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Tom Penick [email protected] www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 1 of 36

ENGINEERING ACOUSTICS EE 363N

INDEX

(p,q,r) modes ....................28

2HP half-power beamwidth.........................................16

A absorption......................27

a absorption coefficient ....21

absorption..........................27average.........................27

measuring.....................27absorption coefficient ..21, 28

measuring.....................21acoustic analogies................8

acoustic impedance........3, 10acoustic intensity ...............10acoustic power...................10

spherical waves ............11

acoustic pressure..............5, 9effective..........................5

adiabatic ........................7, 36adiabatic bulk modulus........6

ambient density................2, 6

amp......................................3amplitude.............................4analogies..............................8

anechoic room ...................36arbitrary direction planewave ..................................9

architectural absorption

coefficient........................28area

sphere...........................36average absorption.............27average energy density ......26

axial pressure.....................19

B bulk modulus..................6

band

frequency......................12bandwidth..........................12bass reflex..........................19Bessel J function..........18, 34binomial expansion............34

binomial theorem...............34bulk modulus.......................6C compliance......................8c speed of sound .................3

calculus..............................34capacitance ..........................8center frequency ................12characteristic impedance....10

circular source ...................15cocktail party effect ...........30coincidence effect ..............22complex conjugate.............33

complex numbers ..............33compliance ..........................8condensation................2, 6, 7conjugate

complex........................33contiguous bands ...............12coulomb...............................3

Cp dispersion ....................22Cramer's rule .....................23critical gradient..................32

cross product .....................35

curl ....................................36D(r) directivity function...16

D() directivity function..14,15, 16

dB decibels.............2, 12, 13dBA...................................13

decibel.....................2, 12, 13del......................................35density.................................6

equilibrium.....................6

dependent variable.............36diffuse field .......................28diffuse field mass law........22dipole.................................14

direct field ...................29, 30directivity function14, 15, 16dispersion ..........................22displacement

particle .........................10

divergence .........................35dot product ........................35double walls ......................23

E energy density...............26

E(t) room energy density..26

effective acoustic pressure...5electrical analogies..............8electrical impedance..........18

electrostatic transducer......19energy density ...................26

direct field ....................29

reverberant field...........30enthalpy.............................36

entropy ..............................36equation of state ..............6, 7

equation overview...............6equilibrium density..............6Euler's equation.................34even function.......................5expansion chamber......24, 25

Eyring-Norris ....................28far field..............................16farad ....................................3

fc center frequency............12

fl lower frequency.............12flexural wavelength...........22flow effects........................25focal plane .........................16

focused source...................16Fourier series.......................5Fourier's law for heatconduction .......................11

frequencycenter............................12

frequency band..................12frequency band intensitylevel.................................13

fu upper frequency ............12gas constant .........................7general math......................33glossary .............................36

grad operator .....................35

gradient

thermoacoustic.............32gradient ratio .....................32graphing terminology........36

H enthalpy........................36

h specific enthalpy............36half-power beamwidth ......16harmonic wave..................36heat flux ............................11Helmholtz resonator..........25

henry ...................................3Hooke's Law........................4horsepower..........................3humidity............................28

hyperbolic functions..........34

I acoustic intensity10, 11, 12

If spectral frequency density........................................13

IL intensity level...............12

impedance .....................3, 10

air 10due to air......................18

mechanical ...................17plane wave ...................10radiation.......................18spherical wave .............11

incident power...................27independent variable .........36inductance ...........................8inertance..............................8

instantaneous intensity ......10instantaneous pressure.........5intensity.......................10, 11intensity (dB) ..............12, 13

intensity spectrum level.....13

intervalsmusical.........................12

Iref reference intensity.......12

isentropic...........................36ISL intensity spectrum level

........................................13isothermal..........................36

isotropic.............................28joule ....................................3k wave number ...................2

k wave vector .....................9kelvin ..................................3

L inertance..........................8Laplacian...........................35line source .........................14linearizing an equation ......34

LM mean free path.............28

m architectural absorptioncoefficient........................28

magnitude..........................33

massradiation.......................18

mass conservation ...........6, 7material properties.............20

mean free path...................28mechanical impedance......17

mechanical radiation

impedance .......................18modal density....................28modes................................28modulus of elasticity........... 9

momentum conservation . 6, 8monopole ..........................13moving coil speaker ..........17

mr radiation mass .............18mufflers.......................24, 25

musical intervals ...............12

N fractional octave ...........12

n number of reflections.... 28

N(f) modal density............28

nabla operator ...................35natural angular frequency....4

natural frequency ................4newton.................................3Newton's Law .....................4

noise..................................36

noise reduction..................30NR noise reduction ..........30

number of reflections........28octave bands......................12odd function........................ 5

p acoustic pressure .........5, 6

Pa ........................................3particle displacement...10, 22partition............................. 21pascal .................................. 3

paxial axial pressure ........... 19Pe effective acousticpressure ............................. 5

perfect adiabatic gas............7

phase ................................. 33

phase angle..........................4phase speed .........................9phasor notation..................33

piezoelectric transducer.....19pink noise..........................36plane wave

impedance.................... 10

velocity.......................... 9plane waves.........................9polar form ........................... 4power ..........................10, 11

