Ece 598 the speech chain
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ECE 598: The Speech Chain. Lecture 4: Sound. Today. Ideal Gas Law + Newton’s 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.

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ECE 598: The Speech Chain

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Ece 598 the speech chain

ECE 598: The Speech Chain

Lecture 4: Sound


Today

Today

  • Ideal Gas Law + Newton’s 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


Speech units cgs

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 (1L=1000cm3)

    • 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


Why does sound happen

Why Does Sound Happen?

A

  • Consider three blocks of air in a pipe.

    • Boundaries are at x (cm) and x+dx (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)

x

x+dx


Step 1 middle block squished

Step 1: Middle Block Squished

A

  • Velocity of air is v or v+dv (cm/s)

    • In the example above, dv is a negative number!!!

  • In dt seconds, Volume of middle block changes by

    dV = A ( (v+dv) - dv ) dt = A dv dt

    cm3 = (cm2) (cm/s) s

  • Density of middle block changes by

    dr = m/(V+dV) - 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

v

v+dv


Step 2 the pressure rises

Step 2: The Pressure Rises

A

  • 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 c2>R

    • “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

v

v+dv


Step 3 pressure x area force

Step 3: Pressure X Area = Force

A

  • Force acting on the air between  and :

    F = pA - (p+dp)A = -A dp

    dynes = (cm2) (d/cm2)

p

p+dp


Step 4 force accelerates air

Step 4: Force Accelerates Air

A

  • Newton’s second law:

    F = m dv/dt = (rAdx) dv/dt

    -dp/dx = r dv/dt

    (d/cm2)/cm = (g/cm3) (cm/s)/s

Air velocity has changed here!


Acoustic constitutive equations

Acoustic Constitutive Equations

  • Newton’s 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)


Solution forward and backward traveling waves

Solution: Forward and Backward Traveling Waves

  • Wave Number k:

    k = w/c

    (radians/cm) = (radians/sec) / (cm/sec)

  • Forward-Traveling Wave:

    p(x,t) = p+ej(wt-kx) = p+ej(w(t-x/c))

    d2p/dt2= c2 d2p/dx2

    -w2p+ = -(kc)2p+

  • Backward-Traveling Wave:

    p(x,t) = p-ej(wt+kx) = p-ej(w(t+x/c))

    d2p/dt2= c2 d2p/dx2

    -w2p- = -(kc)2p-


Forward and backward traveling waves

Forward and Backward Traveling Waves


Other wave quantities worth knowing

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)


Air particle velocity

Air Particle Velocity

  • Forward-Traveling Wave

    p(x,t)=p+ej(w(t-x/c)), v(x,t)=v+ej(w(t-x/c))

  • Backward-Traveling Wave

    p(x,t)=p-ej(w(t+x/c)), v(x,t)=v-ej(w(t+x/c))

  • Newton’s 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


Volume velocity

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)


Acoustic waves

Acoustic Waves

  • Pressure

    p(x,t) = ejwt (p+e-jkx + p-ejkx)

  • Velocity

    v(x,t) = ejwt (1/rc)(p+e-jkx - p-ejkx)


Standing waves resonances in the vocal tract

Standing Waves: Resonances in the Vocal Tract


Boundary conditions

Boundary Conditions

L

0


Boundary conditions1

Boundary Conditions

  • Pressure=0 at x=L

    0 = p(L,t) = ejwt (p+e-jkL + p-ejkL)

    0 = p+e-jkL + p-ejkL

  • Velocity=0 at x=0

    0 = v(0,t) = ejwt (1/rc)(p+e-jk0 - p-ejk0)

    0 = p+e-jk0 - p-ejk0

    0 = p+- p-

    p+= p-


Two equations in two unknowns p and p

Two Equations in Two Unknowns (p+ and p-)

  • Two Equations in Two Unknowns

    0 = p+e-jkL + p-ejkL

    p+= p-

  • Combine by Variable Substitution:

    0 = p+(e-jkL +ejkL)


And now more useful trigonometry

…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(ejkL+e-jkL)

    sin(kL) = -0.5 j (ejkL-e-jkL)


Two equations in two unknowns p and p1

Two Equations in Two Unknowns (p+ and p-)

  • Two Equations in Two Unknowns

    0 = p+e-jkL + p-ejkL

    p+= p-

  • Combine by Variable Substitution:

    0 = p+(e-jkL +ejkL)

    0 = 2p+cos(kL)

  • Two Possible Solutions:

    0 = p+ (Meaning, amplitude of the wave is 0)

    0 = cos(kL) (Meaning…)


Resonant frequencies of a uniform tube closed at one end open at the other end

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, …

    k1=p/2L, w1=pc/2L, F1=c/4L

    k2=3p/2L, w2=3pc/2L, F2=3c/4L

    k3=5p/2L, w3=5pc/2L, F3=5c/4L


Resonant frequencies of a uniform tube closed at one end open at the other end1

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/L=2000Hz, and

    F1=500Hz

    F2=1500Hz

    F3=2500Hz


Closed at one end open at the other end quarter wave resonator

Closed at One End, Open at the Other End = “Quarter-Wave Resonator”

  • The wavelength are:

    l1=4L

    l2=4L/3

    l3=4L/5


Standing wave patterns

Standing Wave Patterns

  • The pressure and velocity are

    p(x,t) = p+ ejwt (e-jkx+ejkx)

    v(x,t) = ejwt (p+/rc)(e-jkx-ejkx)

  • Remember that p+ encodes both the magnitude and phase, like this:

    p+=P+ejf

    Re{p(x,t)} = 2P+cos(wt+f) cos(kx)

    Re{v(x,t)} = (2P+/rc)cos(wt+f) sin(kx)


Standing wave patterns1

Standing Wave Patterns

  • The “standing wave pattern” is the part that doesn’t depend on time:

    |p(x)| = 2P+cos(kx)

    |v(x)| = (2P+/rc) sin(kx)


Standing wave patterns quarter wave resonators

Standing Wave Patterns: Quarter-Wave Resonators

Tube Closed at the Left End, Open at the Right End


Half wave resonators

Half-Wave Resonators

  • If the tube is open at x=0 and at x=L, then boundary conditions are p(0,t)=0 and p(L,t)=0

  • If the tube is closed at x=0 and at x=L, then boundary conditions are v(0,t)=0 and v(L,t)=0

  • In either case, the resonances are:

    F1=0, F2=c/2L, F3=c/L, , F3=3c/2L

  • Example, vocal tract closed at both ends:

    F1=0, F2=1000Hz, F3=2000Hz, , F3=3000Hz


Standing wave patterns half wave resonators

Standing Wave Patterns: Half-Wave Resonators

Tube Closed at Both Ends

Tube Open at Both Ends


Schwa and invv the vowels in a tug

Schwa and Invv (the vowels in “a tug”)

F3=2500Hz=5c/4L

F2=1500Hz=3c/4L

F1=500Hz=c/4L


Front cavity resonances of a fricative

Front Cavity Resonances of a Fricative

/s/: Front Cavity Resonance = 4500Hz

4500Hz = c/4L if

Front Cavity Length is L=1.9cm

/sh/: Front Cavity Resonance = 2200Hz

2200Hz = c/4L if

Front Cavity Length is L=4.0cm


Summary

Summary

  • Newton’s 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 (p+e-jkx + p-ejkx)

    v(x,t) = ejwt (1/rc)(p+e-jkx - p-ejkx)

  • Resonances

    • Quarter-wave resonator: F1=c/4L, F2=3c/4L, F3=5c/4L

    • Half-wave resonator: F1=0, F2=c/2L, F3=c/L

  • Standing-wave Patterns

    |p(x)| = 2P+cos(kx)

    |v(x)| = (2P+/rc) sin(kx)


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