# Diffraction Effects in Sonar Array Master Thesis in - UiO UNIVERSITY OF OSLO
Department of Physics
Diffraction Effects
in Sonar Array
Master Thesis in
Electronics and
Computer
Technology
Kiyani
September 2014
Diffraction effects in sonar array
Friday 5th September, 2014
ii
Abstract
Humans can localize a sound source with the help of three effects, Head
Related Transfer Function (HRTF), Interaural Time Difference (ITD) and
Interaural Level Difference (ILD). With help of these three effects humans
can sense where in the space the signal is coming from. The space can be described by three planes: vertical-, horizontal- and median plane. Sound localization is described by three dimensions: azimuth angle, elevation angle
and distance or velocity detection for static or moving source.
Diffraction of sound by human head is described by the diffraction formula.
The sound is diffracted by the human head if the dimension of the head is
smaller compared to 2λ/3.
Sonar means Sound Navigation and Ranging, and has its roots from as early
as the beginning of World War I. Sonar technology was actively used under
World War I and World War II, and had an increase of interest among the
scientists after this period. Sound is pressure perturbations that travels as
a wave spreads spherically or cylindrically in the water by describing the
decrease of the signal. Sound propagation is affected by absorption, refraction, reflection and scattering. There are three types of sonar equations
described in this thesis, the active sonar equation for noise background,
the active sonar equation for reverberation background and the passive
sonar equation.
In sonar the distance to the sound source is calculated by the travel time and
the sound velocity of the incoming sound wave. Sound velocity in water is
divided into four different regions and is temperature dependent. For circular and spherical arrays, (just like the human head), the sound wave travels
direct to the receiver as long as the elements has a "direct" path to the receiver. Otherwise the sound signal is diffracted and travels along the surface of the transducer (with different sound velocity from as in water) until
it has a "direct" path to the receiver. This has its limitations, and for some
rotational angle on the transducer the sound wave is not possible to detect.
This angle limitation depends on the normalized frequency µ = ka = 2πa
λ
value. where a is radius of the circular transducer. In this thesis I’m going
to study at which rotational angle α the signal disappears when sending a
sound signal pulse with frequency f = 100k H z and a normalized frequency
µ = 25, and compare it with human listening.
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Contents
I
Introduction
1
1
Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
3
II
Background
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2 Introduction to human listening
2.1 Sound diffraction around human ear
2.1.1 Diffraction formula . . . . . . .
2.2 Sound localization . . . . . . . . . . . .
2.2.1 Horizontal localization . . . .
2.2.2 Vertical localization . . . . . .
2.2.3 Distance perception . . . . . .
2.2.4 Motion detection . . . . . . . .
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4 The sonar equation
4.1 Sonar parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Three types of sonar equations . . . . . . . . . . . . . . . . . .
4.2.1 The active noise background sonar equation . . . . .
4.2.2 The active reverberation-background sonar equation
4.2.3 The passive sonar equation . . . . . . . . . . . . . . . .
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3 Introduction to sonar
3.1 Sonar history . . . . . . . . . . . . . . . . . . . . . . .
3.2 Basic physics of sonar . . . . . . . . . . . . . . . . . .
3.2.1 The Decibel unit . . . . . . . . . . . . . . . .
3.2.2 Spherical- vs cylindrical spreading . . . . .
3.3 Underwater sound propagation . . . . . . . . . . . .
3.3.1 Absorption . . . . . . . . . . . . . . . . . . . .
3.3.2 Refraction and sound velocity in sea water
3.3.3 Reflection . . . . . . . . . . . . . . . . . . . .
3.3.4 Scattering . . . . . . . . . . . . . . . . . . . .
3.4 Sonar principles . . . . . . . . . . . . . . . . . . . . .
3.4.1 Range estimation . . . . . . . . . . . . . . . .
3.4.2 Bearing estimation . . . . . . . . . . . . . . .
3.4.3 Imaging sonar . . . . . . . . . . . . . . . . . .
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III
The Experiment
33
5 Background for the experiment
5.1 Equipments . . . . . . . . . . . . .
5.1.1 The Tank . . . . . . . . . .
5.1.2 Transducer . . . . . . . . .
5.1.3 The tube . . . . . . . . . .
5.1.4 The hydrophone . . . . .
5.2 Simulation process in the tank .
5.3 Input signal . . . . . . . . . . . . .
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6 Geometry
6.1 Distances from source to receiver . . . . . . . . . . . . . . . .
6.2 Sound propagation path . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Sound path for angles, α ∈ [−αt ang ent¡, αt ang ent ] . . .¢ .
6.2.2 Sound path for angles, α ∈ [αt ang ent , 360◦ − αt ang ent ]
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7 Pre-Experiments
51
7.1 Experiment I - Finding the beam direction . . . . . . . . . . . . 51
7.2 Experiment II - Finding sound velocity . . . . . . . . . . . . . . 55
8 Main experiments
8.1 Experiment III - Front experiment . . . . . . . . . . . . . . . .
8.1.1 Amplitude inspection and beamwidth calculation . .
8.1.2 Beampattern comparison . . . . . . . . . . . . . . . . .
8.1.3 Instrumental delay . . . . . . . . . . . . . . . . . . . . .
8.2 Experiment IV - Back experiment . . . . . . . . . . . . . . . .
8.2.1 Amplitude plot and comparison
from experiment
III
£
¤
◦
8.3 Sound velocity profile for α ∈ αt ang ent , 360 − αt ang ent . . . .
IV
Conclusion
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9 Conclusion and discussion
9.1 Conclusion . . . . . . . . . . . . .
9.1.1 Sound propagation path .
9.2 Discussion . . . . . . . . . . . . .
9.3 Future work . . . . . . . . . . . .
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Appendix
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79
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A
87
vi
List of Figures
2.1
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
Example of IID - Figure is taken from https://cnx.org/content/
m45358/latest/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Example of ITD - Figure is taken from https://cnx.org/
content/m45358/latest/ . . . . . . . . . . . . . . . . . . . . . . . . 8
Diffraction and shadow . . . . . . . . . . . . . . . . . . . . . . . . 9
Sound localization described in 3D plane - Figure is taken
from https://cnx.org/content/m45358/latest/ . . . . . . . . . . . 10
External ear or the pinna - Figure is taken from https://www.
html . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Spherical spreading, figure taken from [?] . . . . . . . . . . . .
One way spreading loss, figure taken from Hansen (2009)[?]
Two way spreading loss, figure taken from Hansen (2009)[?]
Cylindrical spreading, figure taken from [?] . . . . . . . . . . .
Snell’s law, figure taken from Hansen (2009)[?] . . . . . . . .
Sound velocity in the sea as function of depth, figure taken
from Hansen (2009)[?] . . . . . . . . . . . . . . . . . . . . . . .
Scattering from a smooth surface, figure taken from Hansen
(2009)[?] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scattering from a rough surface, figure taken from Hansen
(2009)[?] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Passive sonar, figure taken from Hansen (2009)[?] . . . . . .
Active sonar, figure taken from Hansen (2009)[?] . . . . . . .
Main lobe pattern of a single transducer, figure taken from
Hansen (2009)[?] . . . . . . . . . . . . . . . . . . . . . . . . . . .
Direction of arrival, figure taken from Hansen (2009)[?] . . .
Imaging sonar, figure taken from Hansen (2009)[?] . . . . . .
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4.1
Echo, noise and reverberation as function of range, figure
taken from [?]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.1
5.2
5.3
5.4
5.5
The system diagram . . . . . . . . . . . . . . . . . . . . . . . . .
The connection betweem computer and instruments. . . . .
Electric circuit with a pin connector . . . . . . . . . . . . . . .
The water tank in the lab . . . . . . . . . . . . . . . . . . . . . .
The tank illustration in matlab made by professor Svein Bøe
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36
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5.6 The transducer, Simrad SH90 mounted on the bottom of the
tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Elements of Simrad SH90 . . . . . . . . . . . . . . . . . . . . .
5.8 Vertical beamwidth with ten sensors arranged vertically and
equally spaced . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Horizontal beamwidth with sensor size d . . . . . . . . . . . .
5.10 The transducer covered by the fender . . . . . . . . . . . . . .
5.11 The hydrophone, Teledyne Reson TC4034 . . . . . . . . . . .
5.12 Input signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 38
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40
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Geometry of the distance between transmitter and receiver .
Tangent on the transducer to hydrophone. . . . . . . . . . . .
Beamwidth of the transducer . . . . . . . . . . . . . . . . . . .
Case I: Sound wave travels direct to the receiver. . . . . . . .
Case II: Sound wave travels first along the surface of the
transducer, and then direct to the receiver. . . . . . . . . . . .
6.6 Case III: Sound wave travels on the surface of the transducer
until it has a clear path to the receiver, and the direct to the
receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Cae IV: Sound wave travels the same path as case III, but in
water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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48
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Signal in different angles and different heights
Inspection of maximum signal value I . . . . . .
Inspection of maximum signal value II . . . . .
3D plot of Figure 7.3 . . . . . . . . . . . . . . . .
Amplitude plot as a function of angle . . . . . .
Signal captured at a distance of 21.4 cm . . . . .
Signal captured at a distance of 41.4 cm . . . . .
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52
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8.1
8.2
8.3
8.4
8.5
Signal captured by the oscilloscope . . . . . . . . . . . . . . .
Signal in dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Amplitude vs rotational angle . . . . . . . . . . . . . . . . . . .
Amplitude dB plot . . . . . . . . . . . . . . . . . . . . . . . . . .
Different beampatterns with element size d = 1cm . The one
with magneta color is the pattern in Equation 8.4, the green
is from Equation 8.5, the black is from Equation 8.7, the
dots are from Equation 8.6 and the red one is the measured
amplitude in dbV. . . . . . . . . . . . . . . . . . . . . . . . . . .
Different beampatterns with respectively element size d =
1.1cm , d = 1.2cm , d = 1.3cm , d = 1.4cm , d = 1.5cm and
d = 1.6cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Signals captured in different rotational angle . . . . . . . . .
Signal captured by the oscilloscope . . . . . . . . . . . . . . .
Signal in dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Signal output for experiment III . . . . . . . . . . . . . . . . .
Signal output for experiment IV . . . . . . . . . . . . . . . . .
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60
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6.1
6.2
6.3
6.4
6.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
viii
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. 48
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. 64
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8.13
8.14
8.15
8.16
8.17
8.18
Amplitude vs rotational angle . . . . . . .
Amplitude vs rotational angle in dB . . .
Amplitude plot of experiment III and IV
Compesated amplitude plot . . . . . . . .
Compensated amplitude plot in dB . . .
Signal output for experiment IV . . . . .
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9.1 Signal response in d B vs angle for different ka values . . . . . 80
9.2 Signal response in d B vs angle for different kav al ues , and
especially for ka = 25 . . . . . . . . . . . . . . . . . . . . . . . . . 80
ix
x
List of Tables
3.1 Sound velocity profile in deep sea water. . . . . . . . . . . . . . 23
3.2 Characteristic impedance of different materials. . . . . . . . . 24
6.1
Parameters of Figure 6.1 . . . . . . . . . . . . . . . . . . . . . . . 47
8.1 Instrumenta delay on different rotational angles . . . . . . . . 67
8.2 Distance to receiver . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.3 Sound velocity for experiment IV . . . . . . . . . . . . . . . . . . 75
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xii
Acronyms
ITD
Interaural time difference
ILD
Interaural level difference
IID
Interaural intensity difference
IPD
Interaural phase difference
MAMA Minimum audible movement angle
CW
Continuous wave
xiii
xiv
Preface
This master thesis was carried out at the Depertment of Physics, Faculty of
Mathematics and Natural Science, University of Oslo (UiO) in the period
January 2013 - September 2014. The thesis is for the grade Master of Science in Electronics and Computer Technology and contributes 60 credits.
Executing the master thesis has been both interesting and challenging. This
project contributes a great experience in life. Since I was one of two first
students to work on this tank system, it made it extra challenging and interesting.
First of all, I want to thank my supervisor, Professor Sverre Holm, for motivating me to work on this project and for providing me all the valuable
and necessary guidance and inspiration. Secondly I want to thank my costudent, Asle Tangen, who worked with me on the tank from the very beginning of my master thesis.
A special thanks goes to senior engineer, Svein Bøe, who helped us understanding the simulation process in the tank. I also want to thank professor,
Andreas Austeng and the whole group of DSB at UiO. You were always supporting and helpful, and it was a pleasure working with you guys. A special
gratitude I give to Kongsberg Maritime AS, who allowed us to borrow the
transducer as well as the tube for the project. Without your help this project
couldn’t have started.
In the end, I want to thank all my friends and family for all their motivation, support and patience throughout my study.
Oslo, 5th September 2014
xv
xvi
Part I
Introduction
1
Chapter 1
Introduction
1.1
Motivation
Humans can sense the direction of the sound source, no matter where the
signal is coming from. Human ear’s can be considered as sensors or antennas, who detects the sound signal and determines the direction of the
sound source with help of the sound level and the time delay between both
ears.
Can a circular or spherical sonar array behave like a human head? How
does the sound diffract around this array? Is it possible to detect signals
coming from behind, or is it possible to receive the signal transmitted from
circular or spherical transducer with the element 180◦ from the receiver?
Can this strengthen the theory of fish finding in water?
1.2
Thesis outline
Chapter 2 gives an introduction to human listenening. How sound is diffracted around human head, and how humans are able to localize sound
sources are discussed in this chapter.
Chapter 3 gives an introduction to sonar. Starting with some history,
some basic physics and then going through how sound propagates in water.
Chapter 4 describes the sonar equation. Discussing different parameters in the equation as well as describing the equation for three scenarios.
Chapter 5 gives some background information about the experiment. I’ll
go through every instrument/equipment used in the experiment as well as
illustrating how the simulation process is done in the tank.
Chapter 6 describes the geometry between the transducer and the hydrophone. Gives some cases for the sound propagation path to the receiver for
different rotational angles.
3
Chapter 7 describes the pre-experiments. Experiment I is done to find
the maximum and minimum values for the sound wave, and to study how
transducer transmits the sound wave. Experiment II is done to calculate
the sound velocity.
Chapter 8 describes the main experiments. Experiment III is done to
find -3dB point of the transmitted signal, and to calculate the instrumental
delay. Experiment III also calculates with help of -3dB point the effective element size on the transducer. Experiment IV is done by rotating the
transducer from 85◦ to 275◦ , to check whether it is possible to receive the
signal from behind. Experiment IV also calculates the sound velocity when
the element doesn’t have a "direct" path to the receiver.
Chapter 9 gives the conclusion for the thesis and discusses some factors
which may have influenced the measured data. It also provides suggestions
for future work.
4
Part II
Background
5
Chapter 2
Introduction to human
listening
In this chapter I’m going to introduce how humans verify or localize sounds
transmitted from a sound source anywhere in the space. How is actually
sound difracted around human head? How are humans able to localize
where the sound is coming from?
2.1
Sound diffraction around human ear
Before sound wave reaches the human ears, it is diffracted by the head
and the diffraction causes the sound to be filtered, which is characterised
by the Head related transfer function (HRTF). One part of HRTF is angle
dependent sound attenuation.This is called Interaural intensity difference
(IID), and Interaural level difference (ILD) when measured in dB. An
example of IID is given in Figure 2.1. The other part is the angle dependent
time difference of arrival of sound at both left and right ears which is
because of their seperation. This is called Interaural time difference (ITD),
and Interaural phase difference (IPD) when measured as the phase shift of
sinusoidal tone. An example of time difference is given in the Figure 2.2.
The Head Related Transfer Function is dispersive, where the phase shift
increases more slowly than linearly by increasing the frequency. The phase
delay decreases as the frequency increases. This is the reason why highfrequency sound waves travels faster around a human head than lowfrequency sound waves.
7
Figure 2.1: Example of IID - Figure is taken from https://cnx.org/content/m45358/latest/
Figure 2.2: Example of ITD - Figure is taken from https://cnx.org/content/m45358/latest/
Diffraction of a sound wave occurs when dimension of the head is
smaller than 2λ/3, and the sound will diffract around the head and cover
the potential shadow region. If the dimension is greater than 2λ/3, then
there will be no diffraction around the head, and we will get sound shadow
region. This shown in Figure 2.3
8
Sound
Plane
wavefronts
2r<2λ/3
2r>2λ/3
λ > 3r
λ < 3r
2.1.1
Diffraction formula
The effects of diffraction by the head, including dispersion, can be
approximated by a diffraction formula for the sound pressure on the surface
of a sphere as shown in Equation 2.1.
µ
pi + p s
po
¶
µ
=
r =a
1
ka
¶2 nmax n+1
X i
(2n + 1)P n (cos θ)
.
0
j n (ka) − i y n0 (ka)
n=0
(2.1)
Symbols p i , p s and p o refer to incident, scattered and free-field
pressures. Factor k = 2π
λ , and a is the radius of the head. Function P is
a Legendre polynomial. And functions j 0 and y 0 are derivatives of spherical
Bessel functions and spherical Neuman functions (Abramovitz and Stegun,
1964). The formula is taken from Constan and Hartmann (2003)[?].