SPL .............................. 29

power absorbed................. 27

Pref reference pressure......13pressure........................... 6, 9progressive plane wave .......9

progressive spherical wave11propagation.........................9propagation constant........... 2

Q quality factor ................29

quality factor.....................29

r gas constant .....................7

R room constant ...............29radiation impedance ..........18

radiation mass ...................18radiation reactance ............18

rayleigh number ................16


2/36


rayls.....................................3

rd reverberation radius......29reflection............................20

reflection coefficient..........20resonance

modal............................28

reverberant field ................30reverberation radius...........29

reverberation room ............36reverberation time..............28

rms.................................5, 34room acoustics...................26room constant ....................29room energy density ..........26room modes.......................28

root mean square ...............34s condensation ................2, 6Sabin formula ....................28sabins.................................27

series..................................34sidebranch resonator..........26simple harmonic motion......4sound ...................................3

sound decay.......................26sound growth.....................26

sound power level..............29sound pressure level (dB)..13

source ..........................13, 14space derivative .................35space-time..........................33speaker...............................17

specific acoustic impedance.........................................10

specific enthalpy................36

specific gas constant............7spectral frequency density.13speed amplitude...................4

speed of sound.....................3sphere ................................36

spherical wave...................11impedance....................11

velocity ........................11spherical wave impedance.11SPL sound power level.....29SPL sound pressure level..13spring constant ....................4

standing waves ..................10Struve function..................18surface density...................21T60 reverberation time.......28

TDS...................................36temperature..........................3temperature effects ............25tesla .....................................3

thermoacoustic cycle.........31thermoacoustic engine.......31

thermoacoustic gradient ....32thin rod ................................9

time constant .....................26time delay spectrometry ....36time-average......................33time-averaged power.........33

TL transmission loss...21, 22

trace wavelength ...............22transducer

electrostatic ..................19

piezoelectric.................19transmission ......................20transmission at oblique

incidence .........................22transmission coefficient ....20

transmission loss ...............21composite walls............22

diffuse field..................22expansion chamber ......25thin partition.................21

trigonometric identities .....34u velocity..................6, 9, 11

U volume velocity..............8vector differential equation35velocity................................6

plane wave .....................9

spherical wave .............11volt ......................................3volume

sphere...........................36

volume velocity...................8w bandwidth.....................12

Wabs power absorbed ........27watt......................................3

waveprogressive...................11spherical.......................11

wave equation .....................6

wave number.......................2

wave vector......................... 9wavelength..........................2

temperature effects.......25

weber...................................3weighted sound levels .......13white noise........................36

Wincident incident power..... 27Young's modulus.................9

z acoustic impedance........10z impedance................10, 11

z0 rayleigh number ...........16ZA elec. impedance due to air

........................................18ZM elec. impedance due to

mech. forces ....................18

Zm mechanical impedance 17Zr radiation impedance.....18

gradient ratio.................32 acoustic power ....... 10, 11 ratio of specific heats....... 6 wavelength......................2p flexural wavelength .....22tr trace wavelength..........220 equilibrium density........ 6

s surface density.............21 time constant .................26 particle displacement....1022

del.................................35 curl.............................362 Laplacian....................35 divergence...................35

DECIBELS [dB]

A log based unit of energy that makes it easier todescribe exponential losses, etc. The decibel means

10 bels, a unit named after Bell Laboratories.

energy10log

reference energyL = [dB]

One decibel is approximately the minimum discernableamplitude difference that can be detected by the human earover the full range of amplitude.

WAVELENGTH [m]

Wavelength is the distance that awave advances during one cycle.

At high temperatures, the speed of

sound increases so changes. Tk istemperature in Kelvin.

2c

f k

= =

343

293

kT

f =

k WAVE NUMBER [rad/m]

The wave number of propagation constantis a component of a wave function

representing the wave density or wavespacing relative to distance. Sometimes

represented by the letter . See alsoWAVE VECTOR p9.

2kc

= =

s CONDENSATION [no units]

The ratio of the change in density to the ambientdensity, i.e. the degree to which the medium hascondensed (or expanded) due to sound waves. Forexample, s = 0 means no condensation or expansionof the medium. s = -means the density is at onehalf the ambient value. s = +1 means the density is at

twice the ambient value. Of course these examplesare unrealistic for most sounds; the condensation willtypically be close to zero.

