same frequency
as that generated by the pistonphone is inserted in series with the micro­
phone and adjusted to give the same output. The response (e/p) at this
frequency is the ratio of this voltage to the applied pressure.
5
6
7
8
Wente. E. C., Phys. Rev .• Vol. 10. No.!, p. 39, 1917.
Wente, E. C.• Phys. Rev., Vol. 19. No.4. p. 333, 1922.
Kaye, G. W. C., Jour. Acous. Soc. Amer., Vol. 7, No.3, p. 174, 1936.
Glover and Baurnzweiger, Jour. Acous. Soc. Amer., Vol. 10, No.3, p. 200, 1939.
MEASUREMENTS
425
b. Thermophone. 9,10,l1-The thermophone consists of one or more strips
of thin gold leaf mounted upon terminal blocks (Fig. 10.IB). In the usual
method the thermophone strip carries a known steady current upon which
a smaller sinusoidal current is superimposed. In this case, the variation of
the pressure in the chamber occurs primarily at the frequency of the alter­
nating current. The cavity of the thermophone is usually filled with
hydrogen. The wavelength in hydrogen is considerably longer than in air
and, as a consequence, the standing waves are shifted to a higher frequency
beyond the useful response range.
The peak alternating pressure developed in the cavity is given by
P=
where D
=
A
=
4KS2
( 1 - wCV
A
. 96SioirE
wmCV ActDl/2
10.2
)2 + (1 + V45Act + 4KSa
4KS2 )2
wC + wCV A
Tct - Y - - l
Tsy -1
m = .:..:..(y_---:-l'-)T...=.s
yPO
- JwCpp
2K
a -
C
io
i
rE
Ts
Ta
K
=
=
=
=
=
=
=
p =
=
=
y=
po =
Cv
Cp
5
=
V
=
=
=
w
j
total thermal capacity of the strip product of the mass in grams
and the specific heat,
steady current, in amperes,
peak value of the alternating current component, in amperes,
total electrical resistance of the strip, in ohms,
mean temperature of the strip, in degrees Kelvin,
mean temperature of the gas in the enclosure, in degrees Kelvin,
thermal conductivity of the gas,
density of the gas, in grams per cubic centimeter,
specific heat of the gas at constant volume,
specific heat of the gas at a constant pressure,
CplC v ,
average pressure of the enclosure, in dynes per square centi­
meter,
total area of one side of the thermo phone foil, in square centi­
meters,
volume of the enclosure, in cubic centimeters,
2"1Tj, and
frequency, in cycles per second.
Arnold and Crandall, Phys. Rev., Vol. 10, No. 1, p. 22, 1917.
Wente, E. C., Phys.Rev., Vol. 19, No.4, p. 333, 1922.
11 Ballantine, S., Jour. Acous. Soc., Amer., Vol. III, No.3, p. 319, 1932.
9
10
426
ACOUSTICAL ENGINEERING
The determination of the ratio e/p is carried out in the same manner as
the pistonphone.
c. Electrostatic Actuator.l2.-The electrostatic actuator consists of an
auxiliary electrode in the form of a grill mounted in front of the microphone
diaphragm, Fig. 10.1e. The actuator is perforated so that it does not
A
PISTON PHONE
c
B
ELECTROSTATIC
THERMOPHONE
ACTUATOR
FIG. 10.1.
Apparatus for obtaining the pressure-frequency characteristic of
a condens~r-type microphone. The pistonphone and'thermophone may be
used for other types of pressure microphones.
appreciably alter the mechanical impedance opposing the motion of the
diaphragm. A large, steady, polarizing voltage is applied to the grill and
microphone diaphragm. Then a sinusoidal voltage is applied, effectively,
in series. The alternating force,13 in dynes per square centimeter of the
grill, assuming no tufting of the electrostatic lines, is
P = 8.85eoe
d2
where eo
=
e=
d=
X
10-7
polarizing voltage, in volts,
alternating voltage, in volts, and
spacing between the actuator and the diaphragm,
meters.
10 3
•
III
centi­
The force developed by the actuator is independent of the frequency.
Therefore, it constitutes a simple system for obtaining the response of a
condenser microphone as a function of the frequency. If the absolute
response is desired this may be obtained by comparison with some known
standard (thermo phone or pistonphone). In the case of some actuator
structures the effective area may be calculated from standard formulas
which correct for tufting.
The determination of the ratio e/p is carried out in the same manner as
the pistonphone.
Ballantine, S., Jour. Acous. Soc. Amer., Vol. 3, No.3, p. 319,1932.
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J.,
1943.
12
13
MEASUREMENTS
427
d. Reciprocity.-The acoustical reciprocity procedure 14 ,15,16,17,18 may be
used to obtain the pressure response frequency characteristic of a micro­
phone. In the reciprocity procedure, three transducers are used to obtain
the pressure response frequency characteristic of a microphone, namely, the
microphone M to be calibrated, a reversible microphone-loudspeaker, 51,
and a loudspeaker, 52, see Fig. 10.2.
The acoustical impedance of the
microphone, M, and the reversible A
transducer, 51, should be the same
in order to obtain accurate results.
Furthermore, the acoustical im­
pedance of the microphone at the
diaphragm should be high. Micro- B
phones satisfying these requirements
are, in general, condenser and piezo­
electric microphones.
The first and second experiments c A-C
INPUT
in the reciprocity procedure are
shown in Fig. 1O.2A and Fig. 1O.2B.
An alternating current is fed to the FIG. 10.2. The three experiments of the
reciprocity procedure for obtaining the
loudspeaker, 52. A sound pressure, pressure calibration of a microphone.
PI, is produced in the volume having A. The open-circuit voltage, es, of the
an acoustical capacitance CA. Let reversible microphone loudspeaker, S1>
the open circuit voltage, in statvolts, when used as a microphone and actuated
by a sound pressure, Pl' B. The open­
of 51 used as a microphone be desig­
circuit voltage, eM, of the microphone, M,
nated as es and the open circuit to be calibrated, when actuated by a sound
voltage output, in statvolts, of the pressure, Pl' C. The open-circuit voltage,
microphone M be designated as eM. eM', of the microphone, M, to be calibrated,
Let Ks = output, in statvolts per when actuated by a sound pressure, P2,
produced by the reversible microphone
dyne per square centimeter, of 51 loudspeaker, S1> used as a loudspeaker
and KM = output, in statvolts per with a current input, i, and a volume
dyne per square centimeter, of M. coupling, CA.
Since the sound pressure, PI, in dynes
per square centimeter, is the same for 51 and M, it is evident that
':;S'~"
,:;&d[:~"
PI
= ~ =
Ks
eM
KM
lOA
In the experiments of Fig. 10.2, it is assumed that the acoustical imped­
ance of the diaphragm of the units 51 and M are large compared to the
Rayleigh, "Theory of Sound," The Macmillan Company, Vol. 1, p. 145.
Ballantine,S., Proc. Inst. Rad. Engrs., Yol. 17, No.6, p. 929,1929.
16 Cook, R. F., Jour. Research, Natl. Bur. Standards, Vol. 25, No.5, p. 489, 1940.
17 McLean, W. R., Jour. Acous. Soc. Amer., Vol. 12, No. 1, p. 140, 1940.
18 " American Standard Method for the Pressure Calibration of Laboratory Standard
Pressure Microphones," American Standards Association, New York, N.Y., z-24.4-1949.
This standard contains an extensive bibliography as well as the reciprocity procedure
for obtaining the pressure response frequency characteristic of a microphone.
14
15
428
ACOUSTICAL ENGINEERING
acoustical impedance of the coupling cavity. In these considerations it
will be assumed that the transducers of 51 and M are of the electrostatic or
condenser type. The microphone described in Sec. 9.2B is one of the
microphone types suitable for use as the units 51 and M. In this case the
transducers are reversible and the volume current1 9 U2, in cubic centimeters
per second, of the diaphragm of 51 due to a current i, in statamperes, applied
to the transducer in experiment 10.2C is given by
U2
Ksi
=
10.5
The sound pressure P2, in dynes per square centimeter, produced by the
loudspeaker 51 in experiment C of Fig. 10.2 is given by
U2
P2 = JW
""'--cA
10.6
Equation 10.6 is valid provided the acoustical impedance of the microphone
M is large compared to the acoustical impedance of the cavity.
The open circuit voltage e'M, in statvolts per dyne per square centimeter,
of the microphone M in experiment C of Fig. 10.2 is given by
KM
=
e'M
P2
10.7
where P2 = sound pressure, in dynes per square centimeter.
From equations to.5, 10.6, and to.7,
K M K s= jwCAe'M
.
z
to.8
From equation lOA,
10.9
From equations to.8 and 10.9,
KM=J~~
esZ
10. to
where es, eM, e'M, and i are obtained from the three experiments A, B, and C
of Fig. to.2. The acoustical capacitance CA is obtained from the dimensions
of the coupling cavity, The voltages are in statvolts and the currents in
statamperes and the acoustical capacitance in (centimeter)5 per dyne.
Equation 10.10 was derived for electrostatic or condenser transducers.
However, equation 10.10 applies to other transducers provided the stipula­
tions in the derivation are satisfied.
2. Field Response.-The field or free-wave response frequency charac­
teristic of a microphone is the ratio elP as a function of the frequency, where
e is the open-circuit voltage generated by the microphone, in volts, and P
19 Olson, "Dynamical Analogies." D. Van Nostrand Company, Princeton, N.J.,
1943.
MEASUREMENTS
429
is the sound pressure, in dynes per square centimeter, in a free progressive
wave prior to the introduction of the microphone.
At the present time the Rayleigh disk and the reciprocity procedure are
the two most common methods in use today for obtaining the field response
frequency characteristic of a microphone. It is the purpose of this section
to describe the calibration of a microphone by means of the Rayleigh disk
and reciprocity methods.
a. Rayleigh Disk. 2o ,21,22,23-Rayleigh observed that when a disk was
suspended by a light fiber it would tend to turn at right angles to the im­
pinging sound wave. Koenig24 developed the formula for the turning
moment of the disk as
M =
~3 pa 3u 2 sin 28
10.11
where M = turning moment acting upon the disk, in dyne centimeters,
p = density of air, in grams per cubic centimeter,
a = radius of the disk, in centimeters,
8 = angle between the normal to the disk and the direction of
propagation of the sound wave, in degrees, and
u = particle velocity of the sound wave, root-mean-square, in
centimeters, per second.
When a sound wave falls upon the disk the angular deflection will be
,p = ~
5
where 5 = moment of torsion of the suspension, in dyne centimeter.
The moment of torsion of the suspension is given by
5 = ;2 [4rr2
where T
=
I
=
+ (log. y)2J
10.12
10.13
periodic time of the suspended disk, in seconds,
moment of inertia of the disk,
1= ma 2/4,
m = mass of the disk, in grams,
a = radius of the disk, in centimeters, and
y = damping factor, the ratio of two successive swings.
From equations 10.11, 10.12, and 10.13 it is possible to determine the
particle velocity u in the sound wave.
The arrangement of a Rayleigh disk for field calibrations of microphones
is shown in Fig. 10.3. The source of sound is a small direct radiator loud­
speaker, with the back completely enclosed, placed halfway between the
Rayleigh, Phil. Mag., Vol. 14, p. 186, 1882.