2.2
Sound localization
In recent years, there has been an increasing of interest in sound localizaton
by human listeners. Sound localization is the process of determining the
location of a sound source. There are many factors that count to localize a
sound by a human brain, such as the strength or intensity of the sound, time
travel, angle of incident, azimuth, elevation etc. Sound localization can be
described by three-dimensional positions: the azimuth or horizontal angle,
the elevation or vertical angle and the distance for static source or velocity
for moving source. The three dimensions is described in Figure 2.4.
9
Figure 2.4: Sound localization described in 3D plane - Figure is taken from https://cnx.org/content/
m45358/latest/
Time difference and intensity difference are two mechanisms that
describes the sound localization process in the horizontal or azimuthal
plane. Spectral cues or the HRTF are the mechanism that describes the
sound localization in the vertical or elevation plane.
2.2.1
Horizontal localization
In the end of the 19th century, Lord Rayleigh made some important series
of observations in the horizonal dimension and reported in his Segwick lecture in 1906 (Rayleigh 1907). He defined localization in terms of interaural
difference cues.
If a sound is coming from the side, and the listener’s head is in the path
of the sound traveling towards the far ear, then the far ear is shadowed and
it would result in level difference. In other words, sounds coming from right
has higher level difference at the right ear than the left ear. Level difference
is frequency dependent and increases with increasing frequency. As mentioned earlier the amount of shadow depends on the wavelength compared
with the dimension of the head.
For frequencies below 800 Hz, the dimensions of the head are smaller than
the half wavelength of the sound wave. And the auditory system can determine the phase delay between both ears without confusion. Level difference is very low in this frequency range, and is therefor negligible. So for
low frequencies below 800 Hz, only time difference or phase difference is
used. For frequencies below 80 Hz, the phase difference between both ears
10
become too small to find the direction of the sound source, thus, it becomes
impossible to use time difference and level difference.
For frequencies above 1600 Hz, the dimension of the head are larger than
the wavelength. The level difference becomes larger, and therefor is used
to find the location of the sound source in this frequency range.
For frequencies between 800Hz and 1600 Hz, there is a transition zone,
where both time difference and level difference plays a part of determing
the location of the sound source.
From all this it is cleared that for localizing the sound source, humans depends on time difference for low frequencies, and on level difference for
high frequencies. This is often referred to as the "duplex" theory.
2.2.2
Vertical localization
If the the head and the ears are symmetrical, a stimulus presented at any
location on the median plane has none interaural differences, and thus, interaural differences provides no cue to the vertical locations of sounds on
the median plane. If the interaural differences, level difference or phase difference are constant, then any point off this median plane falls on a "cone
of confusion".
Batteau(1967,1968) was one of the first to emphasize that the external ear,
specifically the pinna (as shown in Figure 2.5), could be a source of spatial
cues that might be used to localize a sound source. He meant that sound
reflections within the convolutions of the pinna might produce spatial cues.
Figure 2.5: External ear or the pinna - Figure is taken from https://www.nytimes.com/imagepages/
11
The convolutions of the pinna creates echoes that lasts only for a few
microseconds, so most theories interpret the pinna as producing changes
in the spectrum of the sound source that reaches the tympanic membrance.
The pinna’s task is to produce multiple paths to the ear canal, including
a direct path and reflection from the cavum concha of the pinna. The
sum of a direct signal and a delayed version of the same signal produces
a "comb-filtered" spectrum. The length of the reflected path varies with
the elevation of the sound source. Patterns of spectral features associated
with particular location is referred to as "spectral shape cues". They are
often referred to as "pinna cues". Spectral shape cues are the major cues
for vertical localization.
2.2.3
Distance perception
The ability of a human to localize the distance of a sound source is not so
good. But the distance can be judged on the basis of the sound intensity
at the listener’s ear. In 1969, M. B. Gardner showed that the distance of
a person speaking with a conversational tone in anechoic chamber can almost be accurately judged. The distance of the same voice transmitted from
a speaker is almost determined by the sound level of the loudspeaker. Thus,
familiarity to sound source seems to be an important variable. Simpson and
Stanton (1973) have shown that head motion does not improve the judgement of the distance. In 1980, Butler listened to sounds over headphones
and judged their apparant distance. The distance of the source increased as
the low-frequency part of the spectrum increased.
But there is some evidence that states that the listener can better judge the
distance of the source if the surrounding environment is not anechoic. In
an ordinary room, a distant source produces sound energy that reaches the
listener’s ear via direct and indirect paths. Differences in the ratio of these
two energies might produce differences in the quality of the source as a
function of distance. This cue to source distance, however, is strongly influenced by the specific reflections of the particular listening environment,
(Middlebrooks and Green, 1991)[?].
2.2.4
Motion detection
Another consideration ragarding human listening is how humans detect
sounds which are in motion or moving. In this thesis I will only refer to
change in azimuth and/or elevation of the source, not the change in source
distance. There is no compelling evidence for motion-sensitive systems in
the auditory system. The problem is that there are two intepretations of
sensitivity to source motion. One intepretation is that the auditory system
is sensitive to dynamic aspects of localization cues, such as level differences
or phase differences. The second intepretation is that the nervous system
measures the sound source location at two distinct times and interprets a
change in location as motion. This has been called the "snapshot theory".
The reason why this two alternatives are difficult to resolove is because of
12
the fact that most studies of motion detection tend to confuse the attributes
of duration, velocity, and net change in location.
Most studies have used sound sources in actual motion, some have simulated motion by systematically varying the levels of sinusoids presented
from two loudspeaker. Thresholds have been measured for duration, velocity and change in location. All these thresholds, can be expressed in terms
of Minimum audible movement angle (MAMA), which is the smallest net
change in location of a moving stimulus that can be detected under some
specified set of conditions. The MAMA shares several properties with the
minimum audible angle for static sources. Some of them are: a) MAMAs
in azimuth are smallest for stimuli around 0◦ azimuth and increases with
increasing azimuth. b) MAMAs are smaller for broadband than for tonal
stimuli. c) MAMAs are largest for a range of frequencies around 1300-2000
kHz. This applies when measured with tonal stimuli.
13
14
Chapter 3
Introduction to sonar
Sound Navigation and Ranging (Sonar) is a technique that uses sound
propagation to navigate, communicate with or detect objects on or under
the surface of the water, such as other vessels, (Sonar, 2014, http:// en.
wikipedia.org/ wiki/ Sonar[?]).
3.1
Sonar history
In 1490, Leonardo Da Vinci wrote: "If you cause your ship to stop, and
place the head of a long tube in water and place the outer extremity to your
ear, you will hear ships at great distance from you",(Urick, 1983. Chapter
1, page 2)[?]. From this experiment it’s not possible to find the direction of
the sound source. Anyway, this idea had widespread use as late as World
War I. The direction could be achieved and the bearing of the target could
be determined by adding a second tube between the other ear and a point
in the sea separated from the first point.
As mentioned in Urick (1983), perhaps the first measurements in underwater sound occured in 1827. A swiss physicist, Daniel Colladon, and a
French methematician, Charles Sturm, worked together and measured the
velocity of sound in Lake Geneva in Switzerland. They experimented by
taking the time interval between a flash of light and the striking of a bell
underwater. From this they determined the velocity of sound with some
accuracy.
In the 19th century, it was a great amount of interest among scientists on
underwater acustics. An invention from the 19th century is the carbonbutton microphone, which is still the most sensitive hydrophone device for
underwater sound. Another invention of the 19th century was the submarine bell, which was used by ships for offshore navigation. This system made
it possible for ships to find their distance from a lightship. They took the
time interval between the sound of the bell, which was installed above the
sea surface on the lightship, and a simultaneously sent blast from a foghorn,
which was installed underwater on the same lightship. This method didn’t
became so popular, thus, it was replaced by navigation methods involving
15
Echo ranging bacame very popular in the period before World War I, and
in 1912, five days before the "Titanic" accident, L.F. Richardson did some
research on echo ranging with airborne sound. A month later he used a
directional projector of kilohertz-frequency sound waves and a frequencyselective receiver detuned from the transmitting frequency to compensate
for doppler shift, which was caused by the moving vessel.
In 1914, it was a large amount of research on sonar for military purposes.
The same year, R.A. Fessenden designed and built a new kind of moving coil
transducer for both submarine signalling and echo ranging. It could detect
an iceberg up to 2 miles away. They were installed on all United States submarines during the World War I, so they could signal each other when submerged. In France a young Russian engineer called Constain Chilowsky,
worked together with a physicist called Paul Langevin. They experimented
with a electrostatic projector and a carbon-button microphone placed at
focus of a concav mirror. In 1916, they were able to receive echoes from
bottom and from a sheet of armor plate at a distance of 200 meters. Later,
in 1917 Langevin employed vacuum- tube amplifier, which was the first application of electronics to underwater sound equipment. For the first time
in 1918, echoes was received from submarine at distances as much as 1500
meters.
At the end of World War I, Leonardo’s air tube had been used for passive listening, and was improved by use of two tubes to take advantage of
the binaural directional sense of a human observer. The MV device was
mouted along the bottom of a ship on the port and starboard side. It consisted of two line arrays with 12 air tubes each. The device was steered with
a special compensator. The result gave precise achievement in determining the bearing of a noisy target. Another development of the late stages of
World War I was flexible line array of 12 hydrophones called the "eel". They
were easy to fit on any ship and could be towed away from a noisy vessel on
which it was mounted. Almost three thousand escort craft were installed
with listening devices during World War I.
In 1919, after World War I, the Germans published the first scientific paper about underwater sound. In the paper it was written about bending of
sound rays produced by slight temperature and salinity gradients in the sea,
and their importance in determining sound ranges. This paper was unrecognized for over 60 years. The years after World War I was slow in terms of
underwater sound applications for practical use. Depth sounding by ships
under way was developed, and by 1925, fathometers were available commercially both in United States and Great Britain. The problem of finding
a suitable projector in echo ranging was solved by resorting to magnetostrictive projectors for generating the required amount of acoustic power.
the period between World War I and World War II. Which made it possible
16
to make new technologies such as amplifying, processing, and displaying
sonar information to an observer. Ultrasonic frequencies are frequencies
beyond the limits of which a human ear can sense, and was used for both
listening and echo ranging. The range recorder for echo ranging sonars
was also developed in this period. It provided "memory"of past events and
the streamlined dome to protect the transducer on a moving ship from the
noisy environment. An important achievement in this period was the understanding of sound propagation in the sea. Good signals were received
in the morning and bad signals or none in the evening. Bathythermograph
was built for the first time in 1937 by A.F. Spilhaus. Clear understanding
of absorption of sound in the sea was achieved. And accurate values of absorption coefficients were determined at the ultrasonic frequencies.
By the start of the World War II, a large quantity of sonar sets was produced
in the United States, and a large number of American ships were installed
with both underwater listening and echo ranging. QC equipment was the
standard echo ranging sonar set for surface ships. The operator searched
with it by turning a handwheel and listening for echoes with headphones
or loudspeaker. They noted the echoes range by the flash of a rotating light
or by the range recorder. Submarines was installed with JP listening sets,
which consisted of a rotatable horizontal line hydrophone, an amplifier, a
selectable bandpass filter, and a pair of headphones. The period of World
War II had a huge importance to underwater sound. In United Stated, a
large group of scientists arranged by the National Defense Research Committee began researching on all phases of the subject. Most of nowadays
concepts and applications had their origin from that period. Developments
of World War II period was such as acoustic homing torpedo, the modern
acoustic mine, and scanning sonar sets. Understanding of the factors in the
sonar equation that affects the sonar performance was gained.
Years after World War II had some important developments of underwater sound, both for military and nonmilitary uses. On the military side,
active sonars became larger, and more powerful. They could operate at
lower frequencies than in World War II. Also passive sonars started to operate at lower frequencies so they could take the advantage of the tonal or
line components in the low-frequencies submarine noise spectrum. The
development of complex signal processing, in both time and space, made it
poosible to enable much more information, which can be used for whatever
function the sonar is called up to perform. The research of sound propagation in the sea has led to the exploitation of propagation paths.
Examples of developments of underwater sound for nonmilitary
purposes after the World War II are:
1. Depth sounding
• Conventional depth sounders
• Subbottom profilers
17
• Side-scan sonars
2. Acoustic speedometers
3. Fish finding
4. Fisheries aids
5. Divers’ aids
6. Position marking
• Beacons
• Transponders
7. Communication and telemetry
8. Control
9. Miscellaneous uses
Examples of developmets of underwater sound for military
purposes after World War II are:
1. Pressure mine
2. Acoustic mines
3. Minesweeping
4. Passive detection
5. Homing torpedoes
6. The underwater telephone
7. Neutrally buoyant flexible towed-line array
3.2
Basic physics of sonar
Sound is pressure perturbations that travels as a wave. Sound is also referred to as compressional waves, longitudal waves, and mechanical waves.
A sound wave can be characterized by the following parameters:
• Wave period, T [s]
• Frequency, f = T1 [H z]
• Sound speed, c[m/s]
• Wavelength, λ = cf [m]
18
3.2.1
The Decibel unit
The desibel has been long used for reckoning quantities. Decibel makes
it possible to handle large change in variables, and allows to multiply
quantites by simply adding the decibel value together. Since acoustic
signal strength varies in several orders of magnitude over a typical distance
travelled, the decibel unit is used for sonar purposes. The notation of
decibel is dB, and is defined as:
I d B = 10l og 10 (I )
(3.1)
In Equation 3.1, the I d B is intensity in dB and I is linear intensity. The
decibel unit also makes it easier to see how much two quantites differ in dB.
For example if I 1 and I 2 are two intensities, by taking dB of their ratio such
as, Nd B = 10l og 10 (I 1 /I 2 ), you find that I 1 and I 2 differs by N dB.
3.2.2
Spherical- and cylindrical spreading are used to describe decrease of signal
level as the sound wave propagates away from the source.
Figure 3.1: Spherical spreading, figure taken from [?]
Spherical spreading describes the decrease of signal level when the
sound wave propagates uniformly in all directions, shown in Figure 3.1. The
total power crossing any sphere around the source is given as, P = 4πr 2 I .
And let I 0 be the intensity of the wave at range, r 0 . If there is no absorption
in the medium then the total power crossing each sphere is the same. From
this I get, P = 4πr 02 I 0 = 4πr 2 I . If I solve this for I , I get:
Ã
I = I0
r 02
r2
!
(3.2)
The intensity decreases as the inverse square of the range for spherical
spreading. If I say that r 0 = 1m , than I 0 is just the acoustic source level.
19
Transmission loss is the amount by which the intensity decreases to its level
at the source and is expressed in dB:
T L d B = −10l og 10 (I /I 0 )
= 10l og 10 (r 2 )
= 20l og 10 (r )d B
(3.3)
From this I can conclude that sound intensity I decreases with range
R in inverse proportion to the surface of the sphere as I ~ R12 . This is for
the one way spherical spread as illustrated in Figure 3.2. It is important to
know that the relation between sound intensity I and signal amplitude A is
I = A 2 . Thus Equation 3.2 becomes:
Ã
2
A =
⇒
A 20
r 02
!
r2
r0
A
=
A0
r
(3.4)
Figure 3.2: One way spreading loss, figure taken from Hansen (2009)[?]
For two way propagation, as illustrated in Figure 3.3, the wave
propagates spherically to the reflector and the reflector spreads the signal
in all directions and propagates spherically back to the source. The two way
loss becomes, I ~ R12 R12 .
Figure 3.3: Two way spreading loss, figure taken from Hansen (2009)[?]
20
Figure 3.4: Cylindrical spreading, figure taken from [?]
Cylindrical spreading is just made out of the fact that sound cannot
travel uniformly in all directions forever. At some point the wave will hit
the sea surface or the sea floor. An approximation for spreading loss in
that case is to assume that the wave propagates cylindrically away from the
source with radius equal to range, r and height equal to depth, H , as shown
in Figure 3.4. The total power crossing a cylinder around the source is given
as, P = 2πr H I . As same as spherical spreading, the total power crossing
each cylinder is equal, so I get: P = 2πr 0 H I 0 = 2πr H I . Solving for I gives:
I = I0
³r ´
0
r
(3.5)
The intensity decreases as the inverse power of the range for cylindrical
spreading. As same as spherical spreading, I say r 0 = 1m , and I 0 is the
source level, than the transmission loss in dB becomes:
T L d B = 10l og 10 (I /I 0 )
= 10l og 10 (r )d B
3.3
3.3.1
(3.6)
Underwater sound propagation
Absorption
Seawater is a dissipative medium through viscosity and chemical processes.
Acoustic absorption in seawater is frequency dependent. Low frequencies
reaches longer than high frequencies, (Hansen, 2009) [?]. The absorption
coefficient is given in Equation 3.7, and is a function of temperature,
salinity, depth (pressure) and pH value of the water along with frequency.
The absorption coefficient formula given in Fisher and Simmons (1977) [?],
is as follows:
21
α=
A1P1 f1 f 2
f 12 + f 2
+
A2P2 f2 f 2
f 22 + f 2
+ A3P3 f 2
(3.7)
The parameter α is the absorption coefficient, and the first term in the
equation is absorption caused by Boric Acid, the second term is absorption
caused by magnesium sulfate, and the third term is absorption in pure
water. Parameters P 1 , P 2 and P 3 are effect of pressure. Frequency
dependency is shown by f 1 and f 2 , which are the relaxation frequencies
of Boric Acid and magnesium sulfate. The frequency f is simply the sound
wave frequency. Values of A 1 , A 2 and A 3 depends on water properties, such
as temperature, salinity and pH of water.
