Resonance in acoustic tubes 1. wavelength 2. plain wave

Resonance in acoustic tubes
1. wavelength
2. plain wave propagation
3. reflection
4. phase matching
Two ways to measure the
period of a sine wave.
Time
Frequency = 1 / Time
Sound: pressure fluctuation
that travels through space.
Speed of sound = 35,000 cm/s
Space
Wavelength = spatial period
Wavelength = speed of sound * period duration
λ=c*T
λ=c/f
because f = 1 / T
Space
Wavelength = spatial period
1. wavelength. Sine wave has a spatial
period, peaks and valleys located in space.
Space
sound propagates from source in a
sphere.
However, sound in a tube propagates in
a plane – effectively, no curvature
However, sound in a tube propagates in
a plane – effectively, no curvature
2. Plane wave propagation
3. Reflection
Sound reflects off of surfaces
- more reflection off of hard surfaces
- less reflection off of soft surfaces
- scattered reflection off of uneven surfaces
hard
soft
uneven
Sound traveling in a closed tube
reflects off the ends of the tube.
Sound traveling in an open tube
also reflects off the ends of the tube.
Reflection off of a soft surface
The vocal tract is a tube that is open
at one end and closed at the other.
has two kinds of reflection:
1. hard surface at closed end
2. soft surface at open end
Sound waves traveling though space
interfere with each other.
A
direction
direction
B
Destructive interference: A + B = 0
A
direction
direction
B
A+B
Constructive interference: A + B = AB
A
direction
direction
B
A+B
Constructive interference: A + B = AB
1
A
direction
1
direction
B
2
A+B
Constructive interference: A + B = AB
A
direction
-1
direction
B
-1
A+B
-2
Reflected waves in a tube interfere with
each other.
constructive interference = resonance
destructive interference = nonresonance
Q: What frequencies will resonate in a
tube?
= Q: What sine waves will show
constructive interference?
two factors wavelength and tube length
key: wave must “fit” in tube
fit = reflect in phase
An example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = tube length
An example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = tube length
the reflected
wave is in phase
Δ constructive
interference
An example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = tube length
the reflected
wave is in phase
Frequency of this
resonance:
Δ constructive
interference
f = c/λ
Another example of reflecting “in phase”
- a sine wave that “fits” in a closed tube
wavelength = ½ * tube length
A general formula for calculating the
resonant frequencies of sine waves that will
resonate in a tube closed at both ends:
Fn = nc/2L
n = resonant frequency number (1,2,3, ...)
c = speed of sound (35,000 cm/s)
L = tube length (in cm)
Now consider a tube that is open at
one end, and closed at the other.
Now consider a tube that is open at one
end, and closed at the other.
Reflection from the open end is different.
Phase shift!
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
phase shift at
open end
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
λ
phase shift at
open end
A sine wave that “fits” in a tube that is
open at one end, and closed at the other.
λ
phase shift at
open end
resonant frequency is: f = c/(4/5*L)
Another sine wave that “fits” this tube.
resonant frequency:
f = c/(4/3L)
λ
A general formula for resonant frequencies
of tubes open at one end and closed
at the other:
fn = (2n-1)c/4L
n = resonance number (1,2,3...)
c = speed of sound (35,000 cm/s)
L = tube length (in cm)
the vowel schwa [ә]:
a tube open at one end (lips)
and closed at the other (glottis)
Vocal tract length: ~ 17.5 cm
F1 = c/4L = 35,000/70 = 500 Hz
F2 = 3c/4L = 1500 Hz
F3 = 5c/4L = 2500 Hz
Peter Ladefoged saying [ ә ]:
2500 Hz
1250 Hz
400 Hz