ECE 598: The Speech ECE 598: The Speech ChainChain

Lecture 4: SoundLecture 4: Sound

TodayToday Ideal Gas Law + Newton’s Second = Ideal Gas Law + Newton’s Second =

SoundSound Forward-Going and Backward-Going Forward-Going and Backward-Going

WavesWaves Pressure, Velocity, and Volume Pressure, Velocity, and Volume

VelocityVelocity Boundary Conditions:Boundary Conditions:

Open tube: zero pressureOpen tube: zero pressure Closed tube: zero velocityClosed tube: zero velocity

Resonant Frequencies of a PipeResonant Frequencies of a Pipe

Speech Units: cgsSpeech Units: cgs Distance measured in cmDistance measured in cm

1cm = length of vocal folds; 15-18cm = length vocal tract1cm = length of vocal folds; 15-18cm = length vocal tract Mass measured in grams (g) Mass measured in grams (g)

1g = mass of one cm1g = mass of one cm33 of H of H220 or biological tissue0 or biological tissue 1g = mass of the vocal folds1g = mass of the vocal folds

Time measured in seconds (s)Time measured in seconds (s) Volume measured in liters (1L=1000cmVolume measured in liters (1L=1000cm33))

1L/s = air flow rate during speech1L/s = air flow rate during speech Force measured in dynes (1 d = 1 g cm/sForce measured in dynes (1 d = 1 g cm/s22))

1000 dynes = force of gravity on 1g (1cm1000 dynes = force of gravity on 1g (1cm33) of H) of H2200 Pressure measured in dynes/cmPressure measured in dynes/cm22 or cm H or cm H2200

1000 d/cm1000 d/cm22 = pressure of 1cm H = pressure of 1cm H2200 Lung pressure varies from 1-10 cm HLung pressure varies from 1-10 cm H2200

Why Does Sound Happen?Why Does Sound Happen?

Consider three blocks of air in a pipe.Consider three blocks of air in a pipe. Boundaries are at x (cm) and x+dx (cm).Boundaries are at x (cm) and x+dx (cm). Pipe cross-sectional area = A (cmPipe cross-sectional area = A (cm22)) Volume of each block of air: V = A dx (cmVolume of each block of air: V = A dx (cm33)) Mass of each block of air = m (grams)Mass of each block of air = m (grams) Density of each block of air: Density of each block of air: = m/(Adx) (g/cm = m/(Adx) (g/cm33))

A

x+dxx

Step 1: Middle Block Step 1: Middle Block SquishedSquished

Velocity of air is v or v+dv (cm/s)Velocity of air is v or v+dv (cm/s) In the example above, dv is a negative number!!!In the example above, dv is a negative number!!!

In dt seconds, Volume of middle block changes by In dt seconds, Volume of middle block changes by dV = A ( (v+dv) dV = A ( (v+dv) dv ) dt = A dv dt dv ) dt = A dv dt

cmcm33 = (cm = (cm22) (cm/s) s) (cm/s) s Density of middle block changes byDensity of middle block changes by

dd = m/(V+dV) = m/(V+dV) m/V m/V ≈ ≈ dV/V = dV/V = dt dv/dxdt dv/dx

Rate of Change of the density of the middle blockRate of Change of the density of the middle blockdd/dt = /dt = dv/dx dv/dx

(g/cm(g/cm33)/s = (g/cm)/s = (g/cm33) (m/s) / m) (m/s) / m

A

v+dvv

Step 2: The Pressure RisesStep 2: The Pressure Rises

Ideal Gas Law: p =Ideal Gas Law: p =RT (pressure proportional to RT (pressure proportional to density, temperature, and a constant R)density, temperature, and a constant R)

Adiabatic Ideal Gas Law: dp/dt = cAdiabatic Ideal Gas Law: dp/dt = c22dd/dt/dt When gas is compressed quickly, “T” and “When gas is compressed quickly, “T” and “” ” bothboth

increaseincrease This is called “adiabatic expansion” --- it means that cThis is called “adiabatic expansion” --- it means that c22>R>R