Ballantine, Phys. Rev., Vol. 32, No.6, p. 988, 1929.
22 Olson and Goldman, Electronics, Vol. 4, No.9, p. 106, 1931­
23 Sivian, L. j., Bell Syst. Tech. Jour., Vol. 10, No.1, p. 96, 1931­
24 Koenig, Ann. d. Physik, Vol. 43, p. 43, 1891.
20
21
430
ACOUSTICAL ENGINEERING
disk and the microphone. A small loudspeaker is used so that a spherical
wave will be emitted. If a velocity microphone is used no correction need
be made for the spherical wave because the Rayleigh disk also measures
the particle velocity. If a pressure microphone is used the appropriate
correction for the accentuation in velocity in a spherical wave must be
made (see Sec. LSD and Fig. 1.3). From the geometry of the system of
LAMP
OPTICAL
SYSTEM
FIG. 10.3. Arrangement of apparatus for obtaining the free-field response of
a microphone by means of a Rayleigh disk.
Fig. 10.3 the deflection of the disk can be determined from the deflection of
the light beam on the scale.
b. Reciprocity.-The acoustical reciprocity theorem was originally
enunciated by Helmholtz and Rayleigh. 25. Ballantine 26 established
reciprocity theorems for mechanoacoustic, electromechano, and electro­
mechanoacoustic systems. Ballantine also carried out a generalized dis­
cussion to show that a microphone may be calibrated in terms of electrical
standards by the use of the extended reciprocity relations. Later other
investigators 27 ,28,29 extended the applications of reciprocity in both closed
and field systems. It is the purpose of this section to outline the reciprocity
procedure for the field calibration of microphones.
For the application of the reciprocity principle to the calibration of a
microphone, three transducers are used as follows: the microphone, M, to
be calibrated, a reversible microphone loudspeaker 51, and a loudspeaker 52.
For the reversible microphone loudspeaker it is convenient to use a small
back-enclosed loudspeaker.
The first and second experiments are shown schematically in Fig. lOAA
and Fig. lOAB. An alternating current is fed to the loudspeaker 52. A
sound pressure PI is produced at a distance d. Let the open-circuit voltage,
in abvolts, of 51 used as a microphone be designated as es and the output of
the microphone M be designated as eM. Let Ks = output, in abvolts per
Rayleigh, "Theory of Sound," The Macmillan Company, Vol. I., p. 145.
Ballantine, S., Proc. Inst. Rad. Engrs., Vol. 17, No.6, p. 929,1929.
27 Cook, R. K., Jour. Research, Natl. Bur. Standards, Vol. 25, No.5, p. 489, 1940.
28 McLean, W. R., Jour. Acous. Soc. Amer., Vol. 12, No.1, p. 140, 1940.
29 Olson, H. F., RCA Review, Vol. 6, No. 1. p. 36,1941.
25
26
431
MEASUREMENTS
dyne per square centimeter of 51, and, KM = output, in abvolts per dyne
per square centimeter, of M. Since the sound pressure PI, in dynes per
square centimeter, is the same for 51 and M 1 , it is evident that
es
eM
Ks
KM
10.14
P1=-=-
The voltage output,30 m abvolts, of the microphone loudspeaker 51
used as a microphone is
es = ElX1
10.15
where B = flux density in the air gap, in gausses,
l = length of the conductor, in centimeters, and
Xl = velocity of the voice coil, in centimeters per second.
Sa
B
A-C
IN PUT
----r-i/1
-L----""J
.
C
S,
A-C~
I N PUT
-----L---"'J
I·
d
~e~
-----;·~I
FIG. 10.4. The three experiments of the reciprocity procedure
for obtaining the free-field calibration of a microphone. A. The
open-circuit voltage, ea, of the reversible microphone loudspeaker,
Slo when used as a microphone and actuated by a sound pressure,
PI. B. The open-circuit voltage, eM, of the microphone, M, to
be calibrated, when actuated by a sound pressure, Pl' C. The
open-circuit voltage, eM', of the microphone, M, to be calibrated
when actuated by a sound pressure produced by the reversible
microphone loudspeaker, S l' used as a loudspeaker with a current
input, i, and a spacial separation, d.
The velocity, in centimeters per second, of the vibrating system of 51 as
a microphone is
Xl =
PIA
10.16
ZM
where p = actuating sound pressure, in dynes per square centimeter,
A = area of the diaphragm, in square centimeters, and
ZM = mechanical impedance of the vibrating system, in mechanical
ohms.
30
Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J.,
1943.
432
ACOUSTICAL ENGINEERING
From equations 10.14, 10.15, and 10.16
~
PI
=
ElA = Ks
10.17
ZM
The third experiment is shown in Fig. 1O.4C. The velocity, 31 in centi­
meters per second, of the diaphragm and voice coil of S1 for a current i
in abamperes, in the voice coil is
.
Eli
X=­
10.18
ZM
The sound pressure, p, at M, in dynes per square centimeter, at a distance
d, in centimeters, produced by S1 in the range where the dimensions are
small compared to the wavelength, from equations 2.1 and 2.4, is
p = pckAx
4-rrd
10.19
where A = area of the diaphragm, in square centimeters,
x = velocity of the diaphragm, in centimeters per second,
p = density of air, in grams per cubic centimeter, and
k = 2'TT/>",
>.. = wavelength, in centimeters, and
c = velocity of sound.
From equations 10.18 and 10.19,
p=
pcKABli
4-rrdzM
10.20
pckiKs _ rAiKs
4-rrd U>..
10.21
From equations 10.17 and 10.20,
P -_
where r A = pc.
The sound pressure, p, in dynes per square centimeter, at M in terms of
the constant KM and the open-circuit voltage e'M, in abvolts, is
10.22
From equations 10.21 and 10.22,
e'M
KM
rAiKs
2d)'
10.23
From equation 10.14,
10.24
31 Olson, "Dynamical Analogies," D. Van Nostrand Company, Princeton, N.J.,
1943.
MEASUREMENTS
433
When Ks is eliminated from equations 10.23 and 10.24, the response of
the microphone M, in abvolts per dyne per square centimeter, is
KM
=
J2d)..e~e' M
r AleS
10.25
where es, eM, e'M, and i are obtained from the experiments of Fig. 10.4.
The units are as follows: Voltages in abvolts, currents in abamperes, dis­
tances in centimeters, wavelengths in centimeters, and r A = pc = 41.5.
The calibration of microphones by the Rayleigh disk and reciprocity
methods should be made under free-field conditions, that is, in a large room
in which the reflections are negligible or outdoors at a great distance from
reflecting surfaces. A free-field sound room suitable for these measure­
ments is described in Sec. 1O.3A4.
A high-quality microphone calibrated by any of the above methods may
be used as a secondary standard for the calibrations of other microphones.
3. Secondary Calibration of Microphones. 32-A secondary calibration of a
microphone is obtained by a comparison of the response of the microphone
to be calibrated with the response of a microphone that has been calibrated
by primary means described in the preceding sections. Response frequency
characteristics are obtained on the two microphones. The ratio response
of the microphone being calibrated to the response of the calibrated micro­
phone yields the calibrated response of the microphone being calibrated.
4. Artificial Voice.-The proximity of the head in close talking speech
type microphones influences the response frequency characteristics. There­
fore, in testing microphones of this type it is desirable to provide testing
means 33 which simulate actual operating conditions. The artificial voice
consisting of a small loudspeaker unit mounted in the head of a manikin,
as shown in Fig. 10.5, provides a means for obtaining the response frequency
characteristics of close talking microphones. Resonances in the tube
connecting the loudspeaker unit and the mouth are eliminated by the intro­
duction of series and shunt mechanical resistances. The response frequency
characteristic shown in Fig. 10.5 can be obtained by a suitable choice of
constants of the mechanical system.
5. Artificial Throat.-Throat microphones have been described in Sec.
8.7. Throat microphones are actuated by sound waves transmitted through
the throat. An artificial throat 34 for testing throat microphones consists of
a mass-controlled system driven by a voice coil located in magnetic field.
Specifically the voice coil is coupled to a massive platform. The centering
system is made very compliant to insure mass control. In order to maintain
constant velocity with respect to frequency the driving oscillator and
amplifier are compensated so that the current through the voice coil is
proportional to the frequency. The platform system, which the voice coil
32 American Standards Association,
"Secondary Calibration of Microphones,"
z-24.11- 1954.
33 Inglis, Gray, and Jenkins, Bell Syst. Tech. Jour., Vol. 11, No.2, p. 293,1932.
34 Greibach, E. H., Elec. Eng., Vol. 65, No.4, p. 184, 1946.
43+
ACOUSTICAL ENGINEERING
drives, is coupled to the throat microphone under test by means of a filter
pad made of material with high damping-as, for example, Viscoloid.
B. Directional Characteristic.-The directional characteristic of a micro­
phone is an expression of the variation of the behavior of the microphone
20
OJ
015
"'Z
<II
oQ.
10
_I--..
1/
f­
r--­
<II
"'
a: 5
4
8 103
2
810 4
fREQUENCY
SECTIONAL
PERSPECTIVE VIEW
VIEW
FIG. 10.5. Perspective view, sectional view, mechanical network, and response
frequency characteristic of an artificial voice. In the mechanical network, mD = the
mass of the diaphragm and suspension of the small loudspeaker unit. YMD and
C MD = the mechanical resistance and compliance of the suspension system of the
small loudspeaker unit. C MO = the compliance of the air chamber behind the
diaphragm. mI' YM • • • m5' YM5 = the mass and mechanical resistances of the series
elements in the pipe. YMI', C MI . . . YM4', C M4 = the mechanical resistances and
compliances of the shunt elements of the line. mu and YMU = the mass and mechan­
ical resistance of the air load on the mouth. The response frequency characteristic
depicts the free-field sound pressure at a distance of 2 inches.
OSCILLATOR
CATHODE - RAY
10.6.
OSCILLOGRAPH
POTENTIOMETERS
Schematic arrangement of the apparatus employing a cathode­
ray tube with a long persistence screen as a polar directional characteristic
indicator and recorder.
FIG.
MEASUREMENTS
435
with respect to direction. A polar diagram showing the output variation
of the microphone with direction is usually employed.
The directional characteristics should be obtained at representative
frequencies . In order to obviate any errors due to reflections the directional
measurements should be made under free-field conditions. Obviously,
very slight reflections will introduce considerable error for the angles in
which the response is very low.
A cathode-ray tube with a long persistence screen may be used to obtain
the directional characteristic of a microphone or loudspeaker. The apparatus
of Fig. 10.6 is arranged to obtain the directional characteristic of the micro­
phone. The directional characteristic of the loudspeaker may be obtained
by placing the loudspeaker upon the rotating shaft and keeping the micro­
phone fixed in position. The sound is picked up by a microphone and
amplified. The output of the amplifier is detected and fed to a low-pass
filter. The output of the filter is amplified by a d-c amplifier, the output of
which is fed to two potentiometers. The arms of the potentiometers are
spaced at 90°. The potentiometer arms and microphone shaft are rotated
by a motor. The length of the radius vector of the spot is proportional to
the output of the microphone. The angular displacement flf the spot is
synchronized with the microphone shaft. From this it will be seen that the
cathode-ray beam traces the polar directional characteristic of the micro­
phone. In case it is desirable to record the characteristic, this may be
done photographically or by tracing the curve left upon the screen.
