### Acoustics of Strings and Tubes

```Comments, Quiz # 1
So far:
• Historical overview of speech technology
- basic components/goals for systems
• Quick overview of pattern recognition basics
• Quick overview of auditory system
Next talks focus on the nature of the signal:
• Acoustic waves in small spaces (sources)
• Acoustic waves in large spaces (rooms)
Acoustic waves
- a brief intro
A way to bridge from thinking about EE to
•
•
•
•
Acoustic signals are like electrical ones,
only much slower …
Pressure is like voltage
Volume velocity is like current
(and impedance = Pressure/velocity)
For wave solutions, c is a lot smaller for
sound (106)
To analyze, look at constrained models of
common structures: strings and tubes
String and tube models
Vibrating Strings – excitation
Violin – bowed or plucked
Guitar – plucked
Cello – bowed or plucked
Piano – struck
Acoustic tube – excitation
Trumpet – lip vibrations
Clarinet - reed
Human voice – glottal vibration
String model assumptions
• No stiffness
• Constant tension S throughout
• Constant mass density εthroughout
• Small vertical displacement
• Ignore gravity, friction
Vibrating string geometry
δy/δx
F = ma
Fy = S (tan Φ2 - tan Φ1 )
S (δ2y/δx2) dx = ε dx δ2y/δt2
Let c = √S/ ε
c2 (δ2y/δx2) = δ2y/δt2
δy/δx + (δ2y/δx2) dx
String wave equation
So c2 δ2y δ2y
= 2
2
δx
δt
is the wave equation for
transverse vibration (vibration
perpendicular to wave motion
direction) on a string
Where c can be derived from the properties of
the medium, and is the wave propagation speed
Solutions to wave
equation
• Solutions dependent on boundary conditions
• Assume form f(t – x/c) for positive x direction
(equivalently, f(ct – x) )
• Then f(t + x/c) for negative x direction
(or, f(ct + x) )
• Sum is A f(t - x/c) + B f(t +x/c)
(or, A f(ct –x) + B f(ct + x) )
Traveling -> standing waves
• Let g = sin(λx – ct), q = sin(λx + ct)
• sin u + sin v = 2 sin ((u+v)/2)cos((u-v)/2)
• g + q = 2 sin(λx)cos(ct)
• Fixed phase in x dimension, timevarying amplitude (with max fluctuation
determined by position); a “standing
wave”
• Basic phenomenon in strings, tubes,
rooms
Uniform tube, source on one end, open on the other
Excitation
Open
end
x
0
L
Assumptions
• Plane wave propagation for frequencies below ~4
kHz;
as λ increases, plane assumption is better
Since c = fλ, and c ≈ 340 m/s
f = 3400 Hz  λ= .1m = 10 cm
• No thermal conduction losses
• No viscosity losses
• Rigid walls
• Cross-sectional area is constant
Further:
Using Newton’s second law, mass
conservation, and assume pressure change is
proportional to air density change
δ 2p
c2 −
δx2
δ 2p
= −
2
δt
δ 2u
δ 2u
δx
δt
c2 − = −
2
2
Solving, we can show that c = speed of sound
Solutions to wave eqn
• Assume the form f(x – ct) [rightward
wave]
•2
nd
time derivative is c2 times 2nd space
derivative, so it works
• Same for f(x + ct) [leftward wave]
• For sinusoids, sum gives a standing
wave (as before)
Resonance in acoustic
tubes
• Velocity: u(x,t) = u (x-ct) – u (x+ct)
• Pressure: p(x,t) = = Z [ u (x-ct) + u (x+ct)]
• Let u (x-ct) = A e
), u (x-ct) = B e
• Assume u(0,t) = e , p(L,t) = 0
+
-
0
+
jω(t - x/c
jωt
+
-
-
jω (t +x/c)
Problem: Find A and B to match boundary conditions
u(0,t) = ejωt = A e jω (t - 0/c) - B e jω (t + 0/c)
Solve for A and B (eliminate t)
p(L,t) = 0 = A e jω (t - L/c) + B e jω (t + L/c)
Now you can get equation 10.24 in text, for excitation U(ω ) ejωt :
u(x,t) = cos [ω(L-x)/c] U(ω) ejωt
cos [ωL/c]
Poles occur when:
ω = (2n + 1)πc/2L
f = (2n + 1)c/4L
First 3 modes of an acoustic tube open at one
end
Example
Human vocal tract during phonation of neutral
vowel (vocal tract like open tube) – average
male values
c ≈ 340 m/s L = 17 cm, so 4L = 68 cm = .68m
f1= 340/.68 = 500 Hz, f2 = 1500 Hz, f3 = 2500
Hz
Similar to measured resonances
Effect of losses in the tube
• Upward shift in lower resonances
• Poles no longer on unit circle - peak
values in frequency response are finite
Effect of
nonuniformities in the
tube
• Impedance mismatches cause
reflections
• Can be modeled as a succession of
smaller tubes
• Resonances shift - hence the different
formants for different speech sounds
X-ray tracing and area
function for /i/
“small acoustics”
summary
• Voice, many instruments, modeled by
tubes
• Traveling waves in both directions yield
standing waves
• Standing waves correspond to
resonances
• Variations from the idealization give the
variety of speech sounds, musical timbre
Homework #2
(due next
Wednesday)
• Problems 9.2, 9.4, 10.1, 10.5, 14.4, 14.5
from the book (as noted in the email)
• Also the problem: “Describe phase
locking in the auditory nerve. Over
roughly what frequencies does this take
place?”