3.3.2
Refraction and sound velocity in sea water
Sound refraction occurs between two media or layers with different sound
velocity. As shown in Figure 3.5, some of the sound hitting the surface of
the other media is reflected and some is refracted.
Figure 3.5: Snell’s law, figure taken from Hansen (2009)[?]
How much bend the refracted sound wave will get depends on the
incident angle and velocity at both layers:
sin θ1 sin θ2
=
c1
c2
(3.8)
The sound velocity in sea water depends on temperature T , salinty S
and depth D , and is given in Hansen (2009)[?] as:
c = 1449.2+4.6T −0.055T 2 +0.00029T 3 +(1.34 − 0.010T ) (S − 35)+0.016D (3.9)
Speed of sound in sea water is approximately between 1450m/s and
1500m/s .
22
Figure 3.6: Sound velocity in the sea as function of depth, figure taken from Hansen (2009)[?]
As shown in Figure 3.6, the sea water sound velocity is divided into four
different regions:
The surface layer
The seasonal thermocline
The permanent thermocline
The deep isothermal layer
In this layer the sound velocity is exposed by daily and local changes of heating, cooling and wind actions. This layer
may contain a mixed layer of isothermal
water. Sound tends to be trapped or
channeled in this mixed layer. The mixed
layer disappears under sunny conditions,
and is replaced by water where temperature decreases with depth.
Characterized with negative thermal or
velocity gradient that varies with the season. It means that temperature and velocity decreases with depth
Affected only slightly by seasonal
changes.
In this layer, occurs the
major increase in temperature.
Has constant temperature about 3.89◦C .
In this layer the sound velocity increases
with depth because of the effect of pressure on sound velocity.
Table 3.1: Sound velocity profile in deep sea water.
23
In the area between the permanent thermocline and the deep isothermal layer, the sound travelling at great depths tends to bent by refraction.
3.3.3
Reflection
Figure 3.5 shows refraction and reflection. And as same as refraction,
reflection occurs at the interface between to different media. At normal
incidence, the reflection coefficient is given as:
Z − Z0
Z + Z0
(3.10)
2Z0
= 1−V
Z + Z0
(3.11)
V=
The transmission coefficient is:
W=
In Equation 3.10 and Equation 3.11, parameters Z and Z0 are the
characteristic impedances of the two media and are given by the sound
velocity and density:
Z0 = ρc
(3.12)
Characteristic impedance for different materials is given in Table 3.2.
From Equation 3.10, we can calculate the reflection coefficient between air
and seawater. The result will be V = −1, which means that the sea surface
is a "perfect" reflector.
Material
Air
Seawater
Sand
Sandstone
Steel
Impedance
415
1.54 x 106
5.5 x 106
7.7 x 106
47 x 106
Table 3.2: Characteristic impedance of different materials.
3.3.4
Scattering
Figure 3.7: Scattering from a smooth surface, figure taken from Hansen (2009)[?]
24
Figure 3.8: Scattering from a rough surface, figure taken from Hansen (2009)[?]
Scattering can be of two categories:
1. Surface scattering from the sea surface or the sea floor.
2. Volume scattering from ocean fluctuations, marine life or objects.
If the surface is smooth, as in Figure 3.7, then we will get specular reflection. And if the surface is rough as in Figure 3.8, then some part of
the reradiated energy will be scattered diffusely in random directions. The
more rough the surface is, the more energy will be scattered diffusely.
"A criterion for roughness or smoothness of a surface is given by the
Rayleigh parameter, defined as R = k H sin θ , where k is the wave number
2π/λ, H is the rms "wave height", and θ is the grazing angle. When R ¿ 1,
the surface is primarly a reflector and produces a coherent reflection at the
specular angle equal to the incident angle. When R À 1, the surface acts as
a scatterer, sending incoherent energy in all directions". Urick (1983) [?].
3.4
Sonar principles
We have two types of sonar systems, passive sonar and active sonar.
Passive sonar is where a noise source is radiated by the target and received
by the sonar. Active sonar is where the sonar transmits the signal, and
the signal hitting the target reflects back, and is received by the sonar. In
other words, passive sonar only receives the signal, and active sonar both
transmits and receives the signal. When active sonar is used to measure
distance from the transducer to the bottom, it is called "echo sounding".
Active sonar is used for measuring distance between two transducers or a
combination of hydrophones. Passive sonar are used for military settings
and in science applications such as detecting fish in the ocean. Examples of
Passive- and active sonar is shown in Figure 3.10 and Figure 3.9.
25
Figure 3.9: Passive sonar, figure taken from Hansen (2009)[?]
Figure 3.10: Active sonar, figure taken from Hansen (2009)[?]
3.4.1
Range estimation
Radial distance between sonar and reflector is defined as the range.
Assuming that we are transmitting a short pulse with pulse duration, T p .
The receiver records the signal until the reflected echo is received and
estimates the time delay, τ from the time series. The range to the target
is then defined as:
R=
cτ
2
(3.13)
Where c is the sound velocity in the water.
If the purpose is to transmit two echoes, and in order to detect those two
echoes, they has to be seperated by a minimum distance or range defined
as the range resolution:
δR =
cT p
2
(3.14)
Shorter pulses gives better resolution, but shorter pulses have less energy in the pulse, which again gives shorter propagation range. Alternatively, we can phase code the pulse and resolution becomes:
δR =
c
2B
26
(3.15)
Where B is the bandwidth of the acoustic signal, and relates to the pulse
duration as, B = T1p for gated Continuous wave (CW) signals.
3.4.2
Bearing estimation
By bearing estimation, it means estimating the direction of sonar pulses. It
has two key elements involved:
1. The electro-acoustic transducer and its size.
2. The grouping of transducers and its size.
It is said that a sonar source (e.g. transducer, antenna or loudspeakers)
is directive if the size of the source is large compared to the wavelength of
the signal. The directivity pattern of a source, with diameter or size D , has
a main lobe with a -3 dB beamwidth or field of view as:
sin β ≈
λ
D
(3.16)
Figure 3.11: Main lobe pattern of a single transducer, figure taken from Hansen (2009)[?]
If we have a source with N equally spaced (spacing distance of d s )
elements with size D , then beamwidth or field of view becomes:
sin β ≈
λ
N D + N ds
(3.17)
It has to be marked that β is derieved in radians and not in degree.
For small angles, it is mathematically proved that sin β ≈ β. From Equation 3.16 and Equation 3.17 it is clear that the beamwidth depends on frequency (since λ = cf ) and the total length of the array. So higher frequency
or larger antenna array size gives narrower beam.
The bearing from a reflected signal is estimated from the time difference of
arrival δt between two different receivers with distance L apart (illustrated
in Figure 3.12):
θ = sin−1
27
µ
cδt
L
¶
(3.18)
Figure 3.12: Direction of arrival, figure taken from Hansen (2009)[?]
3.4.3
Imaging sonar
Imaging in sonar is to estimate the reflectivity for all calculated ranges and
in all directions. There are several methods to do so, such as delay and sum
(backprojection) and wavenumber-domain beamformers. Imaging sonar is
described in Figure 3.13.
Figure 3.13: Imaging sonar, figure taken from Hansen (2009)[?]
As shown in Figure 3.13, the field of view is given by the angular width
of each element as β = dλ . The angular or azimuth resolution is given by the
array length as δβ = λL , and the range resolution is given by bandwidth of
c
the system as δr = 2B
.
28
Chapter 4
The sonar equation
The sonar equation was first formulated during the World War II for calculation of maximum range of sonar equipments. The sonar equation shown
in this thesis are taken from Urick (1983). In Urick (1983) it states that,
"the main phenomena and effects peculiar to underwater sound produce a
variety of quantitative effects on the design and operation of sonar equipments". Sonar parameters are units of these effect, and are related to the
sonar equations. The basic foundation of sonar equation is the quality
between desired (signal) and undesired (background) parts of the recieved
signal. A portion of the total acoustic field at the receiver is said to be the
signal, and rest is said to be the background. For sonar to successfully detect a signal, the signal level has to be larger than the background level.
There are two types of background levels that masks the signal, noise background and reverberation background. Noise background or noise level
is isotropic sound generated from wind, waves etc. Reverberation background or reverberation level is the slowly decaying portion of the backscattered sound from one’s own acoustic input. Figure 4.1 shows the echo-,
noise- and reverberation level as a function of range. It is shown that the
echo level and reverberation level falls with increasing range, but the noise
level is constant for all ranges. The echo level falls off much before the
reverberation level, and they intersect each other at range R r . The echo
level intersect the noise level after range R r , it is not marked in the figure
but it is the point where echo level is equal to the noise level. If the reverberation is high, the range is said to be reverberation limited. And if
by any reason the noise level rises to the level shown by the dashed lines,
the echoes will rather die away into noise background rather than reverberation background. The noise limited range R n will then be smaller than
reverbeartion limited range R r and the range will become noise limited.
29
Figure 4.1: Echo, noise and reverberation as function of range, figure taken from [?].
4.1
Sonar parameters
The sonar parameters are the effects from the medium, target and the
equipment.
Parameters determined by the medium are:
• Tansmission loss: TL
• Reverberation level: RL
• Ambient-Noise level: NL
Parameters determined by the target are:
• Target strength: TS
• Target source level: SL
Paramaters determined by the equipment are:
• Projector source level: SL
• Self-Noise level: NL
• Receiving directivity index: DI
• Detection threshold: DT
30
A source produces a source level SL , which is the acoustic intensity of
the signal 1 meter away from the source. As the signal propagates towards
the target the intensity is reduced by the range, and the loss of intensity is
called transmission loss T L . T S is the target strength of the reflected signal
1 meter from the target. On the way back the signal again is reduced by the
transmission loss T L , and thus the echo level at the transducer becomes
SL − 2T L + T S . On the background side, the background level is simply N L ,
and is reduced by the directivity index D I . So the echo-to-noise ratio at the
transducer terminals becomes:
SL − 2T L + T S − (N L − D I )
4.2
(4.1)
Three types of sonar equations
There are three types of sonar equations:
• The active sonar equation for noise background.
• The active sonar equation for reverberation background
• The passive sonar equation
4.2.1
The active noise background sonar equation
This is the most commonly used sonar equation. And the signal-to-noise
ratio in Equation 4.1 defines whether the target is absent or present. If the
signal-to.noise ratio is less than the detection threshold DT , then the target
is said to be absent. But if the signal-to-noise ratio equals the detection
threshold, the target is said to be present. The sonar equation becomes:
SL − 2T L + T S − (N L − D I ) = DT
SL − 2T L + T S = N L − D I + DT
4.2.2
(4.2)
The active reverberation-background sonar equation
In this case we replace a noise background with reverberation background.
Thus, the parameter D I is not needed, and the term N L − D I is replaced by
reverberation level RL . The active sonar equation then becomes:
Sl − 2T L + T S = RL + DT
(4.3)
Here it has to be marked that DT value in reverberation is different from
the noise background DT .
31
4.2.3
The passive sonar equation
In this case, the target itself creates the signal , and the SL now refers to
the level of the radiated noise of the target 1 meter away. The T S becomes
irreleveant, and the transmission loss is now one-way not two-way loss. The
passive sonar equation becomes:
SL − T L = N L − D I + DT
32
(4.4)
Part III
The Experiment
33
Chapter 5
Background for the
experiment
The purpose of the experiments or the concept of this thesis is to find
whether it is possible to detect a sound signal in water which is transmitted
from a circular transducer with an incident angle of 180◦ to the receiver.
And to study how sound propagates in this situation. Does it travel along
the surface of the transducer or does it bend in water to reach the receiver?
Can this be compared to human listening?
All experiments was done in the DSB lab at ifi on the 4th floor. All equipments used in the experiments are described in this chapter as well as all
assumptions that has been made. I will also go through a quick section of
how simulation was done on Matlab.
5.1
Equipments
Equipments or tools used in the experiments were:
• Tank in DSB lab
• Transducer as a transmitter
• The tube which protects the electronics in the transducer from water
• Connector to connect transducer to the generator
• Oscilloscope
• Preamplifier
• Signal generator
• JBL sound amplifier
• Programming language, Matlab
35
Figure 5.1: The system diagram
Figure 5.2: The connection betweem computer and instruments.
In Figure 5.1, I have illustrated how the system is connected together.
Both the signal generator and the oscilloscope are connected with the
computer, and controlled by a matlab program which I will come back to
later in this chapter. The signal generator sends a signal to the amplifier,
which sends it to the transducer. The signal from the transducer is then
36
received by the hydrophone, which is connected to a preamplifier, and then
to the oscilloscope. First of all, to connect the transducer we had to buy a
pin connector, and then make a electronic circuit to connect the transducer
with the amplifier. The electronic circuit is shown in Figure 5.3, and is
connected with pin connector in the bottom.
Figure 5.3: Electric circuit with a pin connector
5.1.1
The Tank
Tank which was used is mounted on the DSB lab at ifi on the 4th floor. It
is a small tank with dimensions, along the sink wall, x = 149.7cm and the
long side, y = 187.4cm . The picture of the tank is shown in Figure 5.4 and
in Figure 5.5 you can see an illustration of the tank. The coloured boxes on
the top is obstacles such as pipes, light bulbs and other obstacles.
Figure 5.4: The water tank in the lab
37
Figure 5.5: The tank illustration in matlab made by professor Svein Bøe
There are mounted two probes in the tank, as you can see in Figure 5.5.
On one of them we have installed the transducer, and on the other we have
installed the hydrophone. How they are used is described later. The tank is
filled up with water at a height of 120 cm, and to avoid as much reflections
as possible, the source is positioned in the middle of the tank with a depth
of 60 cm. The receiver is placed 21.4 cm away from the source, and has to
be placed within the beamwidth of the source (which is shown in the next
section).
5.1.2
Transducer
Figure 5.6: The transducer, Simrad SH90 mounted on the bottom of the tube.
38
Most challenging part of my thesis was to find the correct transducer.
I am using a circular transducer named Simrad SH90 which is borrowed
from Kongsberg Maritime AS. As you can see in Figure 5.6 the transducer is the red instrument mounted on the bottom of the tube. Further technical information is given in the Kongsberg Maritimes’ webpage,
The transducer consists of 480 channels or elements which are uniformly
distributed around the transducer. Simrad SH90 consists of 8 sectors,
which again consists of 6 stripes each. Assuming that all elements are
480
equally spaced and distributed then each stripe consists of 68 = 10 elements/channels. In this thesis I have assumed that this 10 sensors are equally
distributed vertically at the transducer on each stripe. It has to be noted
that in the experiments I have only used one strip, which is stripe nr. 3 on
sector nr. 2. Further study of the Simrad SH90, gives us that every stripe
360◦
◦
is spaced to each other by an angle of 48pi
ns = 7.5 . Stripe nr. 1 in sector 2
◦
is located at 3.5 , and stripe nr. 3 in sector 2 is located at 18.5◦ and stripe
nr. N is located at (3.5◦ + (7.5◦ (N − 1)))◦ . How elements are distributed in the
horizontal and vertical plane is illustrated in Figure 5.7.
Figure 5.7: Elements of Simrad SH90
Dimensions of the transducer
The Simrad SH90 transducer has a diameter of d = 12cm , which gives the
radius, r = 6cm . This gives the circumference (perimeter) of the transducer
as, O = 2πr = 2π × 6cm = 37.7cm . The height of the transducer is 21cm ,
and the elements are located at the vertical extent of 11cm and the rest is
covered by other things such as, electronic parts, wires etc. It is markable to
note that in this thesis I have assumed that each element has same depth,
height and width equal to 1cm . I am assuming this because there are
no given information about the dimension of any element/channel in the
39
specifications. But we will see later in this thesis that my assumption of size
is almost correct.
Vertical beamwidth
Since I’m only using 10 elements, which are equally spaced and distributed in the vertical direction, and as mentioned above, the range where
these elements are located is L = 11cm . From Equation 3.17, we know that,
L = N D +N d s where N = 10 in this case. Since D = 1cm , we get that the space
between elements are d s = 0.1cm . This is shown in Figure 5.8.
It is later shown that the wavelength, λ = 1.488cm , thus from Equation 3.16
(for small angles) the vertical beamwidth becomes:
λ 1.488cm
=
L
11cm
(5.1)
βd eg =
= 7.73◦
π
(5.2)
In degrees:
The beamwidth from the transducer is illustrated below:
Figure 5.8: Vertical beamwidth with ten sensors arranged vertically and equally spaced
Horizontal beamwidth
In the horizontal plane, the beamwidth is dependent only on one single
element. This means that the beamwidth is supposed to be larger since the
sensor size becomes smaller. An approximation of how it will look like is
illustrated in Figure 5.9.
40
{
β=λ/d
d
Figure 5.9: Horizontal beamwidth with sensor size d
But this is not the case here. The size of the sensor element is d = 1cm ,
which gives λ > d . The sin β > 1, which is an invlaid value for sinus. The
beam is larger than 90◦ , and covers the whole area around the transducer.
This is further studied in experiment III on chapter 8.