C. Nonlinear Distortion Characteristic.-The harmonic distortion tests
are intended to show the spurious harmonics generated by the microphone
FIG. 10.7.
Arrangement of apparatus for measuring the non­
linear distortion generated by a microphone. (After Phelps.)
when it is actuated by a pure tone. The plot of the total distortion, in per
cent of the fundamental, is termed the distortion characteristic. It is also
common practice to plot the individual components in per cent as the
distortion characteristics.
It is difficult to obtain a sound source which will generate an intense
sound wave of very low distortion in free space. The arrangement 35 shown
in Fig. 10.7 provides a simple means of obtaining a sound wave free from
35
Phelps, W. D., Jour. Acous. Soc. Amer.• Vol. 11, No.2. p. 219. 1939.
436
ACOUSTICAL ENGINEERING
distortion. A stationary wave is obtained in the tube by moving the piston
until the maximum pressure is obtained. A pressure of 1000 dynes per
square centimeter can be obtained with a fraction of a watt input to the
loudspeaker. For the determination of the second harmonic the microphone
is placed at a second harmonic node. Under these conditions the second
harmonic component at the microphone is very small. The second harmonic
component is then measured by means of a harmonic analyzer (see Sec.
lO.3C). For the third harmonic the microphone is placed at a third har­
monic node. Either pressure or velocity microphones may be tested, the
only difference being in the position in the tube.
D. Phase Distortion Characteristic.-The phase distortion characteristic
of a microphone is a plot of the phase angle between the voltage output
of the microphone with respect to some reference voltage as a function of
the frequency. A microphone such as the velocity microphone (see
Sec. 8.3B), in which the output is in phase with the particle velocity (its
output is also in phase with the pressure in a plane sound wave), may be
INCIDENT
SOUND
~
CATHODE-RAY
OSCILLOGRAPH
FIG. 10.8. Schematic arrangement of apparatus for
measuring the phase characteristic of a microphone.
used as the reference standard. The standard microphone and the micro­
phone to be tested may be placed side by side in a plane progressive wave
in free space, Fig. 10.8. The outputs of the two microphones are amplified
by separate identical amplifiers and connected to the vertical and horizontal
plates of a cathode-ray oscillograph. The resultant Lissajou figure indicates
the phase relations between the output of the two microphones. The two
microphones are shifted relative to each other in a line parallel to the direction
of propagation until the outputs of the two microphones are in phase. The
phase angle, in degrees, between the output of the two microphones is
4> =
where d
=
A=
~A 360
0
10.26
distance between the two microphones in the direction of propa­
gation, in centimeters, and
wavelength of the sound, in centimeters.
Phase distortion is of importance in combination microphones such as
the unidirectional microphone (see Sec. 8.4).
MEASUREMENTS
437
E. Electrical Impedance Frequency Characteristic.-The electrical im­
pedance frequency characteristic of a microphone is the electrical impedance
at the output terminals as a function of the frequency. Any convenient
method for measuring electrical impedance may be used for determining
the electrical impedance frequency characteristic.
F. Transient Response Characteristic.-For measurement of transient
response, see Sees. 1O.3G and 8.14.
G. Measurement of Wind Response of Microphones.-The wind response
of microphones is an important characteristic when microphones are used
FIG. 10.9. A wind generator for obtaining the wind response
of microphones.
outdoors under wind conditions. It is very difficult to test the wind response
under actual wind conditions because it is impossible to find constant wind
conditions. Wind consists of a steady flow of air with superimposed pulses
of air. A wind generator 36 which delivers a steady flow of air with super­
imposed pulses is shown in Fig. 10.9. Almost any practical value of wind
velocity and pulses can be obtained by adjusting the speed of rotation and
36
Olson, Preston, and Bleazey, Unpublished Report.
438
ACOUSTICAL ENGINEERING
the angles of the paddles. Comparison of the results obtained with actual
wind with those of the wind generator shows very good correlation with
respect to the distribution of the frequency components. This indicates
that the wind components in the two cases are the same.
In another method 37 for obtaining the wind response of a microphone,
the microphone is suspended at the end of a pendulum about ten feet in
length. In this way maximum wind velocities up to 20 miles an hour can be
obtained. In addition, there is a variation in velocity from zero to the
maximum velocity.
10.3. Testing of Loudspeakers. 38 ,39-Many different measurements are
required to determine the performance of a loudspeaker. The most
important characteristics which depict the performance of a loudspeaker
are as follows:
1. Response frequency characteristic
2. Directional characteristic.
3. Nonlinear distortion characteristic
4. Efficiency frequency characteristic
5. Phase distortion characteristic
6. Electrical impedance characteristic
7. Transient response characteristic
A. Response Frequency Characteristic.-l. Pressure Response.-The pres­
sure response of a loudspeaker is a measure of the sound pressure produced
at a designated position in the medium with the electrical input, frequency,
and acoustic conditions specified.
Absolute response is the ratio of the sound pressure (at a specified point
in space) to the square root of the apparent electrical power input. It is
given by the equation
Absolute response
= -pe- =
PVZE
--
e
10.27
VZE
where P = measured sound pressure, in dynes per square centimeter,
e = effective voltage applied to the voice coil, in volts, and
ZE = absolute value of the electrical impedance of the voice coil, in
ohms (ZE is usually a function of frequency).
The absolute response characteristic is obtained by measuring the sound
pressure p, as a function of frequency with constant voltage, e, on the voice
coil, and measuring the electrical impedance, ZE, as a function of the fre­
quency, and correcting the measured sound pressure for the measured
electrical impedance in accordance with the equation. The resulting
Carrell, R. M., Jour. Acous. Soc. Amer., Vol. 3, No.2, p. 102, 1955.
American Standards Association, Loud Speaker Testing, C. 16.4, 1942.
39 Standards on Electroacoustics, Institute of Radio Engineers, 1938.
87
38
MEASUREMENTS
439
characteristic represents the sound pressure as a function of the frequency
which would be obtained from the speaker if fed from the generator which
would automatically deliver constant apparent power, e2 jzE, to the voice
coil over the frequency range.
The response may be expressed by a value equal to the above ratio or
may be expressed in decibels relative to an arbitrary value of response
corresponding to 1 volt, 1 ohm, and 1 dyne per square centimeter.
Absolute response
=
p
ejvzE
20 loglO - 1 -
= 20
PVZE
10glO -e-
10.28
1/0
The apparatus and methods employed for obtaining the response fre­
quency characteristics of loudspeakers will be described in the sections
which follow.
2. Apparatus for Measuring the Sound Pressure Frequency Relationship
of a Sound Source.-An arrangement for obtaining the sound pressure
frequency characteristic by the semiautomatic method 40 is shown in Fig.
10.10.
This method yields a response frequency curve on semilogarithmic paper
in about 3 minutes. Rotation of a condenser governing the beat frequency
HANDLE
FIG:. 10.10. Schematic arrangement of the apparatus for manually recording
the sound pressure-frequency characteristic of a sound source. (After Wolff
and Ringel.)
of the heterodyne oscillator and coupled to a drum on which the paper
record is made gives the abscissas for the curves, values which are pro­
portional to the logarithm of the frequency due to the manner in which the
condenser plates are cut. The drive may be manual or by motor.
A linear or logarithmic detector 41 ,42 may be employed. In the former,
the deflection of the meter is proportional to the sound pressure. In the
Wolff and lUngeJ, Proc. IRE, Vol. 15, No.5, p. 363,1927.
Ballantine, S., Jour. Acous. Soc. Amer., Vol. 5, No.1, p. 10, 1933.
42 Hackley, R. A., Broadcast News, No. 28, p. 20, July, 1938.
40
41
440
ACOUSTICAL ENGINEERING
latter, the deflection of the meter is a logarithmic function of the sound
pressure. The resulting curve is recorded directly in decibels. A variation
of this method is sometimes used in which the recording pen is coupled to a
gain control in the amplifier, the operator manipulating the control in such
a manner that the output indicated by the meter remains constant. Either
a linear or a logarithmic coordinate scale may be obtained by suitable
design of the gain control.
The acoustical level recorder 43 is an automatic device which records the
gain settings required to keep the amplifier output constant as the frequency
of the sound source is varied. Fig. 10.11 shows how a pressure characteristic
can be made with the sound level recorder. A dark-colored tape coated
LOUD SPEAKER
MICROPHONE
FIG. 10.11. Schematic arrangement of the apparatus used in a high-speed
level recorder for automatically recording a sound pressure-frequency charac­
teristic. (After Wente, Bedell, and Swartzel.)
with white wax is moved under a stylus by a motor which changes the value
of the beat frequency generated at the same time. The loudspeaker under
test is connected to the output of the beat frequency generator and the
variations in response are recorded on the paper directly on a decibel scale
by a stylus which scratches through the wax coating on the recording
paper.
The rectifier output incorporates a control circuit which causes direct
current to flow through one circuit when the rectifier current is less than a
certain critical value and through a second circuit when it is greater than
a second critical value. In the first case, the control circuit operates a
magnetic clutch which causes a potentiometer to operate and increases
the voltage. In the second case, the voltage is decreased.
The output of the rectifier is kept balanced to within the voltage change
produced by a change in potentiometer corresponding to the smallest unit
43
Wente, Bedell, and Swartzel, Jour. Acous. Soc. Amer., Vol. 6, No.3, p. 121, 1935.
MEASUREMENTS
441
of the attenuator calibration. The motion of the potentiometer is com­
municated to the stylus which gives a trace on the recording paper. The
same motor which drives the oscillator frequency control moves the poten­
tiometer by means of the magnetic clutches.
The speed with which changes in sound level are recorded may be varied
from 10 to 560 db per second through alteration of the speed of rotation of
the clutches.
In another design 44 of high-speed level recorder a thyratron actuated
reversible motor drives a fountain pen and records directly on graph paper.
The speed is somewhat slower than the clutch system but the conventional
paper record is more convenient to use and file.
A high-speed level recorder45 with a dynamic drive is shown in Fig. 10.12.
The dynamic driving mechanism is used to drive the stylus and the contactor
FIG. 10.12. Schematic arrangement of the apparatus used in a high-speed
level recorder employing a dynamic driving system.
on the potentiometer. The dynamic driving mechanism consists of a voice
coil located in a very long air gap. See Fig. 10.13. The flux for the air gap
is supplied by permanent magnets. Two drive rods extending axially in
both directions are attached to the voice coil. The stylus and contactor are
attached to the drive rod. The useful amplitude range of the system is two
inches. The potentiometer is of the logarithmic type. The drive system
in conjunction with the potentiometer maintains constant input to the
driving amplifier. If there is an unbalance, the amplifier supplies the proper
polarity to the voice coil, and, as a result, the driving system will move to
restore the balance. The loudspeaker under test is connected to the output
of the beat frequency oscillator and the variations in response are recorded
Clark, W. R., A.I.E.E. Trans., Vol. 59, p. 957,1940.