0

0

s

=

= instantaneous density [kg/m3]0 = equilibrium (ambient) density [kg/m3]


3/36


UNITS

A (amp) =

C W J N m V F

s V V s V s s= = = =

C (coulomb) = J N m W s

A s V FV V V

= = = =

F (farad) =2 2

2

C C C J A s

V J N m V V

= = = =

H (henry) = V s

A(note that 2HF s= )

J (joule) =2

2

CN m V C W s A V s F V

F= = = = =

N (newton) =2

J C V W s kg m

m m m s= = =

Pa (pascal) =2 2 3 3

N kg J W s

m m s m m= = =

T (tesla) =2 2 2

Wb V s H A

m m m= =

V (volt) =

W J J W s N m C

A C A s C C F = = = = =

W (watt) =2

1

746

J N m C V F V V A HP

s s s s= = = = =

Wb (weber) = J

H A V sA

= =

Acoustic impedance: [rayls or (Pas)/m]

Temperature: [C or K] 0C = 273.15K

c SPEED OF SOUND [m/s]

Sound travels faster in stiffer (i.e. higher B, lesscompressible) materials. Sound travels faster athigher temperatures.

Frequency/wavelength relation:2

c f

= =

In a perfect gas:0

0

Kc rT

= = P

In liquids:

0

Tc

=B

where T= B B

= ratio of specific heats (1.4 for a diatomic gas) [no units]P0 = ambient (atmospheric) pressure ( 0p P= ). At sea

level,0 101kPaP [Pa]

0 = equilibrium (ambient) density [kg/m3]

r= specific gas constant [J/(kg K)]TK= temperature in Kelvin [K]

B =

0

0

P adiabatic bulk modulus [Pa]

BT= isothermal bulk modulus, easier to measure than the

adiabatic bulk modulus [Pa]

Two values are given for the speed of sound in solids, Barand Bulk. The Bar value provides for the ability of sound todistort the dimensions of solids having a small-cross-sectional area. Sound moves more slowly in Bar material.The Bulk value is used below where applicable.

Speed of Sound in Selected Materials [m/s]

Air @ 20C 343 Copper 5000 Steel 6100

Aluminum 6300 Glass (pyrex) 5600 Water, fresh 20C 1481

Brass 4700 Ice 3200 Water, sea 13C 1500

Concrete 3100 Steam @ 100C 404.8 Wood, oak 4000


4/36


SIMPLE HARMONIC MOTION

Restoring force on a spring(Hooke's Law):

s f sx= and Newton's Law:

F ma=

s

M

yield:

2

2

d x

sx m dt = and

2

2 0

d x s

xdt m+ =

Let2

0

s

m = , so that the system is described by the

equation

22

020

d xx

dt+ = .

0

s

m = is the natural angular frequencyin rad/s.

00

2f

=

is the natural frequencyin Hz.

The general solution takes the form( ) 1 0 2 0cos sin x t A t A t = +

Initial conditions:

displacement: ( ) 00x x= , so 1 0A x=

velocity ( ) 00x u=& , so0

2

0

uA =

Solution: ( ) 00 0 00

cos sinu

x t x t t = +

s = spring constant [no units]x = the displacement [m]

m = mass [kg]u = velocity of the mass [m/s]t= time [s]

SIMPLE HARMONIC MOTION,POLAR FORM

The solution above can be written

( ) ( )0cos x t A t = + ,where we have the new constants:

amplitude:

2

2 00

0

uA x= +

initial phase angle:1 0

0 0

tanu

x

=

Note that zero phase angle occurs at maximum positivedisplacement.

By differentiation, it can be found that the speed of the mass

is ( )0sinu U t= + , where 0U A= is the speed

amplitude. The acceleration is ( )0 0cosa U t= + .

Using the initial conditions, the equation can be written

( )2

2 10 00 0

0 0 0

cos tanu u

x t x t x

= +

x0 = the initial position [m]u0 = the initial speed [m/s]

0

s

m = is the natural angular frequencyin rad/s.

It is seen that displacement lags 90 behind the speed andthat the acceleration is 180out of phase with thedisplacement.

SIMPLE HARMONIC MOTION,displacement acceleration - speed

t0

22

0 32 Phase

Angle

Displacementx

Displacement,Speed,Acceleration

Speedu

aAcceleration

Initial phase angle =0

The speed of a simple oscillator leads the displacement by90. Acceleration and displacement are 180 out of phasewith each other.


5/36


FOURIER SERIES

The Fourier Series is a method of describing acomplex periodic function in terms of the frequenciesand amplitudes of its fundamental and harmonicfrequencies.

Let ( ) ( )f t f t T = + = any periodic signal

where T= the period.