One thing we were recommended by our professors was to protect the
transducer from water when we don’t do any experiments. It was
challenging to find out how we could clear water around the transducer
when we finished the experiments everytime. We could not lift the
transducer up because of the limits of the probe and the obstacles on the
ceiling. So what we did, was to buy a fender (which is used on boats),
with dimensions accordingly for the transducer, the tube and the amount
of water in the tank. By cutting the top, we had an opening that fitted the
dimesions of the transducer. The main idea was to cover the transducer
with the fender while the transducer was in the water, and by using the
Hevert principle we could take out the water inside the fender. Figure 5.10
shows how it looked like when the transducer was covered by the fender:
Figure 5.10: The transducer covered by the fender
41
5.1.3
The tube
The tube was also borrowed by the Kongsberg Maritime AS, and was used
to protect the electronics of Simrad SH90 from the water. The tube was
made water proof and the dimensions of the tube are, height equals 53.5cm
and the diameter is 22cm . A picture of the tube is shown in Figure 5.6.
5.1.4
The hydrophone
The hydrophone used was the Teledyne Reson TC4034. The datasheet
teledyne-reson.com/products/hydrophones/tc-4034/. In the thesis we have
assumed that the elements are located in front at the hydrophone since
there are no information given to us about the location of the elements.
A picture of the hydrophone mounted on the probe is shown in Figure 5.11.
Figure 5.11: The hydrophone, Teledyne Reson TC4034
As same as the transducer, the hydrophone has to be water free when
not used. That was easy in this case since the probe it was mounted on
could be positioned all the way up to the ceiling so the hydrophone could
be placed outside the water.
5.2
Simulation process in the tank
The position system was programmed by professor Svein Bøe in Matlab.
We used the latest version 13.49. The system was programmed in the
manner that we could position the hydrophone by giving its position on
the program. Even rotating was done in the matlab program. But for the
transducer we could only rotate by using matlab, for positioning we had
42
to do it physically, and measure its position. This was a bit challenging,
and because of that we had to do the experiments several times. We did
some changes in the code where we thought it needed changes, such as for
capturing the data from the oscilloscope.
5.3
Input signal
We are generating a sinus pulse through signal generator with cycles,
N = 3. The signal frequency is f = 100k H z with a peak to peak value
Vpk−pk = 500mV . As shown later in the thesis, the sound velocity in water
is c = 1488m/s . The wavelength of the signal in water is given as λ = cf =
1488m/s
100k H z = 1.488cm . I am using a burst period of 1s . As mentioned the
sinus signal is amplified by an audio amplifier, which gave us the signal
in Figure 5.12. From this figure we can see the amplitude of the signal input
is Vpk−pk = 17.2V .
Signal input to transmitter, f=100kHz N=3
10
8
6
4
Voltage in V
2
0
−2
−4
−6
−8
−10
−10
−5
0
5
10
15
Time in µs
Figure 5.12: Input signal
43
20
25
30
35
44
Chapter 6
Geometry
In this chapter I am going to show some geometrical cases and calculations
(such as distance caluclation, tangent caluclation, Sound path cases etc.).
6.1
In this section I am going to calculate the distances from the source to
the receiver while rotating the transducer. The hydrophone is placed at
a distance x 0 = 21.4cm from the transducer source. And as said before the
radius of the transducer is, r = 6cm . The parameters and the geometry
between the transducer and the hydrophone is illustrated in Figure 6.1.
Figure 6.1: Geometry of the distance between transmitter and receiver
The parameters in the figure above are:
• The distance x 0 = 21.4cm
• The radius of the transducer, r = 6cm
45
• The distance D = r + x 0 = 27.4cm
• Angle α is the rotational angle of the transducer
• Angle θ = 180◦ − θp = 180◦ − α − φ
• Angle θp = α + φ is the beam angle of the transmitted signal
¡
¢
• The distance aα = r × sin α
• The distance bα = r × cos α
• The distance x α0 = D − bα
• The distance x α =
q
2
aα
+ x α02
Figure 6.2: Tangent on the transducer to hydrophone.
First of all, I have to find the tangent of the signal transmitting from the
source. That is at which rotating angle α, the signal is not travelling directly
to the receiver. It means that the receiver is unseen by the source.
The Figure 6.2 shows the geometry of finding the tangent.
In this case θ = θp = 90◦ , and α = αt ang ent can be found by:
x αt ang ent =
sin φ =
p
D 2 − r 2 = 26.7cm
r
= 0.22 → φ = 12.64◦
D
⇓
46
αt ang ent = θp − φ = 77.36◦
(6.1)
Now that we know the angle of tangent, I can calculate by some
trigonometry the distances from the source to the receiver by rotating the
transducer from α = 0◦ to αt ang ent . It is not important to derive the distance
for every angle, and therefore I have chosen some angles given in Table 8.1.
α
◦
0
15◦
30◦
45◦
60◦
75◦
a α (cm)
b α (cm)
0
bα
(cm)
x α0 (cm)
x α (cm)
0
1.55
3
4.24
5.2
5.8
6
5.79
5.2
4.24
3
1.55
0
0.21
0.8
1.76
3
4.45
21.4
21.61
22.2
23.16
24.4
25.85
21.4
21.67
22.4
23.54
24.95
26.49
Table 6.1: Parameters of Figure 6.1
The other important geometrical problem is what is the rotational angle
α, when the hydrophone is at the -3dB point of the beamwidth or mainlobe.
This problem is illustrated in Figure 6.3:
Figure 6.3: Beamwidth of the transducer
β
Here, θp = 2 . I will come back to this problem later in chapter 8.
6.2
Sound propagation path
In this section I want to illustrate how sound can travel in water with
signal transmitted when the rotational angle |α| ≤ αt ang ent or with signal
transmitted when rotational angle |α| > αt ang ent . I will give some cases,
which is only assumptions, and try to give explainations from graphs and
the data I have received from the experiments. Some of those cases are
there to show that they are incorrect compared to the data measured.
6.2.1
Sound path for angles, α ∈ [−αt ang ent , αt ang ent ]
In this interval sound travels either direct to the receiver or it travels along
the surface of the transducer for a while and then travels direct to the
receiver. In case I, as illustrated in Figure 6.4, the sound wave travels direct
47
to the receiver. And in case II, as illustrated in Figure 6.5, the sound wave
travels along the surface of the transducer, until it reaches the 0◦ mark and
Case I
Transducer
Hydrophone
Figure 6.4: Case I: Sound wave travels direct to the receiver.
Case II
Transducer
Hydrophone
Figure 6.5: Case II: Sound wave travels first along the surface of the transducer, and then direct to
6.2.2
Sound path for angles, α ∈ [αt ang ent , 360◦ − αt ang ent ]
¡
¢
In this interval I want to study two cases. In case III, as illustrated in
Figure 6.6, the sound wave travels along the surface of the transducer until
it reaches the αt ang ent , and then travels direct to the receiver. In case IV, as
illustrated in Figure 6.7, the sound wave travels in the same way as in case
III, but not on the surface of the transducer, but in water and bends in the
same way as the surface of the transducer.
48
Case III
Transducer
Hydrophone
Figure 6.6: Case III: Sound wave travels on the surface of the transducer until it has a clear path to
Case IV
Transducer
Hydrophone
Figure 6.7: Cae IV: Sound wave travels the same path as case III, but in water.
49
50
Chapter 7
Pre-Experiments
7.1
Experiment I - Finding the beam direction
Before doing the experiment we have to find where the signal is strongest,
and in which way the transducer transmits the signal. This experiment is
done by rotating the transducer and by moving the receiver in different positions in height. I have picked som rotational angles for the transducer and
some positions for hydrophone in matlab.
Angles picked for transducer is as follows:
α = [−35◦ , −28◦ , −21◦ , −14◦ , −7◦ , 0◦ , 7◦ , 14◦ , 21◦ , 28◦ , 35◦ ]
Figure 7.1 shows the signals in different angles and for each angle at different height positions. Figure 7.1 is just showing how the signal look like
in different angles and height positions.
Figure 7.2 shows the amplitude value when moving the receiver in different
height positions. The different graphs shows the different angles for transducer.
Figure 7.3 shows the intensity of the amplitude, which is given as, I = A 2 .
And Figure 7.4 shows a 3D plot of Figure 7.3.
By inspection of Figure 7.2, 7.3 and 7.4, the signal is absolutely strongest
in 0◦ and when the receiver is placed at the same height as the middle of the
transducer. Here -960 mm is equal to 60 cm from tank floor. The height
parameter is for receiver probes head position given in the tank program.
51
0
500 1000
Samples
Signal at −7°
0.5
0
−0.5
0
500 1000
Samples
Signal at −28°
0.5
0
−0.5
0
500 1000
Samples
0
500 1000
Samples
°
Signal at 7
0.5
0
−0.5
0
500 1000
Samples
Signal at −14°
0.5
0
−0.5
0
500 1000
Samples
Amplitude, V
0.5
0
−0.5
Amplitude, V
0.5
0
−0.5
°
Signal at 28
Amplitude, V
°
Signal at 14
Amplitude, V
500 1000
Samples
Amplitude, V
0
Amplitude, V
0.5
0
−0.5
Amplitude, V
Amplitude, V
Amplitude, V
Amplitude, V
Amplitude, V
°
Signal at 35
°
Signal at 21
0.5
0
−0.5
0
Signal at 0
0
500 1000
Samples
Signal at −21°
0.5
0
−0.5
0.5
0
−0.5
500 1000
Samples
Figure 7.1: Signal in different angles and different heights
52
°
0.5
0
−0.5
Signal at −35°
0
500 1000
Samples
0
500 1000
Samples
Amplitude variation in height with spesific angles
1.3
−35°
−28°
1.2
−21°
1.1
−14°
−7°
1
Amplitude, V
0°
0.9
7°
0.8
14°
0.7
28°
21°
35°
0.6
0.5
0.4
0.3
−900
−920
−940
−960
Height, mm
−980
−1000
−1020
Figure 7.2: Inspection of maximum signal value I
Amplitude in dB as a function of angle and height
−1020
2
−1000
0
−2
−4
−960
−6
−940
Amplitude, dBV
Height, mm
−980
−8
−920
−10
−900
−12
−30
−20
−10
0
10
20
30
Angle, °
Figure 7.3: Inspection of maximum signal value II
53
3D plot of Amplitude in dB as a function of angle and height
2
0
5
−2
0
−6
−10
Amplitude, dBV
−4
−5
−8
−15
−850
−900
40
20
−950
He
igh
t, m
m
−10
0
−1000
−12
−20
−1050
le,
−40
°
Ang
Figure 7.4: 3D plot of Figure 7.3
54
The amplitude should be symmetrical around 0◦ , since the distance is
equal on both sides. But as shown in Figure 7.5, the amplitude is not
symmetrically. Reason behind that can be that I didn’t place the receiver in
front of the transducer, or as I assumed the rotational angle 62◦ in matlab
program of the tank may not equal to 0◦ of the transducer. To get the precice
measurements, I could have rotated the transducer by smaller intervals,
and where I would get the maximum value of the amplitude would be my
0◦ .
Amplitude when receiver position is constant, and by rotating transducer
2.5
2
log of Amplitude, dBV
1.5
1
0.5
0
−0.5
−1
−40
−30
−20
−10
0
10
20
30
40
angle,°
Figure 7.5: Amplitude plot as a function of angle
It has to be noticed that after doing the experiment we knew that we
should have picked more values for the rotational angle. We can see from
Figure 7.5, that the amplitude does’nt fall with -3dB which is the point for
the mainlobe. In this experiment the distance between source and receiver
is unknown, and therefore it is not possible to say how many angles we
should take for.
7.2
Experiment II - Finding sound velocity
Sound velocity in water is approximately 1500m/s . To get an exact value of
the sound velocity in the water tank I had to do some measurements with
the transducer sending signals direct to the hydrophone. It means the angle
of incidence, θ = 0◦ . There are two methods for finding the sound velocity,
55
one is by measuring the temperature of the water and using Equation 3.9.
Since we have a small water tank, we don’t have to consider for depth, and
since the experiments are done in pure water, the parameter of salinity is
negligible. Equation 3.9 becomes:
c (T ) = 1402.3 + 4.95T − 0.055T 2 + 0.00029T 3
(7.1)
After filling the tank I placed a thermometer in the water and after some
days I could read that the temperature was constant at T = 19◦C . By using
Equation 7.1 we find that the sound velocity in water is:
¡
¢3
¡
¢
¡
¢2
c 19◦C = 1402.3+4.95×19◦C −0.0055× 19◦C +0.00029× 19◦C = 1478.5m/s
(7.2)
The other method is by measuring time received in two different positions for the hydrophone. First I placed the hydrophone at a distance,
x 1 = 21.4cm from the transducer and received the signal as shown in Figure 7.6. Signal was received at a time, t 1 = 147µs .
Secondly I moved the hydrophone at a distance, x 2 = 41.4cm from the transducer and received the signal as shown in Figure 7.7. Signal was received at
a time, t 2 = 281.4µs .
Hydrophone at a distance d = 21.4 cm.
5
4
3
Voltage (V)
2
1
X: 147
Y: 0
0
−1
−2
−3
−4
−5
80
100
120
140
160
180
t (µs)
200
220
Figure 7.6: Signal captured at a distance of 21.4 cm
56
240
260
Hydrophone at a distance d = 41.4 cm.
5
4
3
Voltage (V)
2
1
X: 281.4
Y: 0
0
−1
−2
−3
−4
−5
220
240
260
280
300
320
t (µs)
340
360
380
400
Figure 7.7: Signal captured at a distance of 41.4 cm
The sound velocity can be found by:
∆x = x 2 − x 1 = 41.4cm − 21.4cm = 20cm
∆t = t 2 − t 1 = 281.4µs − 147µs = 134.4µs
c=
∆x
20cm
=
= 1488.1m/s
∆t
134.4µs
(7.3)
We can see that the sound velocity for this two methods differs by a
factor of 10m/s . That’s not bad, and the reasons for the difference could
either be geometrically (Transducer and hydrophone not placed correct) or
that the temperature measured differed with some value. The velocity I will
use further in the thesis is c = 1488m/s .
57
58
Chapter 8
Main experiments
In this chapter I will show the main experiments of this thesis. Experiment
III (where I rotate my transducer from −90◦ to 90◦ ) is actually done to find
the beamwidth of the transducer. By plotting the amplitude in d B I can
find where the amplitude drops by −3d B and then calculate the effective
element size d e f f . Experiment III is also done to find the instrumental
delay in all instruments, and to prove and disprove some of my geometrical
calculations and cases in Chapter 6. In experiment IV (where I rotate my
transducer from 85◦ to 275◦ ) I will show whether it is possible or not to
capture the signal when the transducer is rotated 180◦ . As well as finding
the sound velocity for the diffracted wave and to prove and disprove some
of the geometrical cases. I will also compensate amplitudes from both
experiments by studying the difference.
8.1
Experiment III - Front experiment
This experiment was done by rotating the transducer from −90◦ to 90◦ with
an interval of 1◦ . At first, the transducer was rotated by manually changing the parameters in Matlab. This took too much time, thus we made a
program in matlab that rotated the transducer automatically. The program
first captured the data from the oscilloscope, plotted it in the matlab and
then rotated the transducer with an angle of 1◦ . It was quite challenging
since the signal sometimes got delayed because of the mechanism.
The program plotted every data in the matlab in one single figure. In the
end we got 181 signals, which is shown in Figure 8.1. Every data/signal has
1000 samples, and is plotted as the received voltage on the hydrophone.
The samples are converted into time with some simple processing, where
the parameters from the oscilloscope is taken in consideration.
Figure 8.2 shows the signal in d B at every angle as function of time. From
the figure it is confirmed that the signal at 0◦ reaches the receiver earlier
than the other signals. As the positive angle increases, the signal gets more
and more delayed. And as the negative angle decreases, the signal gets more
and more delayed. We can also see that the signal intensity decreases by in59
creasing or decreasing the angle from 0◦ . It has two reasons, the first one
is that the range increases as we decrease or increase the angle from 0◦ .
The other one has something to do with the transducers directivity, which
I don’t have so much information about.
Measurements for rotational angle α ∈ [−90°, 90°].
3
2
Voltage (V)
1
0
−1
−2
−3
120
140
160
180
200
220
t (µs)
240
260
280
300
Figure 8.1: Signal captured by the oscilloscope
Decibel of the signals
0
120
140
−5
160
−10
Time, µs
180
200
−15
220
−20
240
260
−25
280
−30
300
−80
−60
−40
−20
0
20
Rotational angle, α
40
Figure 8.2: Signal in dB
60
60
80
8.1.1
Amplitude inspection and beamwidth calculation
Amplitude plot
5.5
5
4.5
4
Amplitude, V
3.5
3
2.5
2
1.5
1
0.5
0
−80
−60
−40
−20
0
20
Rotational angle, α
40
60
80
60
80
Figure 8.3: Amplitude vs rotational angle
Decibel value of the amplitude.
0
X: −35
Y: −3.055
Normalized Amplitude, dBV
−5
X: 37
Y: −3.056
−10
−15
−20
−25
−30
−80
−60
−40
−20
0
20
Rotational Angle, α
Figure 8.4: Amplitude dB plot
61
40
Figure 8.3 shows the amplitude of the sound signal (in voltage, V) as a
function of the rotational angle α. As expected the maximum amplitude
is achieved at α = 0◦ . The amplitude reduces as we rotate the transducer,
which is expected since the rotational angle α is proportional to beam angle
θp . The distance to receiver increases as well by rotating the transducer.