““c” is the speed of sound!!c” is the speed of sound!! ““c” depends on chemical composition (air vs. helium), c” depends on chemical composition (air vs. helium),

temperature (body temp. vs. room temp.), and atmospheric temperature (body temp. vs. room temp.), and atmospheric pressure (sea level vs. Himalayas)pressure (sea level vs. Himalayas)

dp/dt = cdp/dt = c22dd/dt = /dt = cc22 dv/dx dv/dx

A

v+dvv

Step 3: Pressure X Area = Step 3: Pressure X Area = ForceForce

Force acting on the air between Force acting on the air between and and : :

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

dynes = (cmdynes = (cm22) (d/cm) (d/cm22))

A

p+dpp

Step 4: Force Accelerates Step 4: Force Accelerates AirAir

Newton’s second law:Newton’s second law:F = m dv/dt = (F = m dv/dt = (Adx) dv/dtAdx) dv/dt

dp/dx = dp/dx = dv/dt dv/dt

(d/cm(d/cm22)/cm = (g/cm)/cm = (g/cm33) (cm/s)/s) (cm/s)/s

A

Air velocity has changed here!

Acoustic Constitutive Acoustic Constitutive EquationsEquations Newton’s Second Law:Newton’s Second Law:

dp/dx = dp/dx = dv/dt dv/dt Adiabatic Ideal Gas Law:Adiabatic Ideal Gas Law:

dp/dt = dp/dt = cc22 dv/dx dv/dx Acoustic Wave Equation:Acoustic Wave Equation:

dd22p/dtp/dt22 c c22 d d22p/dxp/dx22

pressure/spressure/s22 = (cm/s) = (cm/s)22 pressure/cm pressure/cm22

Things to notice:Things to notice: p must be a function of both time and space: p(x,t)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 c” is a speed (35400 cm/s, the speed of sound at body

temperature, or 34000 cm/s at room temperature)temperature, or 34000 cm/s at room temperature) ““ct” is a distance (distance traveled by sound in t ct” is a distance (distance traveled by sound in t

seconds)seconds)

Solution: Forward and Solution: Forward and Backward Traveling WavesBackward Traveling Waves

Wave Number k:Wave Number k:k = k = /c/c

(radians/cm) = (radians/sec) / (cm/sec)(radians/cm) = (radians/sec) / (cm/sec) Forward-Traveling Wave: Forward-Traveling Wave:

p(x,t) = pp(x,t) = p++eej(j(t-kx) t-kx) = p= p++eej(j((t-x/c))(t-x/c))

dd22p/dtp/dt22 c c22 d d22p/dxp/dx22

22pp++ = = (kc)(kc)22pp++

Backward-Traveling Wave: Backward-Traveling Wave: p(x,t) = pp(x,t) = p--eej(j(t+kx) t+kx) = p= p--eej(j((t+x/c))(t+x/c))

dd22p/dtp/dt22 c c22 d d22p/dxp/dx22

22pp-- = = (kc)(kc)22pp--

Forward and Backward Traveling Forward and Backward Traveling WavesWaves

Other Wave Quantities Worth Other Wave Quantities Worth KnowingKnowing

Wave Number k:Wave Number k:k = k = /c/c

(radians/cm) = (radians/sec) / (cm/sec)(radians/cm) = (radians/sec) / (cm/sec) Wavelength Wavelength ::

= c/f = 2= c/f = 2/k/k

(cm/cycle) = (cm/sec) / (cycles/sec)(cm/cycle) = (cm/sec) / (cycles/sec) Period T:Period T:

T = 1/fT = 1/f

(seconds/cycle) = 1 / (cycles/sec)(seconds/cycle) = 1 / (cycles/sec)

Air Particle VelocityAir Particle Velocity Forward-Traveling Wave Forward-Traveling Wave

p(x,t)=pp(x,t)=p++eej(j((t-x/c))(t-x/c)), v(x,t)=v, v(x,t)=v++eej(j((t-x/c)) (t-x/c))

Backward-Traveling Wave Backward-Traveling Wave p(x,t)=pp(x,t)=p--eej(j((t+x/c))(t+x/c)), v(x,t)=v, v(x,t)=v--eej(j((t+x/c)) (t+x/c))