0 1T

( )f t

t2T

Then ( ) ( )01

1cos sin

2n n

n

f t A A n t B n t

=

= + +

where 2T = = the fundamental frequency

A0 = the DC component and will be zero provided thefunction is symmetric about the t-axis. This is almostalways the case in acoustics.

An = ( )0

0

2cos

t T

t f t n t dt

T

+

An is zero whenf(t) is anodd function, i.e.f(t)=-f(-t),the right-hand plane is amirror image of the left-hand plane provided one ofthem is first flipped aboutthe horizontal axis, e.g.sine function.

Bn = ( )0

0

2sin

t T

t f t n t dt

T

+

Bn is zero whenf(t) is aneven function, i.e.f(t)=f(-t),the right-hand plane is amirror image of the left-hand plane, e.g. cosinefunction.

where0t = an arbitrary time

p ACOUSTIC PRESSURE [Pa]

Sound waves produce proportional changes inpressure, density, and temperature. Sound is usuallymeasured as a change in pressure. See PlaneWaves p9.

0p = P P

For a simple harmonic plane wave traveling in thexdirection,p is a function ofx andt:

( ) ( )j

,t kx

p x t Pe =

P = instantaneous pressure [Pa]

P0 = ambient (atmospheric) pressure ( 0p P= ). At sea


P = peak acoustic pressure [Pa]x = position along thex-axis [m]t= time [s]

Pe EFFECTIVE ACOUSTIC PRESSURE[Pa]

The effective acoustic pressure is the rms value of thesound pressure, or the rms sum (see page 34) of thevalues of multiple acoustic sources.

2e

PP = 2 2 2e

TP P p dt = =

2 2 2

1 2 3eP P P P= + + +L

P = peak acoustic pressure [Pa]

p = 0P Pacoustic pressure [Pa]


6/36


0 EQUILIBRIUM DENSITY [kg/m3]

The ambient density.

00 2 2c c

= =

PBfor ideal gases

0 2

T

c

=

Bfor liquids

The equilibrium density is the inverse of the specific volume.From the ideal gas equation:

0P RT P RT = =

B = ( )0

0

P adiabatic bulk modulus, approximately equal

to the isothermal bulk modulus, 2.18109

for water [Pa]

c = the phase speed (speed of sound) [m/s]

= ratio of specific heats (1.4 for a diatomic gas) [no units]P0 = ambient (atmospheric) pressure ( 0p P= ). At sea


P = pressure [Pa]

= V/m specific volume [m3/kg]V= volume [m3]m = mass [kg]R = gas constant (287 for air) [J/(kg K)]T= absolute temperature [K] (C + 273.15)

0 Equilibrium Density of Selected Materials [kg/m3]Air @ 20C 1.21 Copper 8900 Steel 7700

Aluminum 2700 Glass (pyrex) 2300 Water, fresh 20C 998

Brass 8500 Ice 920 Water, sea 13C 1026

Concrete 2600 Steam @ 100C 0.6 Wood, oak 720

BBADIABATIC BULK MODULUS [Pa]

B is a stiffness parameter. A largerB means thematerial is not as compressible and sound travelsfaster within the material.

0

2

0 0 0c

= = =

PB P

= instantaneous density [kg/m3]0 = equilibrium (ambient) density [kg/m3]c = the phase speed (speed of sound, 343 m/s in air) [m/s]P = instantaneous (total) pressure [Pa or N/m

2]


level, 0 101kPaP [Pa]= ratio of specific heats (1.4 for a diatomic gas) [no units]

BB Bulk Modulus of Selected Materials [Pa]

Aluminum 75109 Iron (cast) 86109 Rubber (hard) 5109

Brass 136109 Lead 42109 Rubber (soft) 1109

Copper 160109 Quartz 33109 Water *2.18109

Glass (pyrex) 39109 Steel 170109 Water (sea) *2.28109

*BT, isothermal bulk modulus

EQUATION OVERVIEW

Equation of State (pressure)

p s= B

Mass Conservation (density)

3-dimensional 1-dimensional

0

s

ut

+ =

v v 0

s u

t x

+ =

Momentum Conservation (velocity)

3-dimensional 1-dimensional

0 0u

pt

+ =

vv

0 0p u

x t

+ =

From the above 3 equations and 3 unknowns (p, s, u)we can derive the Wave Equation

22

2 2

1 pp

c t

=

EQUATION OF STATE - GAS

An equation of state relates the physical propertiesdescribing the thermodynamic behavior of the fluid. Inacoustics, the temperature property can be ignored.

In a perfect adiabatic gas, the thermal conductivity ofthe gas and temperature gradients due to soundwaves are so small that no appreciable thermalenergy transfer occurs between adjacent elements ofthe gas.