However the reduction of the amplitude has to do with the directivity index
of the transducer and the distance between transducer and hydrophone.
Figure 8.4 shows the normalized amplitude in d B plot. It is marked in the
figure where the -3dB point is. It appears at the rotational angles α = 37◦
and α = −35◦ . Since the α has an error of 1◦ , I will consider the -3dB point
to appear at α = 36◦ .
Beamwidth calculation
Now going back to the beamwidth problem from chapter 6. The geometry
is illustrated in Figure 6.3 in chapter 6. It is mentioned there that the beam
angle θp is half the beamwidth angle β:
a 36◦ = r × sin 36◦ = 3.53cm
b 36◦ = r × cos 36◦ = 4.85cm
0
x 36
◦ = D − b 36◦ = 22.55cm
x 36◦ =
sin φ =
q
2
02
a 36
◦ + x 36◦ = 22.82cm
a 36◦
→ φ = 8.9◦
x 36◦
β
= θp = α + φ = 36◦ + 8.9◦ = 44.9◦ → β = 89.8◦
2
(8.1)
This means that our signal has a mainlobe of β = 89.8◦ (in radians,
β = 1.57r ad ), and the theory of omni-directional signal is not the case.
However this can be compared with the beampattern of the transducer.
Finding the effective element size, de f f
There is a possibility that explains why my signal is not a omni-directional
signal. Since my transducer is only connected to one stripe and the other
stripes are unconnected and since they are very closed to each other, there
is some possibility that the extra energy is coming from the neighbouring
62
unconnected elements. When the element vibrates, it may vibrate the
neighboring elements as well. If we take this theory further, we can
calculate the effective element size de f f of the transducer:
sin β =
λ
λ
1.488cm
→ de f f =
=
= 1.488cm → d e f f ≈ λ
de f f
sin β
sin 1.57
(8.2)
This gives the interfering element size di nt from the neighboring
elements as:
d i nt = d e f f − d el ement = 1.488cm − 1cm = 0.488cm
(8.3)
Assuming that interference is the same on both sides, we get the
interference from each element of 0.244 cm.
8.1.2
Beampattern comparison
The effective element size theory can be further studied by simulating or
comparing different beampatterns for the transducer with different element size. In this section I will compare those patterns with my measured
amplitude dB plot, and conclude that the effective element size theory is
more probable than beampattern theory.
In Selfridge, Kino and Khuri-Yakub (1980)[?], it is stated that the angular response pattern of a single element is of the form:
p = p0
sin πd
λ × sin θp
πd
λ
(8.4)
× sin θp
In my simulations I have used that p 0 = 1. In the same article Selfridge,
Kino and Khuri-Yakub (1980)[?] uses an extra term of cos θp to narrow the
beampattern. They used Rayleigh-Sommerfield diffraction formula, which
gave them this result:
p(r, θp ) =
p0d
j (λ)1/2
e
2 j πr /λ
×
sin πd
λ × sin θp
πd
λ
× sin θp
× cos θp
(8.5)
This is also stated in Johnson and Dudgeon p. 37 (1993) [?]. Johnson
and Dudgeon p. 63 (1993) [?] defines a linear aperture function as:
W=
sin k x d /2
k x /2
(8.6)
Where k x = k × sin θp = 2π
λ × sin θp . With some discussion with my
supervisor Sverre Holm, I was motivated to compare with another pattern
1+cos θp
described in Equation 8.7. This pattern gets the extra term of
2
instead of cos θp in Equation 8.5. This pattern is supposed to lie in between
of patterns in Equation 8.4 and in Equation 8.5. My supervisor couldn’t
find any reference for this formula but he remembered it from his earlier
works. However this pattern was simulated just because I was curious.
63
b(θp ) =
sin πd
λ × sin θp
πd
λ
× sin θp
×
1 + cos θp
(8.7)
2
As seen in Figure 8.5, the measured amplitude is close to the first
beampattern in Equation 8.4, but still Equation 8.4 is a bit wider
(Equation 8.4 is equal to Equation 8.6, and thus they have the same pattern
in Figure 8.5. That’s why the Equation 8.4 is not visible). The reason
why my measured amplitude doesn’t get any zero points is because I’m
transmitting a pulsed signal, and that’s obvious that I will not get any
zeros in my amplitude plot. Since the theoretical pattern is wider, and
the measured amplitude does not fully follow this pattern, thus the theory
of effective element size is more probable in my case. In the Figure 8.6
I have plotted the patterns in different element size, and tried to study the
behaviour of the theoretical patterns. The patterns gets narrower and equal
to each other as I increase the element size. This does not satisfy the theory
of effective element size, but is an indicator that for some element size the
amplitude plot is almost equal to the theoretical pattern in Equation 8.4.
Decibel value of the amplitude w/d=1cm.
Normalized Amplitude, dBV
0
−5
sin[(πd/λ)×sin(θp )]
(πd/λ)×sin(θp )
sin[(k x d/2)]
(k x /2)
−10
sin[(πd/λ)×sin(θp )]
(πd/λ)×sin(θp ) ×
cos(θp )
sin[(πd/λ)×sin(θp )]
1+cos(θp )
(πd/λ)×sin(θp ) ×
2
Meassured Amplitude
−15
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 8.5: Different beampatterns with element size d = 1cm . The one with magneta color is the
pattern in Equation 8.4, the green is from Equation 8.5, the black is from Equation 8.7, the dots are
from Equation 8.6 and the red one is the measured amplitude in dbV.
64
−10
−1
0
1
Decibel value of the amplitude w/d=1.3cm.
0
−5
−10
−15
−1
0
1
Decibel value of the amplitude w/d=1.5cm.
0
−5
−10
−15
Normalized Amplitude, dBV
−15
Normalized Amplitude, dBV
−5
−1
0
1
Normalized Amplitude, dBV
Normalized Amplitude, dBV
Normalized Amplitude, dBV
Normalized Amplitude, dBV
Decibel value of the amplitude w/d=1.1cm.
0
Decibel value of the amplitude w/d=1.2cm.
0
−5
−10
−15
−1
0
1
Decibel value of the amplitude w/d=1.4cm.
0
−5
−10
−15
−1
0
1
Decibel value of the amplitude w/d=1.6cm.
0
−5
−10
−15
−1
0
1
Figure 8.6: Different beampatterns with respectively element size d = 1.1cm , d = 1.2cm , d = 1.3cm ,
d = 1.4cm , d = 1.5cm and d = 1.6cm
65
8.1.3
Instrumental delay
Received signal at α = 15°
3
2
2
Voltage (V)
Voltage (V)
Received signal at α = 0°
3
1
0
−1
X: 147.8
Y: 0
−2
−3
1
0
−1
X: 149.4
Y: 0
−2
150
200
t (µs)
250
−3
300
150
3
2
2
1
0
−1
X: 154.8
Y: 0
−2
−3
0
−1
X: 163.4
Y: 0
−2
150
200
t (µs)
250
−3
300
150
200
t (µs)
250
300
Received signal at α = 75°
3
2
2
Voltage (V)
Voltage (V)
Received signal at α = 60°
1
0
X: 172.2
Y: 0
−2
−3
300
1
3
−1
250
Received signal at α = 45°
3
Voltage (V)
Voltage (V)
Received signal at α = 30°
200
t (µs)
1
0
−1
X: 182.2
Y: 0
−2
150
200
t (µs)
250
300
−3
150
200
t (µs)
250
300
Figure 8.7: Signals captured in different rotational angle
Instrumental delay in this case is the total instrumental delay from all instruments used. Equipments such as the generator, transducer and hydrophone have some sort of delays. This is important when I have to calculate
the sound velocity in experiment IV.
I have chosen to investigate the instrumental delay at some angles, and
as shown in Figure 8.7, I have marked their time of arrival T at the hydrophone. Instrumental delay is given as ∆T :
Tα = t α + ∆Tα
(8.8)
Where t α = xcα , Tα is the total time registered in the oscilloscope and ∆Tα
is the instrumental delay. From Equation 8.8 I get the following equation
for the instrumental delay:
∆Tα = Tα − t α
66
(8.9)
By taking Tα from Figure 8.7 and x α from Table 8.1, I get the following
values for ∆Tα :
α
Tα (µs)
t α (µs)
∆Tα (µs)
◦
147.8
149.4
154.8
163.4
172.2
182.2
143.8
145.6
150.5
158.2
167.7
178
4
3.8
4.3
5.2
4.5
4.2
0
15◦
30◦
45◦
60◦
75◦
Table 8.1: Instrumenta delay on different rotational angles
From this it seems that the instrumental delay is about 4µs , but I will
calculate the instrumental delay by taking the means of these values. The
instrumental delay becomes:
∆T =
8.2
∆T0◦ + ∆T15◦ + ∆T30◦ + ∆T45◦ + ∆T60◦ + ∆T75◦
= 4.33µs
6
(8.10)
Experiment IV - Back experiment
This experiment was done by rotating the transducer from 85◦ to 275◦ with
an interval of 1◦ . In this experiment I had to increase the amplitude on the
input signal, since the signal got weaker when operating in this region of
rotation. I could not increase the amplitude much more because of the limitations on the amplifier.
Figure 8.8 shows the plot which was obtained when the experiment was
finished, and Figure 8.9 shows dB value plot at every rotational angle.
The amplitude plot in V and dBV is shown in Figure 8.13 and Figure 8.14.
From Figure 8.9 and Figure 8.13 it is shown that the signal is received upto
α = 120◦ . From α = 120◦ to α = 240◦ the signal could not be received or not
could not be distinguished from the noise. This is further confirmed in Figure 8.10.
67
Measurements for rotational angle α ∈ [−90°, 90°].
1
0.8
0.6
Voltage (V)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
100
150
200
250
300
350
t (µs)
400
450
500
550
Figure 8.8: Signal captured by the oscilloscope
Decibel of the signals
0
100
150
−5
200
−10
250
Time, µs
−15
300
−20
350
400
−25
450
−30
500
−35
550
100
120
140
160 180 200
Rotational angle, α
220
Figure 8.9: Signal in dB
68
240
260
°
0.5
0
−0.5
100 200 300 400 500
t (µs)
Received signal at α = 179°
0.5
0
−0.5
100 200 300 400 500
t (µs)
0.5
0
−0.5
100 200 300 400 500
t (µs)
Received signal at α = 180°
0.5
0
−0.5
100 200 300 400 500
t (µs)
100 200 300 400 500
t (µs)
Received signal at α = 250°
0.5
0
−0.5
100 200 300 400 500
t (µs)
Voltage (V)
0.5
0
−0.5
100 200 300 400 500
t (µs)
Received signal at α = 160
0.5
0
−0.5
100 200 300 400 500
t (µs)
Received signal at α = 181°
0.5
0
−0.5
100 200 300 400 500
t (µs)
°
Voltage (V)
Voltage (V)
Voltage (V)
°
Received signal at α = 200
0.5
0
−0.5
°
Voltage (V)
Received signal at α = 145
Voltage (V)
100 200 300 400 500
t (µs)
Voltage (V)
0.5
0
−0.5
Received signal at α = 110
°
Voltage (V)
Voltage (V)
Voltage (V)
°
Received signal at α = 120
Received signal at α = 215
0.5
0
−0.5
100 200 300 400 500
t (µs)
Received signal at α = 270°
0.5
0
−0.5
100 200 300 400 500
t (µs)
Figure 8.10
69
°
Voltage (V)
100 200 300 400 500
t (µs)
°
Voltage (V)
0.5
0
−0.5
Received signal at α = 90
Voltage (V)
Voltage (V)
Voltage (V)
°
Received signal at α = 85
Received signal at α = 240
0.5
0
−0.5
100 200 300 400 500
t (µs)
Received signal at α = 275°
0.5
0
−0.5
100 200 300 400 500
t (µs)
Figure 8.10 shows the signal in rotational angles 85◦ , 90◦ , 110◦ , 120◦ ,
145◦ , 160◦ , 179◦ , 180◦ , 181◦ , 200◦ , 215◦ , 240◦ , 250◦ , 270◦ and 275◦ .
It becomes more clear now that I have no signal at 180◦ . This can have
many reasons and some of them are discussed in Chapter 9 (conclusion
and discussion). However, the signal seems to be "drowned" in noise. It
has to be noted that I have reduced the noise on my figures, thus the signal
does not appear. That’s alright since the signal is too small, thus I can say
that I’m not receiving any signal at this point. Figure 8.10 also confirms
that the signal seems to be totally reduced after 120◦ .
If I further study the signal from 85◦ to 90◦ and compare it with experiment
III, then it is (as expected) higher in amplitude, but (not expected) reaches
the hydrophone earlier than in experiment III. The reason behind this can
be many, but some of them are discussed in Chapter 9 (conclusion and discussion). Figure 8.11 shows the signal from experiment III and Figure 8.12
shows the signal from experiment IV:
2
Received signal at α = 86°
Voltage (V)
Voltage (V)
Received signal at α = 85°
X: 189.2
Y: 0
0
−2
150
200
t (µs)
250
2
0
−2
300
150
2
0
X: 190.8
Y: 0
150
200
t (µs)
250
150
Voltage (V)
Voltage (V)
X: 192.4
Y: 0
250
X: 191.4
Y: 0
−2
200
t (µs)
250
300
Received signal at α = 90°
0
200
t (µs)
300
0
300
2
150
250
2
Received signal at α = 89°
−2
200
t (µs)
Received signal at α = 88°
Voltage (V)
Voltage (V)
Received signal at α = 87°
−2
X: 190.2
Y: 0
300
2
0
X: 193
Y: 0
−2
150
200
t (µs)
Figure 8.11: Signal output for experiment III
70
250
300
Received signal at α = 85°
Received signal at α = 86°
0.5
Voltage (V)
Voltage (V)
0.5
X: 189
Y: −0.00625
0
−0.5
X: 189
Y: 0
0
−0.5
100
200
300
400
t (µs)
500
100
Received signal at α = 87°
500
0.5
Voltage (V)
Voltage (V)
300
400
t (µs)
Received signal at α = 88°
0.5
X: 189.5
Y: 0.00625
0
−0.5
X: 190.5
Y: 0
0
−0.5
100
200
300
400
t (µs)
500
100
°
200
300
400
t (µs)
500
°
Received signal at α = 89
Received signal at α = 90
0.5
Voltage (V)
0.5
Voltage (V)
200
X: 191
Y: 0
0
−0.5
X: 191.5
Y: 0
0
−0.5
100
200
300
400
t (µs)
500
100
200
300
400
t (µs)
500
Figure 8.12: Signal output for experiment IV
8.2.1
Amplitude plot and comparison from experiment
III
As mentioned earlier, Figure 8.13 and Figure 8.14 are amplitude plots of experiment IV. As it is shown, the amplitude is almost 0V after about α = 120◦
and before about α = 240◦ .
One thing I’m very interested in studying is how to compensate amplitudes
of experiment III and experiment IV. As mentioned the signal input voltage
in experiment IV is higher than in experiment III. To study this I have
plotted the amplitude on both experiments in Figure 8.15, and then compensated for the amplitude difference at α = 85◦ to α = 90◦ and the same
on the other side. That is why I had to do experiment IV from α = 85◦ to
α = 275◦ and not from α = 90◦ to 270◦ . The compensated amplitude plot of
the whole transducer from α = 0◦ to α = 359◦ is shown in Figure 8.16.