Newton’s Second Law:Newton’s Second Law:dp/dx = dp/dx = dv/dt dv/dt

((/c)/c)pp++ = = vv++

((/c)/c)pp-- = = vv--

Characteristic Impedance of Air: zCharacteristic Impedance of Air: z00 = = ccvv++= = pp++//cc

vv--= = pp--//cc

Volume VelocityVolume Velocity In a pipe, it sometimes makes more In a pipe, it sometimes makes more

sense to talk about movement of all sense to talk about movement of all of the molecules at position x, all at of the molecules at position x, all at once.once. A(x) = cross-sectional area of the pipe A(x) = cross-sectional area of the pipe

at position x (cmat position x (cm22)) v(x,t) = velocity of air molecules (cm/s)v(x,t) = velocity of air molecules (cm/s) u(x,t) = A(x)v(x,t) = “volume velocity” u(x,t) = A(x)v(x,t) = “volume velocity”

(cm(cm33/s = mL/s)/s = mL/s)

Acoustic WavesAcoustic Waves

Pressure Pressure p(x,t) = ep(x,t) = ejjt t (p(p++ee-jkx -jkx + p+ p--eejkxjkx))

Velocity Velocity v(x,t) = ev(x,t) = ejjt t (1/(1/c)(pc)(p++ee-jkx -jkx - p- p--eejkxjkx))

Standing Waves: Standing Waves: Resonances in the Resonances in the

Vocal TractVocal Tract

Boundary Conditions Boundary Conditions

L0

Boundary ConditionsBoundary Conditions

Pressure=0 at x=LPressure=0 at x=L0 = p(L,t) = e0 = p(L,t) = ejjt t (p(p++ee-jkL -jkL + p+ p--eejkLjkL))

0 = p0 = p++ee-jkL -jkL + p+ p--eejkLjkL

Velocity=0 at x=0Velocity=0 at x=00 = v(0,t) = e0 = v(0,t) = ejjt t (1/(1/c)(pc)(p++ee-jk0 -jk0 - p- p--eejk0jk0))

0 = p0 = p++ee-jk0 -jk0 - p- p--eejk0jk0

0 = p0 = p++ - p- p--

pp++ = p= p--

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

Two Equations in Two UnknownsTwo Equations in Two Unknowns0 = p0 = p++ee-jkL -jkL + p+ p--eejkLjkL

pp++ = p= p--

Combine by Variable Substitution:Combine by Variable Substitution:0 = p0 = p++(e(e-jkL -jkL +e+ejkLjkL))

……and now, more useful and now, more useful trigonometry…trigonometry…

Definition of complex exponential:Definition of complex exponential:eejkLjkL = cos(kL)+j sin(kL) = cos(kL)+j sin(kL)

ee-jkL-jkL = cos(-kL)+j sin(-kL) = cos(-kL)+j sin(-kL) Cosine is symmetric, Sine Cosine is symmetric, Sine

antisymmetric:antisymmetric:ee-jkL-jkL = cos(kL) - j sin(kL) = cos(kL) - j sin(kL)

Re-combine to get useful equalities:Re-combine to get useful equalities:cos(kL) = 0.5(ecos(kL) = 0.5(ejkLjkL+e+e-jkL-jkL))

sin(kL) = sin(kL) = 0.5 j (e0.5 j (ejkLjkLee-jkL-jkL))

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

Two Equations in Two UnknownsTwo Equations in Two Unknowns0 = p0 = p++ee-jkL -jkL + p+ p--eejkLjkL

pp++ = p= p--

Combine by Variable Substitution:Combine by Variable Substitution:0 = p0 = p++(e(e-jkL -jkL +e+ejkLjkL))

0 = 2p0 = 2p++cos(kL)cos(kL)

Two Possible Solutions:Two Possible Solutions:0 = p0 = p++ (Meaning, amplitude of the wave is (Meaning, amplitude of the wave is

0)0)

0 = cos(kL) (Meaning…)0 = cos(kL) (Meaning…)

Resonant Frequencies of a Resonant Frequencies of a Uniform Tube, Closed at One Uniform Tube, Closed at One End, Open at the Other EndEnd, Open at the Other End

pp++ can only be nonzero (the can only be nonzero (the amplitude of the wave can only be amplitude of the wave can only be nonzero) at frequencies that satisfy…nonzero) at frequencies that satisfy…

