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ECE 598: The Speech Chain

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p must be a function of both time and space: p(x,t) ... Characteristic Impedance of Air: z0 = rc. v = p /rc. v-= -p-/rc. Volume Velocity ... – PowerPoint PPT presentation

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Title: ECE 598: The Speech Chain


1
ECE 598 The Speech Chain
  • Lecture 4 Sound

2
Today
  • Ideal Gas Law Newtons Second Sound
  • Forward-Going and Backward-Going Waves
  • Pressure, Velocity, and Volume Velocity
  • Boundary Conditions
  • Open tube zero pressure
  • Closed tube zero velocity
  • Resonant Frequencies of a Pipe

3
Speech Units cgs
  • Distance measured in cm
  • 1cm length of vocal folds 15-18cm length
    vocal tract
  • Mass measured in grams (g)
  • 1g mass of one cm3 of H20 or biological tissue
  • 1g mass of the vocal folds
  • Time measured in seconds (s)
  • Volume measured in liters (1L1000cm3)
  • 1L/s air flow rate during speech
  • Force measured in dynes (1 d 1 g cm/s2)
  • 1000 dynes force of gravity on 1g (1cm3) of H20
  • Pressure measured in dynes/cm2 or cm H20
  • 1000 d/cm2 pressure of 1cm H20
  • Lung pressure varies from 1-10 cm H20

4
Why Does Sound Happen?
A
x
xdx
  • Consider three blocks of air in a pipe.
  • Boundaries are at x (cm) and xdx (cm).
  • Pipe cross-sectional area A (cm2)
  • Volume of each block of air V A dx (cm3)
  • Mass of each block of air m (grams)
  • Density of each block of air r m/(Adx) (g/cm3)

5
Step 1 Middle Block Squished
A
v
vdv
  • Velocity of air is v or vdv (cm/s)
  • In the example above, dv is a negative number!!!
  • In dt seconds, Volume of middle block changes by
  • dV A ( (vdv) - dv ) dt A dv dt
  • cm3 (cm2) (cm/s) s
  • Density of middle block changes by
  • dr m/(VdV) - m/V
  • -r dV/V -r dt dv/dx
  • Rate of Change of the density of the middle block
  • dr/dt - r dv/dx
  • (g/cm3)/s (g/cm3) (m/s) / m

6
Step 2 The Pressure Rises
A
v
vdv
  • Ideal Gas Law p rRT (pressure proportional to
    density, temperature, and a constant R)
  • Adiabatic Ideal Gas Law dp/dt c2dr/dt
  • When gas is compressed quickly, T and r both
    increase
  • This is called adiabatic expansion --- it means
    that c2gtR
  • c is the speed of sound!!
  • c depends on chemical composition (air vs.
    helium), temperature (body temp. vs. room temp.),
    and atmospheric pressure (sea level vs.
    Himalayas)
  • dp/dt c2dr/dt - rc2 dv/dx

7
Step 3 Pressure X Area Force
A
p
pdp
  • Force acting on the air between ? and ?
  • F pA - (pdp)A -A dp
  • dynes (cm2) (d/cm2)

8
Step 4 Force Accelerates Air
A
Air velocity has changed here!
  • Newtons second law
  • F m dv/dt (rAdx) dv/dt
  • -dp/dx r dv/dt
  • (d/cm2)/cm (g/cm3) (cm/s)/s

9
Acoustic Constitutive Equations
  • Newtons Second Law
  • -dp/dx r dv/dt
  • Adiabatic Ideal Gas Law
  • dp/dt - rc2 dv/dx
  • Acoustic Wave Equation
  • d2p/dt2 c2 d2p/dx2
  • pressure/s2 (cm/s)2 pressure/cm2
  • Things to notice
  • p must be a function of both time and space
    p(x,t)
  • c is a speed (35400 cm/s, the speed of sound at
    body temperature, or 34000 cm/s at room
    temperature)
  • ct is a distance (distance traveled by sound
    in t seconds)

10
Solution Forward and Backward Traveling Waves
  • Wave Number k
  • k w/c
  • (radians/cm) (radians/sec) / (cm/sec)
  • Forward-Traveling Wave
  • p(x,t) pej(wt-kx) pej(w(t-x/c))
  • d2p/dt2 c2 d2p/dx2
  • -w2p -(kc)2p
  • Backward-Traveling Wave
  • p(x,t) p-ej(wtkx) p-ej(w(tx/c))
  • d2p/dt2 c2 d2p/dx2
  • -w2p- -(kc)2p-

11
Forward and Backward Traveling Waves
12
Other Wave Quantities Worth Knowing
  • Wave Number k
  • k w/c
  • (radians/cm) (radians/sec) / (cm/sec)
  • Wavelength l
  • l c/f 2p/k
  • (cm/cycle) (cm/sec) / (cycles/sec)
  • Period T
  • T 1/f
  • (seconds/cycle) 1 / (cycles/sec)

13
Air Particle Velocity
  • Forward-Traveling Wave
  • p(x,t)pej(w(t-x/c)), v(x,t)vej(w(t-x/c))
  • Backward-Traveling Wave
  • p(x,t)p-ej(w(tx/c)), v(x,t)v-ej(w(tx/c))
  • Newtons Second Law
  • -dp/dx r dv/dt
  • (w/c)p wrv
  • -(w/c)p- wrv-
  • Characteristic Impedance of Air z0 rc
  • v p/rc
  • v- -p-/rc