Perfect adiabatic gas:

0 0

= P

P

Linearized: 0p s= P

P = instantaneous (total) pressure [Pa]



= instantaneous density [kg/m3]0 = equilibrium (ambient) density [kg/m3]= ratio of specific heats (1.4 for a diatomic gas) [no units]p = P -P0 acoustic pressure [Pa]

s = 0

0

1

= condensation [no units]


7/36


EQUATION OF STATE LIQUID

An equation of state relates the physical propertiesdescribing the thermodynamic behavior of the fluid. Inacoustics, the temperature property can be ignored.

Adiabatic liquid: p s= B p = P -P0 acoustic pressure [Pa]

B =

( ) 00

Padiabatic bulk modulus, approximately equal

to the isothermal bulk modulus, 2.18109

for water [Pa]

s = 0

0

1

= condensation [no units]

r SPECIFIC GAS CONSTANT [J/(kg K)]

The specific gas constant rdepends on the universalgas constantR and the molecular weightMof the

particular gas. For air ( )287 J/ kgKr .

rM

=R

R = universal gas constant

M= molecular weight

MASS CONSERVATION one dimension

For the one-dimensional problem, consider soundwaves traveling through a tube. Individual particles ofthe medium move back and forth in the x-direction.

tube area

x

A =

x +dx

( )

xuA is called the mass flux [kg/s]

( ) x dx

uA+

is what's coming out the other side (a different

value due to compression) [kg/s]

The difference between the rate of mass entering the centervolume (A dx) and the rate at which it leaves the centervolume is the rate at which the mass is changing in thecenter volume.

( ) ( )( )

x x dx

uAuA uA dx

x+

=

dv is the mass in the center volume, so the rate at which

the mass is changing can be written as

dv Adxt t

=

Equating the two expressions gives

( )uA Adx dx

t x

=

, which can be simplified

( ) 0ut x

+ =

u = particle velocity (due to oscillation, not flow) [m/s]

= instantaneous density [kg/m3]p = P -P0 acoustic pressure [Pa]

A = area of the tube [m2]

MASS CONSERVATION three dimensions

( ) 0ut

+ =

v v

where x y z x y z

= + +

v

and let ( )0 1 s = +

0s ut

+ =

v v

(linearized)


8/36


MOMENTUM CONSERVATION onedimension (5.4)

For the one-dimensional problem, consider soundwaves traveling through a tube. Individual particles ofthe medium move back and forth in the x-direction.

xA = tube areaAP( ) -( )

x+dxAP

x + x dx

( )x

AP is the force due to sound pressure at locationx in

the tube [N]

( ) x dx

A+

P is the force due to sound pressure at location

x + dx in the tube (taken to be in the positive or

right-hand direction) [N]

The sum of the forces in the center volume is:

( ) ( ) x x dx

F A A A dxx+

= =

P

P P

Force in the tube can be written in this form, noting that this

is not a partial derivative:

( )du

F ma Adxdt

= =

For some reason, this can be written as follows:

( ) ( )du u u

Adx Adx udt t x

= +

with the termu

ux

often discarded in acoustics.

P = instantaneous (total) pressure [Pa or N/m2]

A = area of the tube [m2]

= instantaneous density [kg/m3]p = P -P0 acoustic pressure [Pa]


MOMENTUM CONSERVATION three dimensions

0u u

P ut t x

+ + =

and 0u

p u ut

+ + =

vv vv v

Note that u uvv v is a quadratic term and that u

t

vis

quadratic after multiplication

0 0u

pt

+ =

vv(linearized)

ACOUSTIC ANALOGIESto electrical systems

ACOUSTIC ELECTRIC

Impedance:A

pZ

U=

VZ

I=

Voltage: p V I R=

Current: U VIR=

p = P -P0 acoustic pressure [Pa]

U= volume velocity (not a vector) [m3/s]

ZA = acoustic impedance [Pas/m3]

L INERTANCE [kg/m4]

Describes the inertial properties of gas in a channel.Analogous to electrical inductance.

0 xL

A

=

0 = ambient density [kg/m3]x = incremental distance [m]A = cross-sectional area [m

2]

C COMPLIANCE [m6/kg]

The springinessof the system; a higher value meanssofter. Analogous to electrical capacitance.

0

VC=

V= volume [m3]

= ratio of specific heats (1.4 for a diatomic gas) [no units]0 = ambient density [kg/m3]

U VOLUME VELOCITY [m3/s]Although termed a velocity, volume velocity is not avector. Volume velocity in a (uniform flow) duct is theproduct of the cross-sectional area and the velocity.