71
Amplitude plot
1.2
Normalized Amplitude, dBV
1
0.8
0.6
0.4
0.2
0
100
120
140
160
180
200
Rotational Angle, α
220
240
260
240
260
Figure 8.13: Amplitude vs rotational angle
Amplitude plot in dB
0
Amplitude, dBV
−10
−20
−30
−40
−50
100
120
140
160
180
200
Rotational angle, α
220
Figure 8.14: Amplitude vs rotational angle in dB
72
Amplitude of experiments III and IV
6
Experiment III [0° − 90°]
Experiment III [270° − 359°]
5
Experiment IV [85° − 180°]
Experiment IV [181° − 275°]
Amplitude, V
4
3
2
1
0
0
50
100
150
200
Rotational Angle, α
250
300
350
Figure 8.15: Amplitude plot of experiment III and IV
Compensated Amplitude of experiments III and IV
6
Experiment III [0° − 90°]
5
Experiment III [270° − 359°]
Experiment IV [85° − 180°]
Experiment IV [181° − 275°]
Amplitude, V
4
3
2
1
X: 124
Y: 0.01375
0
0
50
100
X: 240
Y: 0.02625
150
200
Rotational Angle, α
250
Figure 8.16: Compesated amplitude plot
73
300
350
Compensated Amplitude in dB of experiments III and IV
Experiment III [0° − 90°]
10
Experiment III [270° − 359°]
Experiment IV [85° − 180°]
0
Amplitude, dBV
Experiment IV [181° − 275°]
−10
−20
−30
−40
−50
0
50
100
150
200
Rotational Angle, α
250
300
350
Figure 8.17: Compensated amplitude plot in dB
8.3
Sound velocity profile for α ∈ αt ang ent , 360◦ − αt ang ent
£
In this section I will calculate the sound velocity when the transducer is
not transmitting the sound signal "direct" to the hydrophone. To do so
I have to calculate the distance from the element to the receiver at some
given rotational angle α. It was found in chapter 6 that the distance
between transducer and receiver (when the rotational angle was αt ang ent =
77.36◦ ≈ 77◦ ) was x αt ang ent = 26.7cm . To calculate the distance by rotating the
transducer by α is given as:
x α = 2πr
|
³ α − αt ang ent ´
360◦
{z
x¯α
+x αt ang ent
}
Since the signal is readable for upto α = 120◦ I will only calculate
the distance upto that point. The distance x¯α is the circular bow on the
transducer. The distances is given in Table 8.2
74
¤
α
◦
85
90◦
95◦
100◦
105◦
110◦
115◦
120◦
x¯α (cm)
x α (cm)
0.84
1.36
1.88
2.4
2.93
3.46
3.98
4.5
27.5
28.06
28.58
29.1
29.63
30.16
30.68
31.2
From Figure 8.11 and Figure 8.12 I know the received time is incorrect
in experiment IV compared to in experiment III. Therefor I am going to
calculate the velocity from a given rotational angle α to α = 90◦ . For that I
need to know the circular bow of that rotational angle α to α = 90◦ . This is
given as:
¯ ◦ = 2πr
x α−90
α − 90◦
360◦
Figure 8.18 shows the received signals at different times at rotational
angles, α = [95◦ , 100◦ , 105◦ , 110◦ , 115◦ , 120◦ ]. The received time at α = 90◦ are
shown in Figure 8.12. The sound velocity is calculated as:
¯ ◦ = Tα − T90◦ = Tα − 191.5µs
t α−90
cα =
¯ ◦
x α−90
¯ ◦
t α−90
The result is given in Table 8.3:
α
◦
95
100◦
105◦
110◦
115◦
120◦
¯ ◦ (cm)
x α−90
¯ ◦ (µs)
t α−90
c α (m/s)
0.52
1.05
1.57
2.09
2.62
3.14
4
8
11.5
16
20
22.5
1300
1312
1365
1306
1310
1395
Table 8.3: Sound velocity for experiment IV
It seems like the sound velocity is almost constant around 1300m/s 1400m/s .
75
Received signal at α = 95°
Received signal at α = 100°
0.5
Voltage (V)
Voltage (V)
0.5
X: 195.5
Y: 0
0
−0.5
X: 199.5
Y: 0
0
−0.5
100
200
300
400
t (µs)
500
100
Received signal at α = 105°
500
0.5
Voltage (V)
Voltage (V)
300
400
t (µs)
Received signal at α = 110°
0.5
X: 203
Y: 0
0
−0.5
X: 207.5
Y: 0
0
−0.5
100
200
300
400
t (µs)
500
100
Received signal at α = 115°
200
300
400
t (µs)
500
Received signal at α = 120°
0.5
Voltage (V)
0.5
Voltage (V)
200
X: 211.5
Y: 0
0
−0.5
X: 214
Y: 0
0
−0.5
100
200
300
400
t (µs)
500
100
200
300
400
t (µs)
Figure 8.18: Signal output for experiment IV
76
500
Part IV
Conclusion
77
Chapter 9
Conclusion and discussion
9.1
Conclusion
As seen on the results in experiment IV, the signal is obtained and received
when the rotational angle is within α ∈ [−120◦ , 120◦ ]. When the rotational
angle is outside this boundary the signal seems to be "drowned" in noise
and is unable to read on the oscilloscope. There are two ways to conclude
why this is the case, and both may be proportional to each other.
The first way is to compare with Figure 9.1 and Figure 9.2 (made by my
supervisor Sverre Holm, code based on Duda and Martens (1998)[?] and
referenced from Morse and Ingard (1968)[?]). These are the theoretical
models for human listening. The signal response is plotted as a function of
angles with different µ = ka values. µ is the normalized frequency, where
k = 2π
λ and a = radius of the head/sphere. The normalized frequency µ in
this thesis is µ = ka = 25.34 ≈ 25.
By first studying Figure 9.1, I can see that my signal should be somewhere
between the green and the black plot. The green starts to oscillate (or has
it ripples) around 140◦ , and a µ = 20. Figure 9.2 is more interesting and
important in my case, since it describes the decrease of the response when
µ = 25. From Figure 9.2 it can be seen that the signal response falls with
about 17 − 18d b , and in my experiments the signal amplitude falls with
about 50 − 55d b . The most important thing is to see that the response starts
to oscillate (or has it ripples) around 120◦ , which is very close to my results.
The second way to look at the result is by the theory given in Chapter 4.
In my thesis the background is indicated either by noise background or reverberation background. However, in both cases the signal/echo level SL
is larger than the noise or reverberation level N L or RL , upto a range R n or
R r . As familiar, the source to receiver range increases by rotating the transducer towards α = 180◦ . So the signal/echo level SL is a function of range as
well as for α. From Chapter 4 it is known that for some range R r or R n the
signal/echo level SL equals the reverberation- or noise level RL or N L . And
for ranges larger than R r or R n the signal gets smaller than the background,
79
and is impossible to distinguish from the background. In my experiments,
there has to be a point right after α = 120◦ where the signal/echo level SL
equals the background level, and therefore is impossible to distinguish from
the background source.
However, both of these cases concludes with that my results are pretty
good, and the signal is not "achievable" after α = 120◦ .
Figure 9.1: Signal response in d B vs angle for different ka values
Magnitude response vs angle of incidence at 10000 sphere radii
30
ka = 25
25
dB, offset 5 dB per curve
ka = 5
20
ka = 2.5
15
ka = 1
10
5
ka = 0.5
0
−5
0
20
40
60
80
100
120
140
Angle from spherical axis [degrees]
160
180
Figure 9.2: Signal response in d B vs angle for different kav al ues , and especially for ka = 25
80
9.1.1
Sound propagation path
In chapter 6 I had given some cases for how the sound signal propagates
in two different α regions.
After the results
in experiment III and IV, I can
£
¤
conclude that for α ∈ −αt ang ent , αt ang ent the sound waves travels directly
towards the hydrophone, hence case I is correct and case II is uncorrect.
When I measured the instrumental delay I assumed that case I was correct, and that gave me almost identically delays at different angles, and that
proves my assumption was correct.
For α ∈ αt ang ent , 360◦ − αt ang ent case III seems to be correct. Since the
measuring of sound velocity in experiment IV gave me almost identically
sound velocities in all angles. The sound velocities in this region seems to
be smaller than sound velocity in water, thus I can conclude that the sound
is travelling along the surface of the transducer until it reaches the tangent
point and then travels directly towards the hydrophone. It is said in Selfridge, Kino and Khuri-Yakub (1980)[?] that: In typical applications, the
transducer element is separated from the water medium by a thin membrance or by a rubberlike material of the same impedance as water. This
means that the sound velocity has to be the same if the density of the material is the same as water. The reason of getting slower sound velocity are
many, and they are discussed in next section.
£
9.2
¤
Discussion
Working on this thesis was very complicated and challenging. In this
section I’m going to discuss some factors which may have played some
part in "bad" measurement results in the experiments. This factors are
important to take in consideration when reading this thesis.
• Deviation in geometry
The hydrophone may have not been placed in front of the transducer, in both horizonal or vertical plane.
Since experiments was done in different days and I had to move the
transducer after every experiment, thus the geometry may not be
same in every experiment. That could have been the reason why
sound wave reached earlier to the receiver in experiment IV.
• Inexperienced in measurements
• Time display in the oscilloscope may have some deviation from real
time.
• Rotation by 1◦ may not rotate 1◦ but with some deviation.
• The radius of the transducer may be different than measured.
• Leakage of water in the tank may give wrong positions for the
transducer according to the amount of water in the tank.
81
• Air bubbles occured on the surface of the transducer and hydrophone
while doing the experiments. This gave some time delay and
reduction of amplitude on the received signal. When the air bubbles
was cleaned, the signal got bigger in amplitude and it was impossible
to compare it with other measurements and the experiment had to be
done over again.
• Since the experiments was done in different days and we don’t have
some cleaning system mounted on the tank, the water may have been
dirty. Some of the dirt may have been placed at the surface of both
transducer and hydrophone.
• Air bubbles and dirt on transducer may have been the reason why
sound velocity along the surface of the transducer was lower than
sound velocity in water.
• Temperature may have also changed as the days past.
• As seen from all experiments, the received signal has some ripples
in the end of the signal. This could have been caused by diffraction
of the sound signal by the hydrophone’s shape or by reflection and
scattering caused by the hydrophone and the probe it is mounted on.
• The result could have been better if I had sent a sound signal with
higher amplitude, or sent with lower frequency. Both increasing the
amplitude and decreasing the frequency was impossible in this thesis,
because of the limitations on the transducer and the JBL amplifier.
9.3
Future work
• Do experiment III and experiment IV over again. Since I have now
got more experienced in measuring in the tank than before.
• Transmitting less or more pulse periods.
• Do the experiment with lower µ value, better if the µ value is
comparable for humans as well. For example with a head of radius a =
6cm , and a frequency range of 20 − 20, 000H z , the µ value for human
head is around 0.02 and 21.66. For getting in this region I’ll have to
reduce my frequency input, and for doing that I have to change my
transducer to another one that matches these limitations. This wasn’t
easy in my case, since there was no choose and throw concept about
the transducer. First of all, I had to take the one that was available.
The second one, I neeed a commercial Simrad transducer for this
experiment. A transducer with high frequency, since the water tank is
too small. Kongsberg Maritime AS had another transducer available,
but that one had to low µ value (Operated in a frequencies about
20k H z -30k H z ). It would have been difficult to do measurements with
that transducer.
82
• Do the experiment with higher amplitude, so the signal level can be
separated from the background when the transducer is rotated 180◦ .
For tis I’ll have to change the audio JBL amplifier with one that fits
these limitations.
• Inserting small fishes in the tank and receiving with a circular
transducer. Check if it’s possible to receive the signal if the transducer
element is 180◦ to the fish.
83
84
Part V
Appendix
85
Appendix A
All the simulation and programming in this thesis has been done in Matlab
software. There has been written a simulation code for the tank by professor Svein Bøe in Matlab which was used and modified during the experiments (I haven’t attached this on here, but can be shown if needed). Three
of the experiments is done with my co-student Asle Tangen (had another
master thesis than me), but the processing of data afterwards is done individually.
Below is my matlab code that I programmed to process my data from all
experiments:
Matlab Code
Code for experiment I:
1
% Experiment I
2
3
H = figure ;
4
5
6
7
8
9
10
11
12
13
14
load ( ’ x27 . mat ’ )
Vpp1 = max( S ) −min ( S ) ;
p l o t ( z , Vpp1 ) ;
hold on
axis tight
t i t l e ( ’ Amplitude v a r i a t i o n i n h e i g h t with s p e s i f i c
angles ’ )
x l a b e l ( ’ Height , mm’ )
y l a b e l ( ’ Amplitude , V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
15
16
clear S
17
18
19
20
load ( ’ x34 . mat ’ )
Vpp2 = max( S ) −min ( S ) ;
p l o t ( z , Vpp2 , ’ r ’ ) ;
87
21
22
23
24
25
26
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Hoyde i mm’ )
y l a b e l ( ’ Spenning i V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
27
28
clear S
29
30
31
32
33
34
35
36
37
38
lo ad ( ’ x41 . mat ’ )
Vpp3 = max( S ) −min ( S ) ;
p l o t ( z , Vpp3 , ’ g ’ ) ;
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Hoyde i mm’ )
y l a b e l ( ’ Spenning i V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
39
40
clear S
41
42
43
44
45
46
47
48
49
50
lo ad ( ’ x48 . mat ’ )
Vpp4 = max( S ) −min ( S ) ;
p l o t ( z , Vpp4 , ’ k ’ ) ;
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Hoyde i mm’ )
y l a b e l ( ’ Spenning i V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
51
52
clear S
53
54
55
56
57
58
59
60
61
62
lo ad ( ’ x55 . mat ’ )
Vpp5 = max( S ) −min ( S ) ;
p l o t ( z , Vpp5 , ’ c ’ ) ;
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Hoyde i mm’ )
y l a b e l ( ’ Spenning i V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
63
64
clear S
65
66
67
68
lo ad ( ’ x62 . mat ’ )
Vpp6 = max( S ) −min ( S ) ;
p l o t ( z , Vpp6 , ’m ’ ) ;
88
69
70
71
72
73
74
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Hoyde i mm’ )
y l a b e l ( ’ Spenning i V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
75
76
clear S
77
78
79
80
81
82
83
84
85
86
load ( ’ x69 . mat ’ )
Vpp7 = max( S ) −min ( S ) ;
p l o t ( z , Vpp7 , ’ * ’ ) ;
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Hoyde i mm’ )
y l a b e l ( ’ Spenning i V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
87
88
clear S
89
90
91
92
93
94
95
96
97
98
load ( ’ x76 . mat ’ )
Vpp8 = max( S ) −min ( S ) ;
p l o t ( z , Vpp8 , ’ . ’ ) ;
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Hoyde i mm’ )
y l a b e l ( ’ Spenning i V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
99
100
clear S
101
102
103
104
105
106
107
108
109
110
load ( ’ x83 . mat ’ )
Vpp9 = max( S ) −min ( S ) ;
p l o t ( z , Vpp9 , ’ s ’ ) ;