0 = cos(kL) 0 = cos(kL)

kL = p/2, 3p/2, 5p/2, …kL = p/2, 3p/2, 5p/2, …

kk11==/2L, /2L, 11==c/2L, Fc/2L, F11=c/4L=c/4L

kk22=3=3/2L, /2L, 22=3=3c/2L, Fc/2L, F22=3c/4L=3c/4L

kk33=5=5/2L, /2L, 33=5=5c/2L, Fc/2L, F33=5c/4L=5c/4L

……

Resonant Frequencies of a Resonant Frequencies of a Uniform Tube, Closed at One Uniform Tube, Closed at One End, Open at the Other EndEnd, Open at the Other End

For example, if the vocal tract is For example, if the vocal tract is 17.5cm long, and the speed of sound 17.5cm long, and the speed of sound is 350m/s at body temperature, then is 350m/s at body temperature, then c/L=2000Hz, andc/L=2000Hz, and

FF11=500Hz=500Hz

FF22=1500Hz=1500Hz

FF33=2500Hz=2500Hz

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

Resonator”Resonator” The wavelength are:The wavelength are:

11==LL

22=4L/3=4L/3

33=4L/5=4L/5

Standing Wave PatternsStanding Wave Patterns

The pressure and velocity areThe pressure and velocity arep(x,t) = pp(x,t) = p+ + eejjt t (e(e-jkx-jkx+e+ejkxjkx))

v(x,t) = ev(x,t) = ejjt t (p(p++//c)(ec)(e-jkx-jkx-e-ejkxjkx))

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

pp++=P=P++eejj

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

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

Standing Wave PatternsStanding Wave Patterns

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

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

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

Standing Wave Patterns: Standing Wave Patterns: Quarter-Wave ResonatorsQuarter-Wave Resonators

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

Half-Wave ResonatorsHalf-Wave Resonators If the tube is open at x=0 and at x=L, If the tube is open at x=0 and at x=L,

then boundary conditions are p(0,t)=0 then boundary conditions are p(0,t)=0 and p(L,t)=0and p(L,t)=0

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

In either case, the resonances are:In either case, the resonances are:FF11=0, F=0, F22=c/2L, F=c/2L, F33=c/L, , F=c/L, , F33=3c/2L=3c/2L

Example, vocal tract closed at both Example, vocal tract closed at both ends:ends:

FF11=0, F=0, F22=1000Hz, F=1000Hz, F33=2000Hz, , F=2000Hz, , F33=3000Hz=3000Hz

Standing Wave Patterns: Half-Standing Wave Patterns: Half-Wave ResonatorsWave Resonators

Tube Closed at Both Ends Tube Open at Both Ends

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

F1=500Hz=c/4L

F2=1500Hz=3c/4L

F3=2500Hz=5c/4L

Front Cavity Resonances of a Front Cavity Resonances of a FricativeFricative

/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

SummarySummary Newton’s Second Law:Newton’s Second Law:

dp/dx = dp/dx = dv/dt dv/dt Adiabatic Ideal Gas Law:Adiabatic Ideal Gas Law:

dp/dt = dp/dt = cc22 dv/dx dv/dx Acoustic Wave Equation:Acoustic Wave Equation:

dd22p/dtp/dt22 c c22 d d22p/dxp/dx22Pressure Pressure Solutions Solutions

p(x,t) = ep(x,t) = ejjt t (p(p++ee-jkx -jkx + p+ p--eejkxjkx))

v(x,t) = ev(x,t) = ejjt t (1/(1/c)(pc)(p++ee-jkx -jkx - p- p--eejkxjkx))

ResonancesResonances Quarter-wave resonator: FQuarter-wave resonator: F11=c/4L, F=c/4L, F22=3c/4L, F=3c/4L, F33=5c/4L=5c/4L Half-wave resonator: FHalf-wave resonator: F11=0, F=0, F22=c/2L, F=c/2L, F33=c/L=c/L

Standing-wave PatternsStanding-wave Patterns|p(x)| = 2P|p(x)| = 2P++cos(kx)cos(kx)

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