V dU S uS

t d t

= = =

V= volume [m3]

S = area [m2

]u = velocity [m/s]

= particle displacement, the displacement of a fluidelement from its equilibrium position [m]


9/36


PLANE WAVES

PLANE WAVES (2.4, 5.7)A disturbance a great distance from the source isapproximated as a plane wave. Each acousticvariable has constant amplitude and phase on anyplane perpendicular to the direction of propagation.The wave equation is the same as that for adisturbance on a string under tension.

There is noy orz dependence, so 0y z

= =

.

One-dimensional wave equation:2 2

2 2 2

1p p

x c t

=

General Solutionfor the acousticpressure of aplane wave:

( ) ( ) ( )j j

propagating in propagating inthe +x direction the -x direction

,t kx t kx

p x t Ae Be += +14243 14243


A = magnitude of the positive-traveling wave[Pa]

B = magnitude of the negative-traveling wave[Pa] = frequency[rad/s]t= time[s]k= wave number or propagation constant[rad./m]x = position along thex-axis[m]

PROGRESSIVE PLANE WAVE (2.8)A progressive plane wave is a unidirectional planewaveno reverse-propagating component.

( ) ( )j

,t kx

p x t Ae =

ARBITRARY DIRECTION PLANE WAVEThe expression for an arbitrary direction plane wavecontains wave numbers for thex,y, andzcomponents.

( )( )j

, x y zt k x k y k z

p x t Ae =

where

2

2 2 2

x y zk k k

c

+ + =

u VELOCITY, PLANE WAVE [m/s]

The acoustic pressure divided by the impedance, alsofrom the momentum equation:

0 0p u

x t

+ =

0

p pu

z c= =


z = wave impedance [rayls or (Pa s)/m]0 = equilibrium (ambient) density [kg/m3]c = dx

dtis the phase speed (speed of sound) [m/s]

k= wave number or propagation constant [rad./m]r= radial distance from the center of the sphere [m]

PROPAGATION (2.5)

F( tx-c )a disturbance

xx

c =x t

, 0 x dx

c tt dt

=

c = dxdt

is the phase speed (speed of sound) at which Fis

translated in the +x direction. [m/s]

kv

WAVE VECTOR [rad/m or m-1]

The phase constant kis converted to a vector. For

plane waves, the vector kv

is in the direction of

propagation.

x y zk k x k y k z= + +v

where

2

2 2 2

x y zk k k

c

+ + =

THIN ROD PROPAGATIONA thin rod is defined as a ? .

rod radiusa =

a

c = dx


= Young's modulus, or modulus of elasticity, acharacteristic property of the material[Pa]

0 = equilibrium (ambient) density [kg/m3]


10/36


STANDING WAVES

Two waves with identical frequency and phasecharacteristics traveling in opposite directions willcause constructive and destructive interference:

( ) ( ) ( )1 2moving right moving left

,j t kx j t kx

p x t p e p e += +

14243 14243

forp1=p2=10, k=1, t=0:

forp1=5,p2=10, k=1, t=0:

z SPECIFIC ACOUSTIC IMPEDANCE[raylsor(Pa s)/m] (5.10)

Specific acoustic impedance or characteristicimpedancez is a property of the medium and of thetype of wave being propagated. It is useful incalculations involving transmission from one mediumto another. In the case of a plane wave,z is real and

is independent of frequency. For spherical waves theopposite is true. In general,z is complex.

0

pz c

u= = (applies to progressive plane waves)

Acoustic impedance is analogous to electricalimpedance:

pressure voltsimpedance

velocity amps= =

In a sense this is reactive,in that this value representsan impediment to propagation.

In a sense this is resistive,i.e. a loss since the wavedeparts from the source.

z = r+ xj

0c Characteristic Impedance, Selected Materials (bulk) [rayls]Air @ 20C 415 Copper 44.5106 Steel 47106

Aluminum 17106 Glass (pyrex) 12.9106 Water, fresh 20C 1.48106

Brass 40106 Ice 2.95106 Water, sea 13C 1.54106

Concrete 8106 Steam @ 100C 242 Wood, oak 2.9106

v

PARTICLE DISPLACEMENT [m]The displacement of a fluid element from itsequilibrium position.

ut

=

vv

0

p

c =

uv

= particle velocity [m/s]

ACOUSTIC POWER [W]Acoustic power is usually small compared to thepower required to produce it.

SI d s = v v

S = surface surrounding the sound source, or at least the

surface area through which all of the sound passes [m2]I= acoustic intensity [W/m2]

I ACOUSTIC INTENSITY [W/m2]

The time-averaged rate of energy transmissionthrough a unit area normal to the direction of

propagation; power per unit area. Note thatI= puTis a nonlinear equation (Its the product of twofunctions of space and time.) so you can't simply use

jtor take the real parts and multiply, see Time-Averagep33.