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Height , mm’ )
y l a b e l ( ’ Amplitude , V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
111
112
clear S
113
114
115
116
load ( ’ x90 . mat ’ )
Vpp10 = max( S ) −min ( S ) ;
p l o t ( z , Vpp10 , ’ o ’ ) ;
89
117
118
119
120
121
122
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Height , mm’ )
y l a b e l ( ’ Amplitude , V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
123
124
clear S
125
126
127
128
129
130
131
132
133
134
135
lo ad ( ’ x97 . mat ’ )
Vpp11 = max( S ) −min ( S ) ;
p l o t ( z , Vpp11 , ’ d ’ ) ;
axis tight
% t i t l e ( ’ 3 4 grader ’ )
x l a b e l ( ’ Height , mm’ )
y l a b e l ( ’ Amplitude , V ’ )
g r i d on
s e t ( gca , ’ x d i r ’ , ’ r e v e r s e ’ ) ;
legend ( ’ − 35{^\ c i r c } ’ , ’ −28{^\ c i r c } ’ , ’ − 21{^\ c i r c } ’ , ’
− 14{^\ c i r c } ’ , ’ − 7{^\ c i r c } ’ , ’ 0{^\ c i r c } ’ , ’ 7{^\ c i r c } ’ , ’
14{^\ c i r c } ’ , ’ 21{^\ c i r c } ’ , ’ 28{^\ c i r c } ’ , ’ 35{^\ c i r c } ’ )
136
137
138
Y = figure ;
139
140
Vpp = [ Vpp1 ’ Vpp2 ’ Vpp3 ’ Vpp4 ’ Vpp5 ’ Vpp6 ’ Vpp7 ’ Vpp8 ’
Vpp9 ’ Vpp10 ’ Vpp11 ’ ] ;
141
142
143
imagesc ( t h e t a , x , db ( Vpp ) )
t i t l e ( ’ Amplitude i n dB as a f u n c t i o n o f a n g l e and
height ’ )
144
145
146
147
148
x l a b e l ( ’ Angle , {^\ c i r c } ’ )
y l a b e l ( ’ Height , mm’ )
hd= c o l o r b a r ;
s e t ( g e t ( hd , ’ y l a b e l ’ ) , ’ S t r i n g ’ , ’ Amplitude , dBV ’ , ’
R o t a t i o n ’ , − 90.0) ;
149
150
W = figure ;
151
152
153
154
155
156
157
158
s u r f ( t h e t a , x , db ( Vpp ) )
t i t l e ( ’ 3D p l o t o f Amplitude i n dB as a f u n c t i o n o f
a n g l e and h e i g h t ’ )
x l a b e l ( ’ Angle , {^\ c i r c } ’ )
Xlab=g e t ( gca , ’ x l a b e l ’ ) ;
s e t ( Xlab , ’ r o t a t i o n ’ ,20)
y l a b e l ( ’ Height , mm’ )
Ylab=g e t ( gca , ’ y l a b e l ’ ) ;
90
159
160
161
s e t ( Ylab , ’ r o t a t i o n ’ , − 25)
hd= c o l o r b a r ;
s e t ( g e t ( hd , ’ y l a b e l ’ ) , ’ S t r i n g ’ , ’ Amplitude , dBV ’ , ’
R o t a t i o n ’ , − 90.0) ;
162
163
164
165
166
1
s a v e a s (H, ’ amplitude_hoyde ’ , ’ epsc ’ )
s a v e a s ( Y , ’ hoyde_vinkelAmplitude ’ , ’ epsc ’ )
s a v e a s (W, ’ hoyde_vinkelAmplitude3D ’ , ’ epsc ’ )
theta = [ −35:7:35];
2
3
4
5
load ( ’ x27 . mat ’ ) ;
S1 = S ( : , 1 4 ) ;
Vpp_S1 = max( S1 ) −min ( S1 ) ;
6
7
8
9
load ( ’ x34 . mat ’ ) ;
S2 = S ( : , 1 4 ) ;
Vpp_S2 = max( S2 ) −min ( S2 ) ;
10
11
12
13
load ( ’ x41 . mat ’ ) ;
S3 = S ( : , 1 4 ) ;
Vpp_S3 = max( S3 ) −min ( S3 ) ;
14
15
16
17
load ( ’ x48 . mat ’ ) ;
S4 = S ( : , 1 4 ) ;
Vpp_S4 = max( S4 ) −min ( S4 ) ;
18
19
20
21
load ( ’ x55 . mat ’ ) ;
S5 = S ( : , 1 4 ) ;
Vpp_S5 = max( S5 ) −min ( S5 ) ;
22
23
24
25
load ( ’ x62 . mat ’ ) ;
S6 = S ( : , 1 4 ) ;
Vpp_S6 = max( S6 ) −min ( S6 ) ;
26
27
28
29
load ( ’ x69 . mat ’ ) ;
S7 = S ( : , 1 4 ) ;
Vpp_S7 = max( S7 ) −min ( S7 ) ;
30
31
32
33
load ( ’ x76 . mat ’ ) ;
S8 = S ( : , 1 4 ) ;
Vpp_S8 = max( S8 ) −min ( S8 ) ;
34
35
36
37
load ( ’ x83 . mat ’ ) ;
S9 = S ( : , 1 4 ) ;
Vpp_S9 = max( S9 ) −min ( S9 ) ;
38
91
39
40
41
lo ad ( ’ x90 . mat ’ ) ;
S10 = S ( : , 1 4 ) ;
Vpp_S10 = max( S10 ) −min ( S10 ) ;
42
43
44
45
lo ad ( ’ x97 . mat ’ ) ;
S11 = S ( : , 1 4 ) ;
Vpp_S11 = max( S11 ) −min ( S11 ) ;
46
47
Vpp = [ Vpp_S1 Vpp_S2 Vpp_S3 Vpp_S4 Vpp_S5 Vpp_S6
Vpp_S7 Vpp_S8 Vpp_S9 Vpp_S10 Vpp_S11 ] ;
48
49
50
51
52
53
54
p l o t ( t h e t a , smooth ( Vpp ) )
F = f i g u r e , p l o t ( t h e t a , smooth ( db ( Vpp ) ) )
g r i d on
t i t l e ( ’ Amplitude when r e c e i v e r p o s i t i o n i s c o n s t a nt ,
and by r o t a t i n g t r a n s d u c e r ’ )
x l a b e l ( ’ angle , { ^ \ c i r c } ’ )
y l a b e l ( ’ l o g o f Amplitude , dBV ’ )
55
56
saveas (F ,
’ Amplitude_angle_height960 ’ , ’ epsc ’ )
Code for experiment II:
1
2
3
4
5
6
lo ad ( ’ lydhast21_4cm . mat ’ ) ;
X1 = S ;
lo ad ( ’ lydhast41_4cm . mat ’ ) ;
X2 = S ;
7
8
9
vpp1 = max( X1 ) − min ( X1 )
vpp2 = max(X2) − min (X2)
10
11
12
13
t1 = [67.2:0.2:267];
t2 = [ 2 0 4 . 8 : 0 . 2 : 4 0 4 . 6 ] ;
14
15
16
17
18
19
20
21
22
f i g u r e ( 1 ) , p l o t ( t 1 , X1 )
a x i s ( [ 6 7 267 −5 5 ] )
g r i d on
t i t l e ( ’ Hydrophone a t a d i s t a n c e d = 2 1 . 4 cm . ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
23
24
25
f i g u r e ( 2 ) , p l o t ( t2 , X2 , ’ r ’ )
a x i s ([204 405 −5 5 ] )
92
26
27
28
29
t i t l e ( ’ Hydrophone a t a d i s t a n c e d = 4 1 . 4 cm . ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
g r i d on
Code for experiment III:
1
2
load ( ’ exp_front181p_d21_4cm . mat ’ ) ;
load ( ’ exp_front181p_d21_4cm_Vpp ’ ) ;
3
4
5
t1 = [106:0.2:305.8];
angle = [ −90:1:90];
6
7
8
9
10
11
12
f i g u r e ( 1 ) , plot ( t1 , S)
a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Measurements f o r r o t a t i o n a l a n g l e { \ alpha } { \ i n
} [ − 90{^{\ c i r c } } , 90{^{\ c i r c } } ] . ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
13
14
15
16
17
18
19
f i g u r e ( 2 ) , p l o t ( angle , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
a x i s ([ − 90 90 −30 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude . ’ )
x l a b e l ( ’ R o t a t i o n a l Angle , { \ alpha } ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
20
21
22
23
24
25
26
f i g u r e ( 3 ) , p l o t ( angle , smooth ( Vpp ) )
a x i s ([ − 90 90 0 5 . 5 ] )
t i t l e ( ’ Amplitude p l o t ’ )
x l a b e l ( ’ R o t a t i o n a l Angle , { \ alpha } ’ )
y l a b e l ( ’ Amplitude , V ’ )
g r i d on
27
28
29
30
31
32
33
figure (4) , subplot (3 ,2 ,1) ;
p l o t ( t 1 , S ( : , 9 1 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 0{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
34
35
36
37
38
39
40
subplot (3 ,2 ,2) ,
p l o t ( t 1 , S ( : , 1 0 6 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 15{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
93
41
42
43
44
45
46
47
subplot (3 ,2 ,3) ;
p l o t ( t 1 , S ( : , 1 2 1 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 30{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
48
49
50
51
52
53
54
subplot (3 ,2 ,4) ;
p l o t ( t 1 , S ( : , 1 3 6 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 45{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
55
56
57
58
59
60
61
subplot (3 ,2 ,5) ;
p l o t ( t 1 , S ( : , 1 5 1 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 60{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
62
63
64
65
66
67
68
subplot (3 ,2 ,6) ;
p l o t ( t 1 , S ( : , 1 6 6 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 75{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
69
70
71
72
73
74
hold o f f
75
76
77
78
79
80
81
82
figure (5) ,
subplot (3 ,2 ,1) ;
p l o t ( t 1 , S ( : , 1 7 6 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 85{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
83
84
85
86
87
88
subplot (3 ,2 ,2) ;
p l o t ( t 1 , S ( : , 1 7 7 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 86{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
94
89
y l a b e l ( ’ V o l t a g e (V) ’ )
90
91
92
93
94
95
96
subplot (3 ,2 ,3) ;
p l o t ( t 1 , S ( : , 1 7 8 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 87{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
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98
99
100
101
102
103
subplot (3 ,2 ,4) ;
p l o t ( t 1 , S ( : , 1 7 9 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 88{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
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subplot (3 ,2 ,5) ;
p l o t ( t 1 , S ( : , 1 8 0 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 89{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
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112
113
114
115
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subplot (3 ,2 ,6) ;
p l o t ( t 1 , S ( : , 1 8 1 ) ) , a x i s ( [ 1 0 6 306 −3 3 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 90{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
118
119
figure (6) ,
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121
122
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imagesc ( angle , t 1 , db ( S/max( S ( : ) ) ) ) ;
colorbar ;
t i t l e ( ’ D e c i b e l o f the s i g n a l s ’ )
y l a b e l ( ’ Time , { \mu} s ’ ) ;
x l a b e l ( ’ R o t a t i o n a l angle , { \ alpha } ’ ) ;
Beampattern code for Chapter 8:
1
load ( ’ exp_front181p_d21_4cm_Vpp . mat ’ )
2
3
4
r = 6E − 2;
x0 = 2 1 . 4E − 2;
5
6
7
L = 0.01;
L1 = 0 . 0 1 1 ; L2 = 0 . 0 1 2 ; L3 = 0 . 0 1 3 ; L4 = 0 . 0 1 4 ; L5 =
0 . 0 1 5 ; L6 =0.016;
95
8
lambda = 0.01488;
9
10
11
12
13
14
15
16
alpha = [ − p i / 2 : p i /180: p i / 2 ] ;
a = r . * s i n ( alpha ) ;
b = r . * cos ( alpha ) ;
b_p = r − b ;
x_p = x0 + b_p ;
x = s q r t ( ( a . ^ 2 ) + ( x_p . ^ 2 ) ) ;
phi = a s i n ( a . / x ) ;
17
18
theta_p = alpha + phi ;
19
20
21
22
23
s = ( ( s i n ( ( p i * L/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L/ lambda
) . * s i n ( theta_p ) ) ) ;
s2 = ( ( s i n ( ( p i * L/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i / lambda )
. * s i n ( theta_p ) ) ) ;
%p l o t ( t h e t a , b )
24
25
26
figure (1) ;
g1 = p l o t ( theta_p , db ( s ) , ’m ’ )
27
28
29
30
hold on
g2 = p l o t ( theta_p , db ( s2 /max( s2 ) ) , ’ o ’ )
31
32
33
34
35
36
a x i s ([ − p i /2 p i /2 − 15 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude w/d=1cm . ’ )
x l a b e l ( ’ Beam Angle , { \ t h e ta _ {p } } i n r a d i a n s ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
37
38
39
S = ( ( s i n ( ( p i * L/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L/ lambda
) . * s i n ( theta_p ) ) ) . * cos ( theta_p ) ;
40
41
g4 = p l o t ( theta_p , db ( S ) , ’ g ’ )
42
43
S1 = ( ( s i n ( ( p i * L/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L/
lambda ) . * s i n ( theta_p ) ) ) . * ( ( 1 + cos ( theta_p ) ) . / 2 ) ;
44
45
46
g5 = p l o t ( theta_p , db ( S1 ) , ’ k ’ )
g3 = p l o t ( theta_p , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
47
48
h = legend ( ’ $\ f r a c { s i n [ ( \ p i d /\ lambda ) \ times s i n ( \ t he t a _ { p } ) ] } { ( \ p i d /\ lambda ) \ times s i n ( \ t h et a _ {p } ) }$ ’ , ’ $\ f r a c { s i n [ ( k_x d / 2) ] } { ( k_x / 2) }$ ’ , ’ $\ f r a c { s i n [ ( \ p i d /\ lambda ) \ times s i n ( \ t he t a _ { p } ) ] } { ( \ p i d 96 /\ lambda ) \ times s i n ( \ th e ta _ {p } ) } \ times cos ( \ th e t a_ {p } )$ ’ , ’ $\ f r a c { s i n [ ( \ p i d /\ lambda ) \ times s i n ( \ the t a _ { p } ) ] } { ( \ p i d /\ lambda ) \ times s i n ( \ t h et a _ {p } ) }\ times \ f r a c {1+ cos ( \ th e ta _ {p } ) } { 2 }$ ’ , ’ Meassured
Amplitude ’ ) ;
set (h , ’ Interpreter ’ , ’ latex ’ )
s e t ( gca , ’ f o n t s i z e ’ , 1 0 )
49
50
51
s e t ( g1 ,
s e t ( g2 ,
s e t ( g3 ,
s e t ( g4 ,
s e t ( g5 ,
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53
54
55
56
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’
’
’
’
’
,
,
,
,
,
2) ;
2) ;
2) ;
2) ;
2) ;
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s_1 = ( ( s i n ( ( p i * L1 / lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L1 /
lambda ) . * s i n ( theta_p ) ) ) ;
s2_1 = ( ( s i n ( ( p i * L1 / lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i /
lambda ) . * s i n ( theta_p ) ) ) ;
S_1 = ( ( s i n ( ( p i * L1 / lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L1 /
lambda ) . * s i n ( theta_p ) ) ) . * cos ( theta_p ) ;
S1_1 = ( ( s i n ( ( p i * L1 / lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L1 /
lambda ) . * s i n ( theta_p ) ) ) . * ( ( 1 + cos ( theta_p ) ) . / 2 ) ;
64
65
66
67
figure (2) ;
subplot (3 ,2 ,1) ;
g1_1 = p l o t ( theta_p , db ( s_1 ) , ’m ’ )
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69
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71
hold on
g2_1 = p l o t ( theta_p , db ( s2_1 /max( s2_1 ) ) , ’ o ’ )
72
73
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77
a x i s ([ − p i /2 p i /2 − 15 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude w/d = 1 . 1cm . ’ )
x l a b e l ( ’ Beam Angle , { \ t h e ta _ {p } } i n r a d i a n s ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
78
79
g4_1 = p l o t ( theta_p , db ( S_1 ) , ’ g ’ )
80
81
82
g5_1 = p l o t ( theta_p , db ( S1_1 ) , ’ k ’ )
g3_1 = p l o t ( theta_p , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
83
84
%
h1 =legend ( ’ $\ f r a c { s i n [ ( \ p i d /\ lambda ) \ times s i n ( \ the t a _ { p } ) ] } { ( \ p i d /\ lambda ) \ times s i n ( \ t h et a _ {p } ) }$ ’ , ’ $\ f r a c { s i n [ ( k_x d /2) ] } { ( k_x /2) }$ ’ , ’ $\ f r a c { s i n [ ( \ p i d /\ lambda ) \ times s i n ( \ th e ta _ {p } ) ] } { ( \ p i d /\ 97 85 86 % % lambda ) \ times s i n ( \ t h et a _ {p } ) } \ times cos ( \ t h e ta _ {p } )$ ’ , ’ $\ f r a c { s i n [ ( \ p i d /\ lambda ) \ times s i n ( \ t h et a _ { p } ) ] } { ( \ p i d /\ lambda ) \ times s i n ( \ t h e ta _ {p } ) } \ times \ f r a c {1+ cos ( \ t h et a _ {p } ) } { 2 }$ ’ , ’ Meassured Amplitude
’) ;
s e t ( h1 , ’ I n t e r p r e t e r ’ , ’ l a t e x ’ )
s e t ( gca , ’ f o n t s i z e ’ , 1 0 )
87
88
89
90
91
92
s e t ( g1_1 ,
s e t ( g2_1 ,
s e t ( g3_1 ,
s e t ( g4_1 ,
s e t ( g5_1 ,
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’
’
’
’
’
,
,
,
,
,
2) ;
2) ;
2) ;
2) ;
2) ;
93
94
95
96
97
98
99
100
101
s_2 = ( ( s i n ( ( p i * L2/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L2
/ lambda ) . * s i n ( theta_p ) ) ) ;
s2_2 = ( ( s i n ( ( p i * L2/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i /
lambda ) . * s i n ( theta_p ) ) ) ;
S_2 = ( ( s i n ( ( p i * L2/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L2/
lambda ) . * s i n ( theta_p ) ) ) . * cos ( theta_p ) ;
S1_2 = ( ( s i n ( ( p i * L2/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L2/
lambda ) . * s i n ( theta_p ) ) ) . * ( ( 1 + cos ( theta_p ) ) . / 2 ) ;
subplot (3 ,2 ,2) ;
g1_2 = p l o t ( theta_p , db ( s_2 ) , ’m ’ )
102
103
104
105
hold on
g2_2 = p l o t ( theta_p , db ( s2_2 /max( s2_2 ) ) , ’ o ’ )
106
107
108
109
110
111
a x i s ([ − p i /2 p i /2 − 15 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude w/d =1 . 2cm . ’ )
x l a b e l ( ’ Beam Angle , { \ t h e ta _ {p } } i n r a d i a n s ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
112
113
g4_2 = p l o t ( theta_p , db ( S_2 ) , ’ g ’ )
114
115
g5_2 = p l o t ( theta_p , db ( S1_2 ) , ’ k ’ )
116
117
g3_2 = p l o t ( theta_p , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
118
119
120
121
122
123
s e t ( g1_2 ,
s e t ( g2_2 ,
s e t ( g3_2 ,
s e t ( g4_2 ,
s e t ( g5_2 ,
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’
’
’
’
’
,
,
,
,
,
2) ;
2) ;
2) ;
2) ;
2) ;
98
124
125
126
127
128
129
130
s_3 = ( ( s i n ( ( p i * L3/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i *
L3/ lambda ) . * s i n ( theta_p ) ) ) ;
s2_3 = ( ( s i n ( ( p i * L3/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i /
lambda ) . * s i n ( theta_p ) ) ) ;