4. Bruel and Kjar, Tech. Rev., No.3, 1952.
44
#2
ACOUSTICAL ENGINEERING
POLE
SECTIONAL VIEW
FIG. 10.13. Sectional view of the dynamic drive
used in the high-speed level recorder of Fig. 10.12.
on the wax-coated paper directly on a decibel scale by the stylus which
scratches through the wax coating on the paper and thereby leaves a visible
MICRO­
PHONE
AMPLIriER
LOW­
PASS
F"lLTER
AMPLIF"IER
110 V.A.C.
PERSISTENCE IMAGE
CATHODE -RAY OSCILLOGRAPH
FIG. 10.14. Schematic arrangement of the apparatus employing a cathode-ray
tube with a long persistence screen as a pressure response frequency indicator
and recorder. (After Hackley.)
trace. The maximum recording speed is about 900 db per second with a
potentiometer having a 60 db range.
A cathode-ray tube,46,47 with a long persistence screen, may be used as
a response indicator and recorder, Fig. 10.14. A motor drives the beat
frequency oscillator and a potentiometer. The potentiometer varies the
46
47
Hackley, R. A., Broadcast News, No. 28, p. 20, July, 1938.
Sherman, J. B., Proc. IRE, Vol. 26, No. 16, p. 700,1938.
443
MEASUREMENTS
voltage on the horizontal deflection plate of the cathode-ray tube and
thereby drives the cathode-ray beam across the tube in synchronism with
the oscillator. A reversing switch changes the direction of the motor travel
at the upper and lower limits of the audio-frequency range. The output of
the oscillator actuates the loudspeaker. The sound is picked up by the
microphone and amplified. The output of the amplifier is detected by a
linear or logarithmic detector and fed to a low-pass filter. The output of
the filter is amplified by a d-c amplifier, the output of which is connected
to the vertical plates of the cathode-ray tube. The cathode-ray beam traces
the response charactenstic upon the persistence image screen. The ordinates
are in decibels when the logarithmic detector is used. The ordinates are
proportional to the sound pressure when the linear detector is used. The
LOUD SPEAKER
MICROPHONE
A~r" ~"'''''I
0:0
2 3 .4 5
6
0:0
2
6
"i!""j"cr"",<I
3
4
5
C~i
0:0
I
2
3
4
5
F"REQUENCY IN KC
6
10.lS. Schematic arrangement of the apparatus employing a
thermal noise generator and a band-pass filter for obtaining the
response frequency characteristic of a loudspeaker. A. Input to
the loudspeaker. B. The response frequency characteristic of the
band-pass filter. C. Response frequency characteristic of the loud­
speaker.
FIG.
time required to trace a response frequency characteristic of a loudspeaker
is about 30 seconds. The apparatus is very useful for development work
because the motor sweeps through the range again and again. The operator
is free to make changes in the equipment under test and note these changes
upon the response. In case it is desirable to record the characteristic, this
may be done photographically or by tracing the curve left upon the screen.
A system 48 for measuring the response of a loudspeaker employing a
thermal noise generator is shown in Fig. 10.15. A diode may be used as
a source of thermal noise. The output is amplified, filtered, and fed to a
loudspeaker. The frequency distribution of the energy fed to the loud­
speaker is shown in Fig. 10.lSA. The output of the loudspeaker is picked
up by the microphone, amplified, and passed through a narrow band-pass
filter. The response characteristic of the filter is shown in Fig. 1O.lSB.
The band width of the filter should be independent of the frequency. The
48
Olney, B., Jour. Acous. Soc. Amer., Vol. 13, No. 1. p. 79, 1942.
444
ACOUSTICAL ENGINEERING
position of the band-pass filter is varied with respect to frequency. The out­
put of the filter is detected and measured by means of a meter. The
response characteristic of a loudspeaker is shown in Fig. 1O.15C.
Apparatus employing thermal noise for obtaining response characteristics
has not been developed to the stage where it may be used with the facility
of other methods. It appears, however, that this type of measurement
will become very important for all types of acoustical measurements when
suitable apparatus has been developed.
3. Calibration of the Sound Measuring Equipment. 49-The microphone
should be calibrated in terms of the pressure in a free progressive sound
wave. The microphone, amplifier, and detector should have a combined
characteristic which is substantially independent of the frequency over the
frequency range under consideration. If it is not substantially constant
over the frequency range the data must be adjusted for known variations.
MEASURING
SYSTEM
FIG. 10.16. Schematic arrangement for obtaining the factor
formula for absolute response of a loudspeaker.
Pie in the
A general schematic circuit arrangement showing one specific way to
obtain the factor Pie in the formula for absolute response (equation 10.20)
is shown in Fig. 10.16. This arrangement has the feature that it does not
require an absolute calibration of the measuring system.
Referring to Fig. 10.16, the absolute response is given by
Absolute response, in decibels,
where A
=
20 10glO ~
=
[A - B - C - DJ
v'ZE
+
10 10glO ZE
10.29
output of measuring system, in decibels, with the microphone
picking up sound from the loudspeaker with S open,
B = output of measuring system, in decibels, with S closed and the
microphone shielded from sound,
C = open-circuit voltage output of the microphone, in decibels
above 1 volt for 1 dyne per square centimeter, in a free
progressive wave,
D = 20 log el/e2 = 20 log (rEi
rE2)/rE2, and
ZE = electrical impedance of loudspeaker, in ohms.
=
+
49
Standards on Electroacoustics, Institute of Radio Engineers, 1938.
MEASUREMENTS
445
should be sufficiently small compared to the electrical impedance of the
microphone or, in other words, the output of the microphone should not
change when rE2 is short circuited. rEl should be so selected as to obtain
a value of B in the range of the values obtained for A.
4. Free-Field Sound Room. 50 ,5l,52-Acoustical measurements under free­
field conditions are required in the development of the major portion of
electroacoustic transducers. The most obvious and direct solution would
seem to be to make the measurements out of doors at a great distance from
all reflecting surfaces. There are several objections to outdoor testing, for
example, interruptions due to wind, rain, and snow; noise, both natural
and man made; difficulty in arranging experiments at sufficient distance
from the earth so that reflections will be negligible. In view of the im­
portance of free-field testing and the objections to outdoor arrangements,
it is obvious that a free-field sound room is an almost indispensable part of
the equipment of an acoustical laboratory. It is the purpose of this section
to describe such a sound room.
The objective in the design of a free-field sound room is to reduce to a
negligible amount all reflections from the boundary surfaces of the room.
This is equivalent to a very small ratio of generally reflected to direct sound.
The ratio of the generally reflected to the direct sound in a room is
rE2
where ER =
ED =
D =
S
V
a
=
=
ER!ED = 1&nD2 (1 - a)jaS
10.30
energy density of reflected sound, in ergs per cubic centimeter,
energy density of the direct sound, in ergs per cubic centimeter,
distance from the source to the observation point, in centi­
meters,
area of absorbing material, in square centimeters,
volume of room, in cubic centimeters, and
absorption coefficient (see Sec. 11.2A).
An examination of equation 10.30 shows that the ratio of reflected to
direct sound may be reduced by decreasing the distance between the source
and observation point, by making the absorption coefficient of the walls
near unity, or by increasing the area of the walls. In other words, free­
field conditions are approached by making the room large and absorption
coefficient of the wall near unity. To satisfy the first requirement, the free­
field room was made as large as seemed practical from an architectural
and constructional standpoint. The dimensions of the free-field sound
room, before acoustical treatment was applied, were as follows: 48 feet
50 The term free-field sound room is used to designate a room in which free-field
sound conditions are obtained, that is, a room in which the reflections from the
boundaries are negligible. These rooms have also been termed anechoic rooms.
The word anechoic is made up of the Greek prefix an, meaning not or without, the
Greek word echo, meaning echo and the adjectival suffix ic, meaning characterized
by (see Beranek, Ref. 40).
51 Olson, H. F., Jour. Acous. Soc. Amer., Vol. 15, No.2, p. 96,1943.
52 Beranek, L., lOUf'. Acous. Soc. Amer., Vol. 18, No. 1, p. 140, 1946.
446
ACOUSTICAL ENGINEERING
long, 36 feet wide, and 36 feet high. The next objective was to obtain an
absorption coefficient as near unity as possible. The high- and low-fre­
quency ranges present the greatest difficulty in attaining this objective.
It is a comparatively simple matter to attain high absorption in the mid­
frequency range. In the high-frequency range the principal difficulty is
reflection from grills, control boxes, and test apparatus. These reflections
can be eliminated by acoustical treatment of these reflecting surfaces. In
the case of the low-frequency range it appears to be an inexorable fact that
the ideal objective can be attained only in a relatively large room with
correspondingly thick absorption material. An examination of existing
rooms indicates that regardless of the form of treatment it appears that
absorption deviates quite rapidly from unity when the thickness of the
treatment is less than a quarter wavelength. In this statement, it is
assumed that thickness of the material is measured to an outside boundary
of relatively high acoustical impedance compared with the characteristic
acoustical impedance of air. It is also assumed that treatment does not
involve resonant systems.
The absorbing system employed in this room is of the baffle type, that
is, strips of absorbing material arranged normal to the walls of the room as
shown in Fig. 10.17. Several years ago a smaller room (22 feet long, 20
feet wide, and 13 feet high) was treated with baffles. The performance of
this room appeared to be comparable to rooms with equivalent thickness
of other types of absorbing material. The advantage of the baffle type of
treatment is the relatively simple construction and lower cost as compared
with more elaborate absorbing systems.
Plan and elevation views of the room are shown in Fig. 10.17. One­
inch Ozite is spaced 1 foot from the walls, ceiling, and floor. One-inch Ozite
baffies, 7 feet in length and spaced 2 feet apart, are placed normal to the
walls, ceiling, and floor. Four-foot baffles of the same material are placed
between the 7-foot baffles. The total thickness of the absorbing material,
measured from the outside wall, is 8 feet. This leaves the inside dimensions
of the room 32 feet long, 20 feet wide, and 20 feet high. A special grill,
12 feet wide and 24 feet long, is supported on vibration-isolated feet. The
ratio of open to total area in the grill is 0.87. This is a relatively open grill
when it is considered that the grill platform will carry a load of 200 pounds
per square foot. The floor level of the grill is located 11 feet above the
floor level of the room. The floor level of the grill coincides with the first
floor level which makes it readily accessible to the adjoining laboratory.
The acoustical merit of the room can be expressed by the deviation in
sound pressure from an inverse distance characteristic. Pressure response
frequency characteristics were obtained at various distances from a small
loudspeaker. The maximum deviation in pressure from an inverse distance
characteristic for various frequencies is shown in Fig. 10.18. It will be
noted that the deviation in the mid-frequency ranges is negligible. The
deviation at the high frequencies is due to the grill, overhead trolley track,
power and signal outlet boxes. These units which if treated, will make
L
Itt
t
~
,~k
~
!!
~ IIi
r
~
I
L
i
i iIH Ii
SECTION
A·Iit
l!l!
