Title: ECE 598: The Speech Chain
1ECE 598 The Speech Chain
2Today
- 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
3Speech 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
4Why 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)
5Step 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
6Step 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
7Step 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)
8Step 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
9Acoustic 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)
10Solution 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-
11Forward and Backward Traveling Waves
12Other 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)
13Air 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
14Volume 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)
15Acoustic Waves
- Pressure
- p(x,t) ejwt (pe-jkx p-ejkx)
- Velocity
- v(x,t) ejwt (1/rc)(pe-jkx - p-ejkx)
16Standing Waves Resonances in the Vocal Tract
17Boundary Conditions
L
0
18Boundary 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-
19Two 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)
20and 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)
21Two 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)
22Resonant 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
23Resonant 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
24Closed at One End, Open at the Other End
Quarter-Wave Resonator
- The wavelength are
- l14L
- l24L/3
- l34L/5
25Standing 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)
26Standing Wave Patterns
- The standing wave pattern is the part that
doesnt depend on time - p(x) 2Pcos(kx)
- v(x) (2P/rc) sin(kx)
27Standing Wave Patterns Quarter-Wave Resonators
Tube Closed at the Left End, Open at the Right End
28Half-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
29Standing Wave Patterns Half-Wave Resonators
Tube Closed at Both Ends
Tube Open at Both Ends
30Schwa and Invv (the vowels in a tug)
F32500Hz5c/4L
F21500Hz3c/4L
F1500Hzc/4L
31Front 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
32Summary
- 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)