S_3 = ( ( s i n ( ( p i * L3/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L3/
lambda ) . * s i n ( theta_p ) ) ) . * cos ( theta_p ) ;
S1_3 = ( ( s i n ( ( p i * L3/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L3/
lambda ) . * s i n ( theta_p ) ) ) . * ( ( 1 + cos ( theta_p ) ) . / 2 ) ;
subplot (3 ,2 ,3) ;
g1_3 = p l o t ( theta_p , db ( s_3 ) , ’m ’ )
131
132
133
134
hold on
g2_3 = p l o t ( theta_p , db ( s2_3 /max( s2_3 ) ) , ’ o ’ )
135
136
137
138
139
140
a x i s ([ − p i /2 p i /2 − 15 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude w/d = 1 . 3cm . ’ )
x l a b e l ( ’ Beam Angle , { \ t h e ta _ {p } } i n r a d i a n s ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
141
142
g4_3 = p l o t ( theta_p , db ( S_3 ) , ’ g ’ )
143
144
g5_3 = p l o t ( theta_p , db ( S1_3 ) , ’ k ’ )
145
146
g3_3 = p l o t ( theta_p , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
147
148
149
150
151
152
s e t ( g1_3 ,
s e t ( g2_3 ,
s e t ( g3_3 ,
s e t ( g4_3 ,
s e t ( g5_3 ,
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’
’
’
’
’
,
,
,
,
,
2) ;
2) ;
2) ;
2) ;
2) ;
153
154
155
156
157
158
159
s_4 = ( ( s i n ( ( p i * L4/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L4
/ lambda ) . * s i n ( theta_p ) ) ) ;
s2_4 = ( ( s i n ( ( p i * L4/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i /
lambda ) . * s i n ( theta_p ) ) ) ;
S_4 = ( ( s i n ( ( p i * L4/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L4/
lambda ) . * s i n ( theta_p ) ) ) . * cos ( theta_p ) ;
S1_4 = ( ( s i n ( ( p i * L4/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L4/
lambda ) . * s i n ( theta_p ) ) ) . * ( ( 1 + cos ( theta_p ) ) . / 2 ) ;
subplot (3 ,2 ,4) ;
g1_4 = p l o t ( theta_p , db ( s_4 ) , ’m ’ )
160
161
162
163
hold on
g2_4 = p l o t ( theta_p , db ( s2_4 /max( s2_4 ) ) , ’ o ’ )
99
164
165
166
167
168
169
a x i s ([ − p i /2 p i /2 − 15 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude w/d =1.4cm . ’ )
x l a b e l ( ’ Beam Angle , { \ t h e ta _ {p } } i n r a d i a n s ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
170
171
g4_4 = p l o t ( theta_p , db ( S_4 ) , ’ g ’ )
172
173
g5_4 = p l o t ( theta_p , db ( S1_4 ) , ’ k ’ )
174
175
g3_4 = p l o t ( theta_p , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
176
177
178
179
180
181
s e t ( g1_4 ,
s e t ( g2_4 ,
s e t ( g3_4 ,
s e t ( g4_4 ,
s e t ( g5_4 ,
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’
’
’
’
’
,
,
,
,
,
2) ;
2) ;
2) ;
2) ;
2) ;
182
183
184
185
186
187
188
s_5 = ( ( s i n ( ( p i * L5/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L5
/ lambda ) . * s i n ( theta_p ) ) ) ;
s2_5 = ( ( s i n ( ( p i * L5/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i /
lambda ) . * s i n ( theta_p ) ) ) ;
S_5 = ( ( s i n ( ( p i * L5/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L5/
lambda ) . * s i n ( theta_p ) ) ) . * cos ( theta_p ) ;
S1_5 = ( ( s i n ( ( p i * L5/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L5/
lambda ) . * s i n ( theta_p ) ) ) . * ( ( 1 + cos ( theta_p ) ) . / 2 ) ;
subplot (3 ,2 ,5) ;
g1_5 = p l o t ( theta_p , db ( s_5 ) , ’m ’ )
189
190
191
192
hold on
g2_5 = p l o t ( theta_p , db ( s2_5 /max( s2_5 ) ) , ’ o ’ )
193
194
195
196
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198
a x i s ([ − p i /2 p i /2 − 15 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude w/d = 1 . 5cm . ’ )
x l a b e l ( ’ Beam Angle , { \ t h e ta _ {p } } i n r a d i a n s ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
199
200
g4_5 = p l o t ( theta_p , db ( S_5 ) , ’ g ’ )
201
202
g5_5 = p l o t ( theta_p , db ( S1_5 ) , ’ k ’ )
203
204
g3_5 = p l o t ( theta_p , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
205
206
207
s e t ( g1_5 ,
s e t ( g2_5 ,
’ l i n e w i d t h ’ , 2) ;
’ l i n e w i d t h ’ , 2) ;
100
208
209
210
s e t ( g3_5 ,
s e t ( g4_5 ,
s e t ( g5_5 ,
’ l i n e w i d t h ’ , 2) ;
’ l i n e w i d t h ’ , 2) ;
’ l i n e w i d t h ’ , 2) ;
211
212
213
214
215
216
217
s_6 = ( ( s i n ( ( p i * L6/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i *
L6/ lambda ) . * s i n ( theta_p ) ) ) ;
s2_6 = ( ( s i n ( ( p i * L6/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i /
lambda ) . * s i n ( theta_p ) ) ) ;
S_6 = ( ( s i n ( ( p i * L6/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L6/
lambda ) . * s i n ( theta_p ) ) ) . * cos ( theta_p ) ;
S1_6 = ( ( s i n ( ( p i * L6/ lambda ) . * s i n ( theta_p ) ) ) . / ( ( p i * L6/
lambda ) . * s i n ( theta_p ) ) ) . * ( ( 1 + cos ( theta_p ) ) . / 2 ) ;
subplot (3 ,2 ,6) ;
g1_6 = p l o t ( theta_p , db ( s_6 ) , ’m ’ )
218
219
220
221
hold on
g2_6 = p l o t ( theta_p , db ( s2_6 /max( s2_6 ) ) , ’ o ’ )
222
223
224
225
226
227
a x i s ([ − p i /2 p i /2 − 15 0 ] )
t i t l e ( ’ D e c i b e l v a l u e o f the amplitude w/d =1.6cm . ’ )
x l a b e l ( ’ Beam Angle , { \ t h e ta _ {p } } i n r a d i a n s ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
228
229
g4_6 = p l o t ( theta_p , db ( S_6 ) , ’ g ’ )
230
231
g5_6 = p l o t ( theta_p , db ( S1_6 ) , ’ k ’ )
232
233
g3_6 = p l o t ( theta_p , smooth ( db ( Vpp/max( Vpp ) ) ) , ’ r ’ )
234
235
236
237
238
239
s e t ( g1_6 ,
s e t ( g2_6 ,
s e t ( g3_6 ,
s e t ( g4_6 ,
s e t ( g5_6 ,
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’ linewidth
’
’
’
’
’
,
,
,
,
,
2) ;
2) ;
2) ;
2) ;
2) ;
Code for experiemnt IV:
1
2
load ( ’ exp_back_85to275_2 . mat ’ )
load ( ’ exp_back_85to275_Vpp_2 . mat ’ )
3
4
5
t = [78:0.5:577.5];
angle = [ 8 5 : 1 : 2 7 5 ] ;
6
7
8
9
figure (1) , plot ( t , S)
a x i s ( [ 7 8 578 −1 1 ] )
101
10
11
12
13
g r i d on
t i t l e ( ’ Measurements f o r r o t a t i o n a l a n g l e { \ alpha } { \ i n
} [ − 90{^{\ c i r c } } , 90{^{\ c i r c } } ] . ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
14
15
16
17
18
19
20
f i g u r e ( 2 ) , p l o t ( angle , smooth ( ( Vpp ) ) , ’ r ’ )
a x i s ( [ 8 5 275 0 1 . 3 ] )
t i t l e ( ’ Amplitude p l o t ’ )
x l a b e l ( ’ R o t a t i o n a l Angle , { \ alpha } ’ )
y l a b e l ( ’ Normalized Amplitude , dBV ’ )
g r i d on
21
22
23
24
25
26
27
28
29
figure (3) ;
subplot (3 ,2 ,1) ;
p l o t ( t , S ( : , 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 85{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
30
31
32
33
34
35
36
subplot (3 ,2 ,2) ;
p l o t ( t , S ( : , 2 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 86{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
37
38
39
40
41
42
43
subplot (3 ,2 ,3) ;
p l o t ( t , S ( : , 3 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 87{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
44
45
46
47
48
49
50
subplot (3 ,2 ,4) ;
p l o t ( t , S ( : , 4 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 88{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
51
52
53
54
55
56
subplot (3 ,2 ,5) ;
p l o t ( t , S ( : , 5 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 89{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
102
57
y l a b e l ( ’ V o l t a g e (V) ’ )
58
59
60
61
62
63
64
subplot (3 ,2 ,6) ;
p l o t ( t , S ( : , 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 90{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
65
66
67
68
69
70
71
72
figure (4) , subplot (5 ,3 ,1) ;
p l o t ( t , S ( : , 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 85{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
73
74
75
76
77
78
79
subplot (5 ,3 ,2) ;
p l o t ( t , S ( : , 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 90{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
80
81
82
83
84
85
86
subplot (5 ,3 ,3) ;
p l o t ( t , S ( : , 2 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 110{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
87
88
89
90
91
92
93
subplot (5 ,3 ,4) ;
p l o t ( t , S ( : , 3 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 120{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
94
95
96
97
98
99
100
subplot (5 ,3 ,5) ;
p l o t ( t , S ( : , 6 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 145{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
101
102
103
104
subplot (5 ,3 ,6) ;
p l o t ( t , S ( : , 7 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
103
105
106
107
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 160{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
108
109
110
111
112
113
114
subplot (5 ,3 ,7) ;
p l o t ( t , S ( : , 9 5 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 179{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
115
116
117
118
119
120
121
subplot (5 ,3 ,8) ;
p l o t ( t , S ( : , 9 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 180{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
122
123
124
125
126
127
128
subplot (5 ,3 ,9) ;
p l o t ( t , S ( : , 9 7 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 181{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
129
130
131
132
133
134
135
subplot (5 ,3 ,10) ;
p l o t ( t , S ( : , 1 1 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 200{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
136
137
138
139
140
141
142
subplot (5 ,3 ,11) ;
p l o t ( t , S ( : , 1 3 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 215{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
143
144
145
146
147
148
149
subplot (5 ,3 ,12) ;
p l o t ( t , S ( : , 1 5 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 240{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
150
151
152
subplot (5 ,3 ,13) ;
p l o t ( t , S ( : , 1 6 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
104
153
154
155
156
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 250{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
157
158
159
160
161
162
163
subplot (5 ,3 ,14) ;
p l o t ( t , S ( : , 1 8 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 270{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
164
165
166
167
168
169
170
subplot (5 ,3 ,15) ;
p l o t ( t , S ( : , 1 9 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 275{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
171
172
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177
178
figure (5) ;
subplot (3 ,2 ,1) ;
p l o t ( t , S ( : , 1 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 95{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
179
180
181
182
183
184
185
subplot (3 ,2 ,2) ;
p l o t ( t , S ( : , 1 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 100{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
186
187
188
189
190
191
192
subplot (3 ,2 ,3) ;
p l o t ( t , S ( : , 2 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 105{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
193
194
195
196
197
198
199
subplot (3 ,2 ,4) ;
p l o t ( t , S ( : , 2 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 110{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
200
105
201
202
203
204
205
206
subplot (3 ,2 ,5) ;
p l o t ( t , S ( : , 3 1 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 1 1 5 { ^ { \ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
207
208
209
210
211
212
213
subplot (3 ,2 ,6) ;
p l o t ( t , S ( : , 3 6 ) ) , a x i s ( [ 7 8 578 − 0.6 0 . 6 ] )
g r i d on
t i t l e ( ’ Received s i g n a l a t { \ alpha } = 120{^{\ c i r c } } ’ )
x l a b e l ( ’ t ( { \mu} s ) ’ )
y l a b e l ( ’ V o l t a g e (V) ’ )
214
215
216
217
218
219
220
221
222
figure (6)
p l o t ( angle , db ( smooth ( Vpp ) ) ) ;
a x i s ( [ 8 5 275 −50 5 . 5 ] )
g r i d on
t i t l e ( ’ Amplitude p l o t i n dB ’ )
y l a b e l ( ’ Amplitude , dBV ’ ) ;
x l a b e l ( ’ R o t a t i o n a l angle , { \ alpha } ’ ) ;
223
224
225
226
227
228
229
figure (7)
imagesc ( angle , t , db ( S/max( S ( : ) ) ) ) ;
colorbar ;
t i t l e ( ’ D e c i b e l o f the s i g n a l s ’ )
y l a b e l ( ’ Time , { \mu} s ’ ) ;
x l a b e l ( ’ R o t a t i o n a l angle , { \ alpha } ’ ) ;
1
2
3
4
lo ad ( ’ exp_front181p_d21_4cm_Vpp . mat ’ ) ;
Vpp_1 = Vpp ( 9 1 : 1 8 1 ) ;
Vpp_2 = Vpp ( 1 : 9 0 ) ;
5
6
7
8
lo ad ( ’ exp_back_85to275_Vpp_2 . mat ’ ) ;
Vpp_3 = Vpp ( 1 : 9 6 ) ;
Vpp_4 = Vpp ( 9 7 : 1 9 1 ) ;
9
10
a n g l e 1 = [ 0 : 1 : 9 0 ] ; angle2 = [ 2 7 0 : 1 : 3 5 9 ] ; angle3 =
[ 8 5 : 1 : 1 8 0 ] ; angle4 = [ 1 8 1 : 1 : 2 7 5 ] ;
11
12
13
14
15
16
17
figure (1) ;
p l o t ( an g le 1
hold on
p l o t ( angle2
p l o t ( angle3
p l o t ( angle4
, ( Vpp_1 ) ) ;
, ( Vpp_2 ) , ’ r ’ ) ;
, ( Vpp_3 ) , ’ g ’ ) ;
, ( Vpp_4 ) , ’ k ’ ) ;
106
18
19
20
21
22
23
a x i s ( [ 0 359 0 6 ] )
t i t l e ( ’ Amplitude o f experiments I I I and IV ’ )
x l a b e l ( ’ R o t a t i o n a l Angle , { \ alpha } ’ )
y l a b e l ( ’ Amplitude , V ’ )
legend ( ’ Experiment I I I [0{^{\ c i r c } } − 90{^{\ c i r c } } ] ’ ,
’ Experiment I I I [270{^{\ c i r c } } − 359{^{\ c i r c } } ] ’ , ’
Experiment IV [85{^{\ c i r c } } − 180{^{\ c i r c } } ] ’ , ’
Experiment IV [ 1 8 1 { ^ { \ c i r c } } − 275{^{\ c i r c } } ] ’ )
g r i d on
24
25
26
27
28
29
figure (2) ;
Vpp1_1 = Vpp_1 ( 1 : 8 6 ) + ( Vpp_3 ( 1 ) − Vpp_1 (86) ) ;
Vpp2_1 = Vpp_2 ( 6 : 9 0 ) − ( Vpp_2 ( 6 ) − Vpp_4 ( 9 5 ) ) ;
Vpp3_1 = Vpp_3 ( 1 : 9 6 ) ;
Vpp4_1 = Vpp_4 ( 1 : 9 5 ) ;
30
31
a n g l e 1 _ 1 = [ 0 : 1 : 8 5 ] ; angle2_2 = [ 2 7 5 : 1 : 3 5 9 ] ;
32
33
34
35
36
37
38
39
40
41
42
43
p l o t ( angle1_1 , smooth ( Vpp1_1 ) ) ;
hold on
p l o t ( angle2_2 , smooth ( Vpp2_1 ) , ’ r ’ ) ;
p l o t ( angle3 , smooth ( Vpp3_1 ) , ’ k ’ ) ;
p l o t ( angle4 , smooth ( Vpp4_1 ) , ’ g ’ ) ;
a x i s ( [ 0 359 0 6 ] )
t i t l e ( ’ Compensated Amplitude o f experiments I I I and IV
’)
x l a b e l ( ’ R o t a t i o n a l Angle , { \ alpha } ’ )
y l a b e l ( ’ Amplitude , V ’ )
legend ( ’ Experiment I I I [0{^{\ c i r c } } − 90{^{\ c i r c } } ] ’ ,
’ Experiment I I I [270{^{\ c i r c } } − 359{^{\ c i r c } } ] ’ , ’
Experiment IV [85{^{\ c i r c } } − 180{^{\ c i r c } } ] ’ , ’
Experiment IV [ 1 8 1 { ^ { \ c i r c } } − 275{^{\ c i r c } } ] ’ )
g r i d on
44
45
46
47
48
49
50
51
52
53
54
figure (3) ;
p l o t ( angle1_1 , db ( smooth ( Vpp1_1 ) ) ) ;
hold on
p l o t ( angle2_2 , db ( smooth ( Vpp2_1 ) ) , ’ r ’ ) ;
p l o t ( angle3 , db ( smooth ( Vpp3_1 ) ) , ’ k ’ ) ;
p l o t ( angle4 , db ( smooth ( Vpp4_1 ) ) , ’ g ’ ) ;
t i t l e ( ’ Compensated Amplitude i n dB o f experiments I I I
and IV ’ )
x l a b e l ( ’ R o t a t i o n a l Angle , { \ alpha } ’ )
y l a b e l ( ’ Amplitude , dBV ’ )
legend ( ’ Experiment I I I [0{^{\ c i r c } } − 90{^{\ c i r c } } ] ’ ,
’ Experiment I I I [270{^{\ c i r c } } − 359{^{\ c i r c } } ] ’ , ’
Experiment IV [85{^{\ c i r c } } − 180{^{\ c i r c } } ] ’ , ’
Experiment IV [ 1 8 1 { ^ { \ c i r c } } − 275{^{\ c i r c } } ] ’ )
107
55
56
g r i d on
a x i s ( [ 0 359 −55 1 6 ] )
57
58
59
60
61
62
63
figure (4) ;
p l o t ( a ng le 1
hold on
p l o t ( angle2
p l o t ( angle3
p l o t ( angle4
, db ( Vpp_1 ) ) ;
, db ( Vpp_2 ) , ’ r ’ ) ;
, db ( Vpp_3 ) , ’ g ’ ) ;
, db ( Vpp_4 ) , ’ k ’ ) ;
108
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