~
Fig. 10.17.
i i Iii !iI
Hi!!--: i
li i d!i H!!d HlHiii 1~ 1:
I
O~~'_~IO
End elevation, plan, and side elevation of a free-field
II
~1=I=~~~WI+l+l-----i-~~ MONORAIL
I
FREE- FIELD SOUND ROOM
ACOUSTICAL ENGINEERING
448
the deviations from an inverse characteristic practically the same as the
mid-frequency range. The deviation at the low frequencies begins when
the thickness of the material is approximately a quarter wavelength. How­
ever, the deviation is only ± 1. 7 db at 40 cycles at a distance of 8 feet. At
40 cycles the thickness of the material is 0.28 of the wavelength.
The absorption coefficient of the walls may be determined from the ratio
of direct to generally reflected sound. These two components may be
BalillH
~ :bJD "IT OJ tml1H
::[11 IHI[' 1111
z
40
100
40
100
1000
Iqooo
40
100
1000
Iqooo
10,000
40
100
1000
CYCLES PER SECOND
10pOO
o
1000
fREQUENCY
IN
FIG. 10.18.
Deviation of the pressure from an inverse distance charac­
teristic for various distances from a sound source in the free-field sound
room.
determined by employing a velocity microphone. Two measurements are
made-one with the normal to the plane of the ribbon passing through the
source and the other with the plane of the ribbon passing through the source.
The absorption coefficient frequency characteristic of the walls of the room
is shown in Fig. 10.19.
A low-noise level is another essential requirement in a free-field sound
room. The noise level in the free-field sound room, when the laboratories
are in normal operation, is about 10 db. At night, when the shops are
---
1.00
I­
~
U
.99
~
-I"'"
..."­
au .98
:3
Z .97
o
j:;
Q.
~ .96
<Il
ID
-<
.9~0
100
fREQUENCY
FIG. 10.19.
room.
IN
CYCLES
1000
PER
10,000
SECOND
Absorption coefficient frequency characteristic of free-field sound
449
MEASUREMENTS
closed down, the noise level is 0 db. This shows that the sound treatment
is also quite effective in absorbing sounds generated outside the room.
The free-field sound room is heated by hot air forced through 48 openings
in the floor. With the blower in operation the noise level in the room is
about 20 db. However, it is not necessary to operate the heater during the
day because the room is very well insulated thermally as well as acoustically.
For example, if the heater is operated 8 hours in every 24 hours, the tempera­
ture variation from 70 0 Fahrenheit is only ±3° Fahrenheit on the coldest
day.
The above data and other measurements show that it is possible to make
measurements in this room under essentially free-field conditions over the
A
FACE VIEW
Ij;,>;,'&~tj;;~\i·dkm?4
X
MASONRY WALL
SECTION A-A'
rY;)"'j[,;?MJ.(!)~;Xtit#~:l1tlMj
Y
MASONRY WALL
SECTION
s- s'
tr:%~i%\~~i~'1!ii'[email protected]@J
Z
MASONRY WALL
SECTION C - c'
FIG. 10.20.
Wall treatments for free-field rooms, X. Baffle type; Y. Pyramid
type; Z. Wedge type.
frequency range above 40 cycles for distances between the source and obser­
vation up to 8 feet. This distance can be increased if either the source or
the microphone, or both, are directional.
Three common types of sound absorbing systems used for free-field or
anechoic rooms are shown in Fig. 10.20. Fig. 10.20, X depicts the baffle
type of sound absorbing system employed in Fig. 10.17 and described in this
section. In the system described in this section the spacing between the
baffies is 12 inches. Each baffle is made up of two layers of i-inch Ozite with
actual over-all thickness of 2 inches because the material was obtained in
the uncompacted form. Employing twice as many baffles would improve
450
ACOUSTICAL ENGINEERING
the absorption in the low-frequency range by a small factor. Such an
improvement is not of a practical significance. In the absorbing system
shown in Fig. 10.20, Y and Z are of the pyramid and wedge types made of
Fibreglass. It appears that in the system of Fig. 10.20, Z exhibits the
greatest absorbing efficiency. In this connection, it should be mentioned
that absorbing efficiency beyond a certain value is of little practical sig­
nificance under actual operating conditions when reflecting surfaces almost
invariably are introduced in any measurement. In any case, the portion
of the frequency range in which it is most difficult to obtain free-field condi­
tions is in the low-frequency range. As previously stated, regardless of the
form and material of the treatment employing existing nonactive materials,
the absorption deviates rapidly from unity when the depth of the material
is less than one-quarter wavelength. Furthermore, free-field conditions can
only be obtained when the dimensions of the room are greater than a wave­
length.
5. Outdoor Response.-If a free-field sound room is not available, free­
field conditions may be obtained outdoors by locating the microphone and
loudspeaker at a sufficient distance from reflecting surfaces so that the level
of the direct sound striking the microphone is at least 20 db above the re­
flected sound. The microphone and loudspeaker may be suspended on
a cable between two high towers. A velocity microphone may be used to
discriminate against the reflected sound if there is only one reflecting sur­
face, as, for example, the earth, by orienting the microphone so that the
plane of the ribbon coincides with the direction of the reflected sound.
Outdoor measurements have the disadvantage of being dependent upon
the weather and noise conditions. For this reason, nearly all development
and routine work on loudspeakers'is carried on in rooms.
6. Small and Partially Deadened Rooms.-When only a small deadened
room or a partially deadened room is available, the distance between the
microphone and loudspeaker must be small in order to reduce reflection
errors. A response frequency characteristic taken under these conditions
is useful in determining system resonance and general smoothness of the
output.
When the distance between the microphone and loudspeaker, in a partially
deadened room, is large, a rotating microphone or warble tone may be used
to reduce reflection errors.
In the case of the rotating microphone, the microphone is revolved in
a circle about 5 feet in diameter. The plane of the circle is inclined at an
angle of 30° toward the horizontal. The microphone is arranged so that
it is always directed toward the source of sound.
In the case of the stationary microphone, a warble frequency (20 cycles
+ 10 per cent of the mean audio frequency as a maximum total band width)
may be used to average out reflection errors. This method tends to average
out very abrupt variations in the loudspeaker response. A check response
frequency measurement taken close to the loudspeaker with no warble
should be made to determine if there are any abrupt variations in its response.
MEASUREMENTS
451
7. Arrangement of Loudspeakers for Test.-In obtaining response fre­
quency characteristic of loudspeakers, the systems may be divided into two
classes-namely, direct radiator, loudspeaker units designed to operate in
some additional structure and complete systems such as direct radiator
mechanisms mounted in cabinets and horn loudspeakers.
In the test of direct radiator, loudspeaker units alone, the unit should
be mounted 1 foot off center in a direction parallel to one side and 6 inches
off center in a right-angle direction in a square and flat baffle 12 by 12 feet.
The baffle should be of sufficient thickness so that no radiation results from
vibration of the baffle. The microphone should be located on the axis
of the radiator 5 feet from the surface of the baffle when the transverse
dimension of the radiator is not more than 2t feet. For larger radiators,
the distance should be the smallest integral multiple by 5 feet, which is
greater than twice the maximum traverse dimension of the radiator and
should be specified with the test.
Complete loudspeaker systems such as direct radiator mechanisms
mounted in cabinets and horn loudspeakers are tested in the same manner
as in the case of direct radiator, loudspeaker units, but without the use of
additional baffles.
8. Living Room Measurements. 53-The performance of a radio receiver
in a living room will be discussed in Sec. 11.2R. The characteristics shown
in Fig. 11.19 were obtained with the cathode-ray response measuring system
described in Sec. 1O.3A2. However, any of the systems described in Sec.
1O.3A2 may be used. It is customary to obtain a large number of charac­
teristics for each position of the receiver in the room.
9. Theater Measurements.-The performance of a loudspeaker in a theater
will be discussed in Sec. 11.2G. The characteristics for the various parts
of the theater may be obtained with any equipment described in Sec. 1O.3A2.
However, the high-speed response measuring equipments are preferable for
this type of work.
10. Automobile Measurements.-The conditions under which an auto­
mobile radio receiver operates differ widely from those of a loudspeaker
in a room. For this reaS(lU it is very important to test the performance
under actual operating conditions. The response frequency characteristic
should be obtained by placing the microphone at the ear position in each
of the normal listening positions in the automobile. In the case of back­
seat measurements persons should be seated in the front seat to simulate
actual conditions. Measurements should be made with the windows open
and closed. In general, the response frequency characteristics will differ
widely for the front and back seats. It is customary to favor the front
seats in determining the optimum response frequency characteristic. In
some radio installations in automobiles an auxiliary loudspeaker is installed
in the deck behind the rear seat. When this loudspeaker is used response
should be obtained with both front and rear loudspeakers in operation.
At high speeds, wind, road rumble, and engine noises are quite high and mask
53
Wheeler and Whitman, Proc. Inst. Rad. Eng., Vol. 23, No.6, p. 610, 1935.
452
ACOUSTICAL ENGINEERING
the reproduced sound. The power output should be sufficient to override
these noises and give intelligible speech. In view of the fact that the sound
level delivered by the loudspeaker is quite high under these conditions, it is
important that the response frequency characteristic be smooth, otherwise
the reproduced sound will be disagreeable.
The response frequency characteristics may be obtained with any equip­
ment described in Sec. lO.3A2.
B. Directional Characteristic.-The directional characteristic of a loud­
speaker is the response as a function of the angle with respect to some axis of
the system. The characteristics may be plotted as a system of polar
characteristics for various frequencies or as response frequency charac­
teristics for various angles with respect to the reference axis.
The directional characteristics of a direct radiator loudspeaker in a very
large baffle may be obtained at a distance of 5 feet. For a small baffle or
cabinet the distance should be at least three times the largest linear dimen­
sion of the system. The directional characteristics of a horn loudspeaker
should be obtained at a distance three or more times the largest dimension
of the mouth.
Obviously, very slight reflections will introduce considerable error for
angles in which the response is very low. For this reason, it is almost
imperative that the measurements be made under free-field conditions.
Apparatus for obtaining the directional pattern of a microphone has been
described in Sec. 1O.2B and depicted in Fig. 10.6. The same system may
be used to obtain the directional pattern of a loudspeaker. In this case
the loudspeaker and microphone are interchanged, that is, the microphone
is fixed and the loudspeaker rotated.
C. Nonlinear Distortion Characteristic.-The nonlinear distortion charac­
teristic of a loudspeaker is a plot of the total distortion in per cent versus
the frequency at a specified input power. A plot of the individual com­
ponents of the distortion in per cent versus frequency is also used to depict
the distortion characteristic of a loudspeaker.
The apparatus and circuit in schematic form for measuring the distortion
produced by a loudspeaker are shown in Fig. 10.21. Great care must be
taken to avoid appreciable harmonics in the sound generating and sound
measuring equipment. To reduce the already low harmonic content in the
signal generator to a negligible amount a variable cutoff low-pass electrical
filter, admitting only the fundamental, should be employed. The microphone
and amplifiers may be the same as those used for response measurements.
The harmonic analyzer may be any of the various types employed in
distortion measurements on amplifiers.
In making the test, the output of the power amplifier is connected to the
loudspeaker. The sound is picked up by the microphone and then amplified
and the measurement of harmonics is carried out in the conventional manner.