( )0

1 T

TT I I t pu pu dt

T= = =

For a single frequency:

{ }1

*2 I p u= Re

For a plane harmonic wave traveling in the +z direction:

1 1 E E x E I c

A t A x t V

= = =

,

2 2

0 02

ep P

Ic c

= =

T= period [s]I(t) = instantaneous intensity [W/m2]p = P -P0 acoustic pressure [Pa]

|p| = peak acoustic pressure [Pa]

u = particle velocity (due to oscillation, not flow) [m/s]Pe = effective or rms acoustic pressure [Pa]

0 = equilibrium (ambient) density [kg/m3]c = dx



11/36


FOURIER'S LAW FOR HEATCONDUCTION, HEAT FLUX

Sound waves produce proportional changes inpressure, density, and temperature. Since theperiodic change in temperature is spread over thelength of a wavelength, the change in temperature perunit distance is very small.

Tq Kx

=

q = heat flux [C/m]

T= temperature [C]

K= a constant

x = distance [m]

SPHERICAL WAVES

SPHERICAL WAVES (5.11)

General solution for a symmetric spherical wave:

( ) ( ) ( )j j

divirging from converging onthe source the source

, t kr t kr A B p r t e er r

+= +14243 14243

Progressive spherical wave: ( ) ( )j

,t krA

p r t er

=

p = P -P0 acoustic pressure [Pa]r= radial distance from the center of the sphere [m]

A = magnitude of the positive-traveling

wave[Pa]B = magnitude of the negative-

traveling wave[Pa] = frequency[rad/s]t= time[s]k= wave number or propagation

constant [rad./m]

r

SPHERICAL WAVE BEHAVIOR

Spherical wave behavior changes markedly for verysmall or very large radii. Since this is also a functionof the wavelength, we base this on the krproduct

where krr/.

For 1kr? , i.e. r

? (far from the source):

In this case, the spherical wave is much like a plane

wave with the impedance 0z c; and withp and u inphase.

For 1kr= , i.e. r = (close to the source):In this case, the impedance is almost purely reactive

0jz r; andp and u are 90out of phase. Thesource is not radiating power; particles are just sloshingback and forth near the source.

IMPEDANCE [rayls or (Pa s)/m]

Spherical wave impedance is frequency dependent:

( )0

1 j /

cpz

u kr

= =






u VELOCITY, SPHERICAL WAVE [m/s]

0

j1

p pu

z c kr

= =


z = wave impedance [rayls or (Pa s)/m]




ACOUSTIC POWER, SPHERICALWAVES [W]

For a constant acoustic power, intensity increasesproportional to a reduction in dispersion area.

SI d s = v v

general definition

{

2

area of asphericalsurface

4 r I = for spherical dispersion

{

2

hemi-sphericalsurface

2 r I = for hemispherical dispersion

S = surface surrounding the sound source, or at least the

surface area through which all of the sound passes [m2]I= acoustic intensity [W/m2]r= radial distance from the center of the sphere [m]


12/36


FREQUENCY BANDS

fu fl FREQUENCY BANDS

The human ear perceives different frequencies atdifferent levels. Frequencies around 3000 Hz appearloudest with a rolloff for higher and lower frequencies.Therefore in the analysis of sound levels, it isnecessary to divide the frequency spectrum intosegments or bands.

1/2 Nu lf f= a 1N

-octave band

fu = the upper frequency in the band [Hz]fl = the lowest frequency in the band [Hz]N= the bandwidth in terms of the (inverse) fractional portion

of an octave, e.g.N=2 describes a -octave band

fc CENTER FREQUENCY [Hz]

The center frequency is the geometric mean of afrequency band.

f

300 1k

cfl fu

10k

log f

c u l f f f =

fu = the upper frequency in the band [Hz]fl = the lowest frequency in the band [Hz]

w BANDWIDTH [Hz]

The width of a frequency band.

1k300

fl

w

uf

10k

log f

( )1 1

2 22 2N Nu l cw f f f = =

CONTIGUOUS BANDS

The upper frequency of one band is the lowerfrequency of the next.

n+1

f

300 1k

ffn+1

cl

fnn

clf

n

u

fn+1

u

10k

log f

11

2Nn

c

n

c

f

f

+

=

Octave bands are the most common contiguous bands:

1

2n

c

n

c

f

f

+

= 2

cl

ff = 2u cf f=

2

cfw =

e.g. forfc = 1000 Hz,fl = 707 Hz,fu = 1414 Hz

STANDARD CENTER FREQUENCIES [Hz]

Octave bands:16, 31.5, 63, 125, 250, 500, 1000, 2000, 4000, 8000

1/3-Octave bands:10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80, 100, 125, 160, 200,

250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500,3150, 4000, 5000, 6300, 8000

MUSICAL INTERVALS [Hz]

Each half-step is 21/12

times higher than the previousnote.Harmonious frequency ratios:

2:1 octave 212/12 = 2.000 2/1 = 2.000

3:2 perfect fifth 27/12 = 1.489 3/2 = 1.500

4:3 perfect fourth 25/16 = 1.335 4/3 = 1.333

5:4 major third 24/12 = 1.260 5/4 = 1.200

IL INTENSITY LEVEL [dB]

Acoustic intensity in decibels. Note thatIL = SPLwhenIL is referenced to 10-12 and SPL is referenced to2010

-6.