14
Volume Velocity
  • In a pipe, it sometimes makes more sense to talk
    about movement of all of the molecules at
    position x, all at once.
  • A(x) cross-sectional area of the pipe at
    position x (cm2)
  • v(x,t) velocity of air molecules (cm/s)
  • u(x,t) A(x)v(x,t) volume velocity (cm3/s
    mL/s)

15
Acoustic Waves
  • Pressure
  • p(x,t) ejwt (pe-jkx p-ejkx)
  • Velocity
  • v(x,t) ejwt (1/rc)(pe-jkx - p-ejkx)

16
Standing Waves Resonances in the Vocal Tract
17
Boundary Conditions
L
0
18
Boundary Conditions
  • Pressure0 at xL
  • 0 p(L,t) ejwt (pe-jkL p-ejkL)
  • 0 pe-jkL p-ejkL
  • Velocity0 at x0
  • 0 v(0,t) ejwt (1/rc)(pe-jk0 - p-ejk0)
  • 0 pe-jk0 - p-ejk0
  • 0 p - p-
  • p p-

19
Two Equations in Two Unknowns (p and p-)
  • Two Equations in Two Unknowns
  • 0 pe-jkL p-ejkL
  • p p-
  • Combine by Variable Substitution
  • 0 p(e-jkL ejkL)

20
and now, more useful trigonometry
  • Definition of complex exponential
  • ejkL cos(kL)j sin(kL)
  • e-jkL cos(-kL)j sin(-kL)
  • Cosine is symmetric, Sine antisymmetric
  • e-jkL cos(kL) - j sin(kL)
  • Re-combine to get useful equalities
  • cos(kL) 0.5(ejkLe-jkL)
  • sin(kL) -0.5 j (ejkL-e-jkL)

21
Two Equations in Two Unknowns (p and p-)
  • Two Equations in Two Unknowns
  • 0 pe-jkL p-ejkL
  • p p-
  • Combine by Variable Substitution
  • 0 p(e-jkL ejkL)
  • 0 2pcos(kL)
  • Two Possible Solutions
  • 0 p (Meaning, amplitude of the wave is 0)
  • 0 cos(kL) (Meaning)

22
Resonant Frequencies of a Uniform Tube, Closed at
One End, Open at the Other End
  • p can only be nonzero (the amplitude of the wave
    can only be nonzero) at frequencies that satisfy
  • 0 cos(kL)
  • kL p/2, 3p/2, 5p/2,
  • k1p/2L, w1pc/2L, F1c/4L
  • k23p/2L, w23pc/2L, F23c/4L
  • k35p/2L, w35pc/2L, F35c/4L

23
Resonant Frequencies of a Uniform Tube, Closed at
One End, Open at the Other End
  • For example, if the vocal tract is 17.5cm long,
    and the speed of sound is 350m/s at body
    temperature, then c/L2000Hz, and
  • F1500Hz
  • F21500Hz
  • F32500Hz

24
Closed at One End, Open at the Other End
Quarter-Wave Resonator
  • The wavelength are
  • l14L
  • l24L/3
  • l34L/5

25
Standing Wave Patterns
  • The pressure and velocity are
  • p(x,t) p ejwt (e-jkxejkx)
  • v(x,t) ejwt (p/rc)(e-jkx-ejkx)
  • Remember that p encodes both the magnitude and
    phase, like this
  • pPejf
  • Rep(x,t) 2Pcos(wtf) cos(kx)
  • Rev(x,t) (2P/rc)cos(wtf) sin(kx)

26
Standing Wave Patterns
  • The standing wave pattern is the part that
    doesnt depend on time
  • p(x) 2Pcos(kx)
  • v(x) (2P/rc) sin(kx)

27
Standing Wave Patterns Quarter-Wave Resonators
Tube Closed at the Left End, Open at the Right End
28
Half-Wave Resonators
  • If the tube is open at x0 and at xL, then
    boundary conditions are p(0,t)0 and p(L,t)0
  • If the tube is closed at x0 and at xL, then
    boundary conditions are v(0,t)0 and v(L,t)0
  • In either case, the resonances are
  • F10, F2c/2L, F3c/L, , F33c/2L
  • Example, vocal tract closed at both ends
  • F10, F21000Hz, F32000Hz, , F33000Hz

29
Standing Wave Patterns Half-Wave Resonators
Tube Closed at Both Ends
Tube Open at Both Ends
30
Schwa and Invv (the vowels in a tug)
F32500Hz5c/4L
F21500Hz3c/4L
F1500Hzc/4L
31
Front Cavity Resonances of a Fricative
/s/ Front Cavity Resonance 4500Hz
4500Hz c/4L if Front Cavity Length is
L1.9cm
/sh/ Front Cavity Resonance 2200Hz
2200Hz c/4L if Front Cavity Length
is L4.0cm
32
Summary
  • Newtons Second Law
  • -dp/dx r dv/dt
  • Adiabatic Ideal Gas Law
  • dp/dt - rc2 dv/dx
  • Acoustic Wave Equation
  • d2p/dt2 c2 d2p/dx2Pressure
  • Solutions
  • p(x,t) ejwt (pe-jkx p-ejkx)
  • v(x,t) ejwt (1/rc)(pe-jkx - p-ejkx)
  • Resonances
  • Quarter-wave resonator F1c/4L, F23c/4L,
    F35c/4L
  • Half-wave resonator F10, F2c/2L, F3c/L
  • Standing-wave Patterns
  • p(x) 2Pcos(kx)
  • v(x) (2P/rc) sin(kx)
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