The output switch is now thrown to the dummy load, the electrical resistance
of which should be the same as the electrical impedance of the loudspeaker
at the measurement frequency. The variable attenuator is adjusted until
453
MEASUREMENTS
the output of the microphone amplifier is the same as that obtained with the
sound. The harmonic content under these conditions should be negligible.
The purpose of this operation is to insure that no distortion is introduced by
the associated measuring equipment. In the above discussion the possible
distortion in the microphone has been neglected. The distortion generated
by the microphone may be measured as outlined in Sec. 1O.2C.
VACUUM
TUBE
VOLTMETER
HARMONIC
ANALVZER
FIG. 10.21. Schematic arrangement of the apparatus for measuring the non­
linear distortion of a loudspeaker.
20
o
v
'" -20
o
~
/
/
....
~-40
oDo
en
~-60
-80
VACUUM
TU8E
VOLTMETER
II
V
"
\
\
r-...
"
-100
6420246
CYCLES OF RESONANCE
SCHEMATIC DIAGRAM
FIG. 10.22.
analyzer.
Schematic arrangement of the elements in a heterodyne-type harmonic
Harmonic distortion measurements should be made in a free-field sound
room or outdoors to eliminate errors due to standing waves. If it is neces­
sary to make these measurements in a room other than a free-field room
they should be made under a sufficient variety of conditions with respect to
frequency and microphone placements to give average values which are not
appreciably affected by the errors associated with room reflections.
The heterodyne analyzer 54 is shown schematically in Fig. 10.22. The
.4
Arguimbau, L. B., General Radio Experimenter, No.8, p. 1. June, July, 1933.
454
ACOUSTICAL ENGINEERING
incoming signal, mixed with a carrier supplied by the heterodyne oscillator
is fed to the modulator. A balanced modulator is usually used so that the
carrier will be suppressed. The heterodyne oscillator is adjusted so that
the sum of its frequency and that of one of the components of the signal
equals the pass band of the highly selective tuned amplifier. The high
selectivity is usually obtained by means of a quartz filter. The upper side
band is passed through the selective tuned amplifier, detected and then
measured on a meter.
A recording harmonic analyzer consists of a heterodyne harmonic analyzer
coupled with a level recorder as shown in Fig. 10.23. The conventional
harmonic analyzer is equipped with a dial calibrated in frequency. This dial
can be coupled to the driving system of a high-speed level recorder. The
HARMONIC
ANALYZER
LEVEL
RECORDER
INPUI~Ti.~::==~~~t======2~C:~==~:::~
FIG. 10.23. Schematic arrangement of the apparatus used in a recording
'larmonic analyzer of the heterodyne type.
electrical output of the heterodyne harmonic analyzer is coupled to the input
of the level recorder. In this manner the components in the complex wave
input to the heterodyne harmonic analyzer may be automatically recorded
by the high-speed level recorder.
A panoramic analyzer 55 is a system which presents the components of a
complex wave input in the form of a response frequency characteristic on
the screen of a cathode-ray tube with a persistence image screen. Fig.
10.24. The system is basically the heterodyne harmonic analyzer of Fig.
10.22. The output of a low-frequency oscillator is converted into a saw­
tooth wave by the saw-tooth generator. This wave is amplified by the
horizontal deflection amplifier. The output of the horizontal deflection
amplifier is coupled to the horizontal deflection system of the cathode-ray
tube. The saw-tooth generator is fed also to the controlled oscillator. This
oscillator supplies the heterodyne signal to the modulator. The complex
wave signal is amplified and fed to the modulator. The output of the
modulator passes through a tuned amplifier with a narrow band-pass charac­
55 Richard, Smith, and Stephens, Trans. IRE, Prof. Group Audio, Vol. AU-3,
No.2, p. 37, 1955.
455
MEASUREMENTS
teristic. The output of the tuned amplifier is amplified by the vertical
deflection amplifier. The output of the vertical deflection amplifier is
coupled to the vertical deflection system of the cathode-ray tube. The
characteristic depicted on the cathode-ray tube represents the output as a
CATHODE RAY TUBE
WITH PERSISTANCE
IMAGE SCREEN
INPUT
~~r~~~~~~ r-----1
AMPLIFIER
10.24. Schematic arrangement of the elements used in a heterodyne analyzer in
which the components of a complex wave are depicted as a response frequency charac­
teristic on a cathode-ray tube.
FIG.
function of the frequency. The vertical scale may be either linear or decibel.
The frequency scale is logarithmic.
The balance bridge for measuring the total distortion is shown sche­
matically in Fig. 10.25. A part of the output of the oscillator is fed to the
apparatus to be tested and another part to the analyzer. The amplitude
and phase relations of the fundamentals from the oscillator and apparatus
VACUUM
TUBE
VOLTMETER
10.25. Schematic arrangement of the elements in a balance bridge
harmonic analyzer.
FIG.
to be tested are adjusted by means of suitable networks so that none of the
fundamentals remains. The remainder is the total harmonic generated by
the system under test. This is measured by means of a root-mean-square
meter.
A cathode-ray oscillograph is often used to depict the wave form and
thus obtain an indication of the departure from a pure sine wave. A
schematic diagram of the apparatus employing a cathode-ray oscillograph
to indicate the extent to which distortion is introduced by a loudspeaker
is shown in Fig. 10.26. When the switch is thrown to the right and the sine
wave generated in the oscillator is sent into the amplifying system through
the attenuator, the cathode-ray oscillograph should show a pure sine wave
456
ACOUSTICAL ENGINEERING
form over the entire audio-frequency range considered. The attenuator is
adjusted to give the same amplitude of the wave pattern on the oscillograph
screen as is secured when the switch is thrown to the left and the power is
supplied to the loudspeaker. With the switch in the latter position the
microphone picks up the sound and the wave form is reproduced upon the
fluorescent screen of the oscillograph. The departure from the pure sine
VACUUM
TUBE
VOLTMETER~--~:::~~____,
CATHODE - RAY
OSCILLOGRAPH
FIG. 10.26. Schematic arrangement of the apparatus employing a cathode­
ray tube for indicating the nonlinear distortion of a loudspeaker.
LOUD SPEAKER
MICROPHONE
r-----------,
800 CYCLE
HIGH PASS
fiLTER
FIG. 10.27. Schematic arrangement of the apparatus for measuring the
nonlinear distortion of a loudspeaker employing the intermodulation
method. (After Hilliard.)
457
MEASUREMENTS
wave is indicated readily by the difference in appearance of the pattern
from the pure sine wave form secured with the switch thrown to the right.
The extent of introduction of harmonics by the loudspeaker can be estimated
from a slight, moderate or very marked change in the wave form.
A
INPUT
SIGNAL
B~-------------------+~~----------------~,
OUTPUT
SIGNAL
pf\Aflflflf\I\Q{)"~"""fl
vrvrv vrVlJ V V~-V V VIJ V
c~V
MODULATED
D
HIGH
----
REGTIFIED
MODULATED
ReSULTANT
FREQUENCY
HIGH
fREQUENCY
INTERMODULA'rION
10.28. Typical wave shapes in the various stages of the inter­
modulation system of distortion measurement. A. The input signal to
the loudspeaker. B. The ontput signal of the loudspeaker. C. The
modulated high-frequency output of the band-pass filter. D. The recti­
fied modulated high frequency of the detector. E. The resultant
intermodulation output of the copper oxide rectifier.
FIG.
A schematic diagram of the apparatus for the intermodulation method 56
of measuring nonlinear distortion is shown in Fig. 10.27. Two tones are
impressed upon the loudspeaker to be tested. The low-frequency tone
may be 40 or 60 or 100 cycles and the high-frequency tone may be 1000 or
7000 or 12,000 cycles. The wave shape of the input signal to the apparatus
under test is shown in Fig. 1O.2SA. The output of the microphone is
56
Hilliard. J. K.. Proc. Inst. Rad. Eng.• Vol. 29. No. 12. p. 614.1941.
458
ACOUSTICAL ENGINEERING
shown in Fig. 1O.28B. This output is fed to an SOO-cycle high-pass electrical
filter. If nonlinear distortion is produced by the equipment under test,
the high-frequency output from the electrical filter will be modulated as
shown in Fig. 10.28C. Beyond the electrical filters the signal is amplified
and impressed upon a full-wave detector. The output of the detector is
shown in Fig. 10.2SD. The output of the detector is passed through a
200-cycle low-pass electrical filter. The output of the low-pass electrical
filter is shown in Fig. 10.2SE. The output of the 100-cycle low-pass
electrical filter is fed through an amplifier which removes the d-c electrical
component. The final resulting a-c electrical intermodulation component
is measured by means of a copper oxide rectifier meter. An approximate
relation between the intermodulation and harmonic terms may be developed.
It appears that, in general, the intermodulation terms are approximately
four times the harmonic terms. For example, if certain apparatus is found
to have 1 per cent total distortion in harmonics, an intermodulation test
will show intermodulation products of 3 to 4- per cent when the amplitude
of the higher frequency is 12 db below the amplitude of the lower frequency.
A more universal method 57 for intermodulation testing consists of three
radio frequency signal generators, one fixed and two variable. The system
can provide one frequency, /I from 20 to 20,000 cycles and a second fre­
quency h higher than!l by a fixed amount which may be anywhere between
o and 10,000 cycles. As the frequency /I is varied, the difference frequency
remains constant. The system can provide one fixed frequency and a
variable higher frequency. The analyzer of Fig. 10.22 may be used as the
detector.
The automatic nonlinear distortion analyzer 58 consists of the conventional
system for obtaining a response frequency characteristic of a loudspeaker
coupled with an automatic means for suppressing the fundamental fre­
quency, see Fig. 10.29. The loudspeaker is supplied by a pure tone from
a low distortion oscillator and power amplifier combination. The sound
output of the loudspeaker is picked up by a calibrated microphone. Both
the loudspeaker and the microphone are located in a free-field room. The
output of the microphone is amplified and fed to a recorder, and a response
frequency characteristic of the loudspeaker is obtained from this system in
the conventional manner. To obtain the distortion frequency characteristic
which depicts the distortion generated by the loudspeaker as a function of
the frequency, the system for automatically suppressing the fundamental is
connected between the microphone, amplifier, and the recorder. Under
these conditions, the voltage applied to the recorder is the root-me an-square
total of the harmonic frequencies generated by the loudspeaker.
In the system described above, the major problem becomes one of attenu­
ating the fundamental frequency in a dependable and fairly rapid manner.
Of the several methods available for eliminating the fundamental, a reliable
and straightforward one is shown in Fig. 10.29. This consists of a series of
57
58
Peterson, A. P. G., Gen. Radio Exp., Vol. 25, No.3, 1950.
Olson and Pennie, RCA Review, Vol. 12, No. 1. p. 35, 1951.
459
MEASUREMENTS
high-pass filters, sequentially interposed between the microphone pickup
and the recording equipment to attenuate the 40- to IS,OOO-cycle sweep
fundamental. The primary advantage of using this method for harmonic
distortion measurements is its dependability. The filters themselves may
be made very rugged. Furthermore, should the filter switching circuit fail
to function properly, the distortion readings will immediately go to 100 per
cent, thus reading fundamental rather than harmonic, and thereby providing
a positive check against a possible switching error.
The useful frequency range of each filter is determined by two frequencies,
namely, fe and f 00" The frequency at which the response is down one
r------------..,
I
I
I
I
I
RECORD
• 1()4
FIG. 10.29.
Schematic arrangement of the apparatus employed in the automatic non­
linear distortion recording system. The graph shows the response-frequency charac­
teristic and the distortion frequency characteristic of a typical commercial 12-inch
loudspeaker.
decibel is fe. This frequency sets the lower limit of the useful pass band of
each high-pass filter when recording distortion to an accuracy of 10 per cent.
The frequency at which the response is down sixty decibels is fr;o This
frequency sets the upper limit of the useful rejection band of each filter
when recording distortion to an accuracy of 10 per cent for a 0.3 per cent,
2nd harmonic distortion value. The response frequency characteristics of
the filters are shown in Fig. 10.30. The fe and fx) overlap characteristic of
adjacent filters are very close at the lower frequencies; and it, therefore,
becomes very important that the frequency at which a filter is switched be
held to a close frequency tolerance if the full possible accuracy of the distor­
tion analysis is to be realized. For this reason, an electronic rather than a
mechanical system for the detection of the switch frequency is used. A
460
ACOUSTICAL ENGINEERING
typical bridge-T network employed for switch frequency detection is shown
in Fig. 10.31. The response frequency characteristic of the network is also
shown in Fig. 10.31.
For a loudspeaker distortion analysis a conventional amplitude versus
frequency response curve is first run with the power amplifier adjusted to
furnish the proper power level to the loudspeaker under test, and with the
distortion analyzer step switch solenoid power turned off. The recorder
preamplifier is adjusted to a level such that the recorder will not go .off scale
for the response curve. The response frequency characteristic is run with
this gain setting. This procedure is repeated with the distortion analyzer
(
2
{
3
4'
6{
5
1/
8
f
9
10'
II
r ,12
I.
{
14
15
-10
-20
m
o
""
~ -30
Z
.
o
III
"'" ·40
\J
-50
-60
~
40
I
80
100
200
I
400
8001000
FREQUENCY IN CYCLES
2000
4000
8000 10000 15000
PER SECOND
FIG. 10.30. Response frequency characteristics of the fourteen high-pass filters used in
the system of Fig. 10.29.
step switch solenoid power turned on and with the gain control of the pre­
amplifier turned some 20 to 40 decibels higher. The resultant character­
istics, with due consideration for the difference in preamplifier settings,
gives the relative distortion frequency characteristic of the loudspeaker
under test, see Fig. 10.29.
D. Efficiency Frequency Characteristic. 59-The efficiency of a loudspeaker
at any frequency is the ratio of the total useful acoustical power radiated
to the electrical power supplied to the load, the current wave of which
exercises a controlling influence on the wave shape of the sound pressure.
The plot of efficiency, in per cent, versus frequency, in cycles, is termed the
efficiency frequency characteristic.
The measurement of efficiency of a loudspeaker may be divided into two
methods, direct and indirect. One direct method depends on measuring
the total energy flow through a spherical surface without reflections. Several
59
Standards on Electroacoustics, Institute of Radio Engineers, 1938.
MEASUREMENTS
461
indirect methods have been developed. The most common of these consists
in measuring the electrical impedance under various conditions of diaphragm
loading. It has been found in practice that these two methods of deter­
mining efficiency are those most widely used at the present time.
1. Direct Determination oj Radiated Power.-The sound power output
from a speaker at a particular frequency may be obtained by measuring
the total flow of sound power through a spherical surface of which the sound
source is the center. The surface of the sphere is divided into incremental
areas and the power transmitted through each area is determined from the
sound pressure and the particle velocity as well as the phase displacement
10
o
CD
o
""\
~
-10
,.-­
~-20
z
:r
.....-SWITCH
LEVEL
(/)
~-30
ELECTRICAL
NETWORK
-50
-6
-3
0
3
6
FREQUENCY
FIG. 10.31. The electrical diagram and response frequency
characteristic of the bridged T network used for frequency
detection in the switching system of Fig. 10.29.
between them. To simplify the process, the measurements may be made at
a distance sufficiently large so that these quantities are in phase. Then,
the radiated power may be determined by measuring the sound pressure or
particle velocity over each incremental area (assuming the measuring
equipment does not disturb the sound field and no standing wave pattern
exists). The total power is equal to the summation of the power transmitted
through the incremental areas and may be expressed as
PA =
where P A
=
p
=
c=
p=
dS =
:c JJ
p2
dS X 10-7
10.31
total acoustical power, in watts,
density of the medium, in grams per cubic centimeter,
velocity of sound in medium, in centimeters per second,
root-mean-square pressure, in dynes per square centimeter,
over the element of areas dS, and
element of area on spherical surface, in square centimeters.
462
ACOUSTICAL ENGINEERING
The input electrical power can be determined from the electrical current,
voltage, and phase angle, while operating under the above conditions.
The efficiency, fL, in per cent, is then
fL
where P A
PE
=
=
=
PA
10.32
P E X 100
total acoustical output, in watts, and
electrical input, in watts.
As previously mentioned, the loudspeaker should be located so that the
reflected energy reaching the measuring equipment is negligible. This
means that the measurements must be made either in a free-field sound room
AMPLIFIERS
r-------------,I
I
:
SQUARE
LAW
DETECTORS
I
I
I
I
I
I
I
I
POWER
AMPLIFIER
I LOUD
ISPEAKER
I
I
I
IL ____________ ....JI
D-C
AMPLIFIER
OSCILLATORr==~~~r===::~~~~J
SHAFT
LEVEL
DRUM
PEN
RECORDER
LOGARITHMIC
AMPLIF'lER
10.32. Schematic arrangement of the apparatus for obtaining the total
sound power output frequency characteristic of a loudspeaker.
FIG.
or in free space. The measurements and computations in this method are
quite laborious. On the other hand, there can be no question as to the
validity of results which are obtained if the test is carefully conducted.
Because of its fundamental nature and validity, the direct method is usually
considered standard for determining loudspeaker efficiency.
The procedure outlined above is quite laborious and time consuming.
Apparatus has been developed in which the total integrated power output
frequency characteristic of a loudspeaker can be obtained in a manner
comparable to that of a response frequency characteristic.
The schematic arrangement of the apparatus used for obtaining the total
output frequency characteristic of a loudspeaker is shown in Fig. 10.32.
The total power output is depicted by a single curve on a graph sheet. The
MEASUREMENTS
463
ordinate scale is in decibels. This apparatus approximates the integration
process of equation 10.31. The microphones are placed on the quadrant of
a circle and arranged to intercept equal areas on the surface of a hemisphere.
The measurement assumes that the directional pattern is symmetrical about
the axis of the loudspeaker. If the pattern is unsymmetrical, the loud­
speaker is mounted in a cradle and continuously rotated about the axis.
The measurement covers one hemisphere. A similar measurement can be
made in the other hemisphere if the radiation in the backward direction is
of any consequence.
2. Indirect Determination of Radiated Power.-There are several methods
for determining loudspeaker efficiency by indirect means. The most
common method is to measure the electrical impedance under various con­
ditions of diaphragm loading.
A one-to-one ratio bridge, capable of measuring the electrical impedance
at the full power output of the speaker, should be used. Care should be
taken that the temperature of the yoice coil does not vary appreciably
during the various measurements. The power supply for driving the speaker
and bridge should be reasonably free from harmonic distortion.
The motional electrical impedance method 60 is generally applied to
moving coil electrodynamic speakers in which the force factor is real. In
case the force factor is imaginary it becomes rather complicated to employ
the motional electrical impedance method.
The efficiency ft, in per cent, by the motional electrical impedance method
is given by
rEM
rEN
ft = -
where rEM
rEN
=
rED
=
=
X
100
10.33
rEN - rED motional electrical resistance, in ohms,
resistive component of the electrical impedance with the
system in the normal state, in ohms, and
damped electrical resistance with the vibrating system
blocked, in ohms.
This equation describes the simplest method of determining the effi­
ciency from motional electrical impedance measurements when the electro­
mechanical coupling factor is real (see Chapters VI and VII). It assumes
that the entire value of the motional electrical resistance may be attributed
to radiation acoustical resistance. This method adds the radiation from
both sides of the diaphram and, therefore, assumes that the radiation from
both sides is useful. It assumes that there are no mechanical losses in the
diaphragm and suspension system. These losses can be determined from
the measurements of the motional electrical impedance in a vacuum. Of
course, in this case, the load on the diaphragm is not normal and the losses
may be quite different from those which obtain under actual operating
60
Kennelly and Pierce, Proc. A. A. A. S., Vol. 48, No.6, 1912.
464
ACOUSTICAL ENGINEERING
conditions. This method also assumes that there are no losses due to viscous
air friction. Since the amplitude of the vibration of a voice coil is normally
small at the higher frequencies, the problem of blocking the voice coil
against motion is not a simple matter. Obviously, any motion will intro­
duce an error in the determination of the efficiency.
E. Phase Distortion Characteristic.-The phase distortion characteristic
of a loudspeaker is a plot of the phase angle between the sound output and
some reference sound as a function of the frequency.
Two microphones and separate amplifiers and a cathode oscillograph
may be used as outlined in Sec. lO.2D, Fig. 10.8. A reference sound may
be set up by a separate loudspeaker, in which the phase shift is small, and
SQUARE
WAVE
GENERATOR
UNDER
TEST
SQUARE
CATHODE­
RAY
OSCILLOGRAPH
-~L----+<>S
I
WAVE
GENERATOR
S
APPARATUS
LOUD SPEAKER
MICROPHONE
CATHODE­
RAY
OSCILLOGRAPH
FIG. 10.33.
Schematic arrangement of the apparatus employing a square
wave generator and a cathode-ray tube for indicating the transient response
characteristics of acoustical apparatus such as microphones or loudspeakers.
picked up by one microphone. A reference voltage source may be substi­
tuted for the reference microphone. The sound from the loudspeaker to be
tested may be picked up on the other microphone. The phase difference
may be determined as outlined in Sec. lO.2D. The phase distortion is of
importance in the overlap region of the multiple channel systems. In this
case the phase shift may be several hundred degrees (equivalent to a sound
path difference of several feet), see Sec. 7AB.
F. Electrical Impedance Frequency Characteristic.-The electrical imped­
ance characteristic of a loudspeaker is the electrical impedance at the input
terminals as a function of the frequency. The plot of the characteristic
should also include the resistive and reactive components of the electrical
impedance.
A one-to-one ratio electrical impedance bridge may be used and should
465
MEASUREMENTS
be capable of measuring the electrical impedance at the full power output
of the speaker. The power input should be included with every electrical
impedance characteristic. If the electrical impedance characteristic varies
with power input, it is desirable to show a series of electrical impedance
frequency curves for various inputs. Other methods may be used as, for
example, the three voltmeter and a known electrical resistance method.
G. Transient Response Characteristic.-The measurements in the pre­
ceding sections have been concerned with steady state conditions. In all
types of sound reproduction the phenomena is of a transient character.
For this reason it is important to measure the response of the system to a
suddenly applied force or voltage. The Heaviside Operational Calculus is
a very powerful tool for predicting the performance of a system to a suddenly
applied force or voltage (see Sec. 6.25.)
The apparatus for investigating the transient response of an audio system
is shown schematically in Fig. 10.33. The output of a square wave generator
is fed to the apparatus to be tested. The output of the apparatus under test
SOUND ROOM
CATHODE RAY
OSCILLOSCOPE
FIG. 10.34.
Schematic arrangement of the apparatus used in the tone burst
system for depicting the transient response of a loudspeaker.
is fed to a cathode-ray oscillograph. The deviation from the square wave
is shown on the screen of the cathode-ray oscillograph. Square waves 61
offer a simple and rapid method of including both phase shift and amplitude
response in a single test.
A measure of the transient response of a loudspeaker may be obtained
by measuring the response to an electrical input in the form of a tone burst. 62
A tone burst is a term usually used to designate a wave with a rectangular
envelope. A schematic diagram 63 of the apparatus for measuring the
response of a loudspeaker to an electrical input in the form of a tone burst
is shown in Fig. 10.34. An oscillator provides the sine wave signal which is
Kallmann, Spencer, and Singer, Proc. Inst. Rad. Eng., Vol. 33, No.3, p. 169, 1945.
Olson, H. F .• Audio Eng.• Vol. 34. No. 10. p. 15. 1950.
63 Corrington. M. S.• Jour. Audio Eng. Soc .• Vol. 3. No. 1. p. 35. 1955.
61
62
466
ACOUSTICAL ENGINEERING
fed to an electronic gate which intermpts the sine wave signal at regular
intervals, thereby producing a series of tone bursts. The tone bursts are
amplified and fed to the loudspeaker. The sound output from the loud­
speaker is picked up on the microphone and amplified and applied to the
vertical deflection system of a cathode-ray oscilloscope. The acoustical
output from the loudspeaker may be compared with the electrical input to
the loudspeaker. Another gate may be employed so that the residue after
the electrical input has ceased may be displayed on the oscilloscope. A
delay network is required to compensate for the time required for the wave
to travel from the loudspeaker to microphone.
H. Subjective Measurements. 64 _A subjective test of a loudspeaker
involves a determination of some of the performance characteristics by
direct listening to the loudspeaker operating under controlled program and
environment conditions. Listening tests play an important part in research,
development, and commercialization of loudspeakers. Listening tests
range in scope from exceedingly simple comparison tests to elaborately
controlled and conducted tests. In many cases the apparatus for making
all the objective tests outlined in the preceding sections are not available.
Furthermore, there is always some difficulty in evaluating the objective
measurements. For this reason a sUbjective test of efficiency, frequency
response, directional characteristics, nonlinear distortion, and transient
response, in which two or more loudspeakers are compared with each other
or with a reference loudspeaker, is widely used. The subjective test
may be used to determine the lumped effects of the following factors:
loudness, frequency range, tone balance, spacial distribution, quality, and
hangover. It is the purpose of this section to describe listening tests of loud­
speakers.
1. Loudspeaker Environment.-The listening test of a loudspeaker should
be conducted in the environment for which it was designed to {)perate.
Specifically, a loudspeaker designed for home-type radio receivers, phono­
graphs, and television receivers should be tested in a room with dimensions
and acoustics similar to those of the average living room in the home. A
loudspeaker for an automobile radio receiver should be tested in an auto­
mobile. A loudspeaker for a sound motion-picture system should be tested
in a typical theater. A loudspeaker for a public address, sound reinforcing,
or paging system should be tested under typical surroundings for these
systems.
2. Loudspeaker Housing, Placement, and Mounting.-The horn, baffle,
housing, or cabinet for loudspeaker listening tests should be similar to those
used under actual operating conditions. The placement and mounting
arrangement in the test invironment should correspond to those used in
actual installations.
3. Signal Sound Level.-The signal sound level produced by a loudspeaker
in a listening test should correspond to the sound level under actual operating
conditions in the field. The use of the proper level is very important in
64
Olson, H. F., Trans. IRE, Prof. Group on Audio, Vol. AD-l, No.5, p. 7, 1953.
MEASUREMENTS
4-67
detennining the balance of high, mid, and low frequencies, the distortion,
the transient response, etc., under actual operating conditions. The upper
sound levels in the description which follows do not necessarily represent the
upper power capabilities of the systems. The signal sound level will be
somewhere between 65 and 75 db for a radio receiver, phonograph, or
television receiver operating in a typical or average living room. The
signal sound level will be somewhere from 65 to 100 db for an automobile
radio receiver. The signal sound level will be between 76 to 85 db for speech
reproduction, and 75 to 95 db for music reproduction for a sound motion­
picture reproducing system operating in a theater. The signal sound level
for a public address, sound reinforcing, or paging system will vary over wide
limits depending upon the application. To summarize, the signal sound
level of the test should correspond to the level under actual operating
conditions.
4. Ambient Noise Level.-The ambient noise under which the listening
test is conducted should correspond to the ambient noise encountered under
actual conditions. This involves two main factors, namely, the sound level
and spectrum of the ambient noise. For example, the average ambient noise
sound level in the average living room is 42 db. The average ambient
noise sound level in a theater is also 42 db. In an automobile, the ambient
noise sound level depends upon the speed, open or closed windows, the road,
etc. In public address, sound reinforcing or paging applications, the noise
sound level will vary over wide limits. This must be taken into account, and
the noise conditions under which the equipment will be operated must be
simulated in the listening tests. It is important that the spectrum of the
noise encountered under the actual operating conditions should be simulated
in the subjective tests as well as the noise level.
5. Signal or Program Material.-The signal or program material used in
listening tests should be similar to that encountered in the field. This is
not so for the material presented under" Frequency Range" and" Power
Handling Capacity." A radio or television receiver should be operated
from typical broadcast or television transmitters. Under certain condi­
tions it may be necessary to use the equivalent of a radio or television
transmitter, as, for example, a modulated signal generator. A phonograph
should be operated from typical commercial records. A sound motion­
picture reproducing system should be operated from typical sound motion­
picture film. Sound reinforcing systems for use with music should be tested
with musical program material. Public address and paging systems should
be tested with speech as the program material.
6. Reference Systems.-Almost all listening tests on loudspeakers are con­
ducted by comparing the loudspeaker under test with a reference loudspeaker.
The reference system is, in general, a loudspeaker which is similar to the
loudspeaker under test. The loudspeakers should be placed behind a light­
opaque, sound-transparent curtain so that it is impossible to identify the
loudspeakers by sight. A suitable indicator should show which loudspeaker
is operating at any time. In general, the procedures in most listening
468
ACOUSTICAL ENGINEERING
tests are not formalized because the tests are conducted to determine the
engineering and commercial aspects of rather small changes in design. If a
jury-type procedure is used, secret ballots should be taken of the preference.
Statistical methods should be employed in planning and conducting such
jury tests.
7. Relative Loudness Efficiency.-The relative loudness efficiency of a
loudspeaker is determined from a loudness balance. High-quality trans­
formers should be used to match each loudspeaker to the appropriate im­
pedance. In some cases it is desirable to include the driving means in
determining the efficiency, because this is important in any practical design.
The input to the loudspeakers should be adjusted so that the loudness levels
of all loudspeakers are the same. The attenuation required to adjust to
the same loudness gives a measure of the relative loudness efficiency. In
these tests, the observers should move around to different locations to
insure that no advantages are given to any loudspeaker due to a better
listening location. For the same reason, the locations of the loudspeakers
should also be interchanged.
8. Relative Directivity.-The relative directivity of a loudspeaker is
determined by listening at observation points removed from the axis. In
order to reduce the effect of the difference in the angle during a comparison,
the following precautions should be observed: Only two loudspeakers should
be used at a time. The loudspeakers should be placed as close together as
possible. The position of the two loudspeakers should be interchanged
during the test. In determining the relative directivity, listening tests should
be conducted along different angles with respect to the axis. This test
indicates the loss in loudness. level and frequency discrimination for observa­
tion points removed from the axis.
9. Frequency Range.-The approximate frequency ranges of loudspeakers
may be determined from listening tests by employing program material
which has a wider frequency range than the loudspeaker under test in
combination with calibrated high- and low-pass filters introduced between
the program source and the loudspeaker. It is very important that
the program material contain adequate frequency components in both the
high- and low-frequency ranges and thereby insure reliable results. The
approximate frequency range can be determined by noting the settings of
the filters for which there is no appreciable frequency discrimination, as
determined by the quality of reproduction. The filters should have at
least three cutoff steps per octave.
10. Power Handling Capacity.-The power handling capacity of a loud­
speaker may be determined by employing a low distortion program source
capable of overloading the loudspeaker without introducing distortion in the
program source which is fed to the loudspeaker. The frequency range of
the system which feeds the loudspeaker should be restricted by means of
filters to correspond to that of the loudspeaker under test. The power level
at which the distortion becomes intolerable may be considered to be the
power handling capacity of the loudspeaker. In this connection intolerable
MEASUREMENTS
469
distortion depends upon the application in which the loudspeaker is to be
used. This requires a high order of judgment by the listener.
The test outlined above for determining the power handling capacity may
appear to be oversimplified in view of the many factors involved. For
example, the power handling capacity of a loudspeaker may be determined
by failure of the diaphragm, the suspension system, the voice coil structure,
and heating of the voice coil. Of course, all these forms of failure will be
manifested as intolerable distortion. Again it should be emphasized that
the crux of this test is the determination of what is considered intolerable
distortion.
11. Response Frequency Contour.-In most completely integrated systems,
such as radio and television receivers and phonographs, there are distinct
economic and technical advantages in employing components which indi­
vidually do not exhibit a uniform response frequency characteristic but
taken collectively do exhibit a uniform response frequency characteristic.
In these applications, listening tests are very useful in checking the objective
measurements for a proper balance of the frequency characteristic. This
type of listening requires great skill obtained through practice. A reference
system which is known to be acceptable is almost a necessity in tests of this
type.
12. Nonlinear Distortion.-Loudspeakers are used with other components
in a sound reproducing system. Therefore, in a properly integrated system
the limitations upon the allowable nonlinear distortion of each element
depends upon the allowable distortion of the system as a whole. For
example, it would be technically and economically unsound to use a wide­
range, high-quality loudspeaker in a reproducing system in which the
components in the remainder of the system were of much lower quality.
The quality of the loudspeaker required for the application can be deter­
mined from listening tests of loudspeakers of various degrees of quality.
In this way it is possible to determine the loudspeaker which introduces
distortion of such magnitude as to be perceptible above the distortion of the
remainder of the system.
13. Transient Response.-Since all speech and music are of a transient
character, the transient response is another important characteristic which
depicts the performance of a loudspeaker. Poor transient response leds to
fuzzy reproduction with poor definition. As a result the character of speech
and music is destroyed. In objective measurements, a deviation in the
sound output from the rapid growth and decay characteristic of an applied
tone burst depicts the transient response of a loudspeaker. See Sec. 10.3G.
A loudspeaker with a nonuniform respo