Intensity Level:

ref

10logI

ILI

=

I= acoustic intensity [W/m2]

Iref= the reference intensity 110-12 in air [W/m

2]


13/36


If SPECTRAL FREQUENCY DENSITY[W/m2]

The distribution of acoustic intensity over thefrequency spectrum; the intensity at frequencyfover a

bandwidth of f. The bandwidth fis normally takento be 1 Hz and may be suppressed.

( )fI

I f

f

=

ISL INTENSITY SPECTRUM LEVEL [dB]

The spectral frequency density expressed in decibels.This is what you see on a spectrum analyzer.

( )( )

ref

1Hz10log

fI f

ISL fI

=

Iref= the reference intensity 10-12 [Pa]

ILBAND FREQUENCY BAND INTENSITYLEVEL [dB]

The sound intensity within a frequency band.

BAND 10log IL ISL w= + ISL = intensity spectrum level [dB]

w = bandwidth [Hz]

SPL SOUND PRESSURE LEVEL [dB]

Acoustic pressure in decibels. Note thatIL = SPLwhenIL is referenced to 10-12 and SPL is referenced to

2010-6. An increase of 6 dB is equivalent to doublingthe amplitude. A spherical source against a planarsurface has a 3 dB advantage over a source in freespace, 6 dB if it's in a corner, 9 dB in a 3-wall corner.

Sound Pressure Level:ref

20log eP

SPLP

=

for Multiple Sources:2

/10

1 1ref

10log 10log 10 iN N

ei SPL

i i

PSPL

P= =

= =

for Multiple Identical Sources:

0 10logSPL SPL N = + Pe = effective or rms acoustic pressure [Pa]Pref= the reference pressure 2010

-6 in air, 110-6 in water[Pa]

N= the number of sources

PSL PRESSURE SPECTRUM LEVEL[dB]

Same as intensity spectrum level.

( ) ( ) in a 1 Hz bandPSL f ISL f SPL= =

SPL = sound pressure level [dB]

dBA WEIGHTED SOUND LEVELS (13.2)

Since the ear doesn't perceive sound pressure levelsuniformly across the frequency spectrum, severalcorrection schemes have been devised to produce amore realistic scale. The most common is the A-weighted scale with units of dBA. From a referencepoint of 1000 Hz, this scale rolls off strongly for lowerfrequencies, has a modest gain in the 2-4 kHz regionand rolls off slightly at very high frequencies. Otherscales are dBB and dBC. Most standards, regulationsand inexpensive sound level meters employ the A-weighted scale.

ACOUSTICAL SOURCES

MONOPOLE (7.1)

The monopole source is a basic theoretical acousticsource consisting of a small (small ka) pulsatingsphere.

( ) ( )j

,t krA

p r t er

=

where2

0 0j A ka cu=

a

eu0

jt

p = P -P0 acoustic pressure [Pa]r= radial distance from the center of the source [m]

= frequency[rad/s]k= wave number or propagation constant [rad./m]





14/36


DIPOLE

The dipole source is a basic theoretical acousticsource consisting of two adjacent monopoles 180 outof phase. Mathematically this approximates a singlesource in translational vibration, which is what wereally want to model.

++

++ ==

++

r

2

r

r

1

x

d

p (r, )= 0

as r

d2

sin 2d

2r r r1

( ) ( ) ( )1 2j j

1 2

,t kr t kr A A

p r e er r

=

where2

0 0j A ka cu= Far from the source, the wave looks spherical:

( ) ( ) ( )j 1

2

directivity functionspherical wave

, j2 sin sint krA

p r e kd r

= 144244314243

p = P -P0 acoustic pressure [Pa]r= radial distance from the center of the source [m]

= frequency[rad/s]k= wave number or propagation constant [rad./m]




LINE SOURCE (7.3)

A line source of lengthL is calculated as follows.

r

L

dx

x

R

z

p (r, )

Let ( ) ( )j

, ,t kR

dx

dx AP R t e

L R

=

where dxP is the pressure at a remote point due to

one tiny segment of the line source,

and2

0 0j A ka cu= .

for r L? , sin R r x ,

( ) ( ), , ,dxL p r P r t dx = (abbreviated form)

( ) ( )j sin

, ,t kr kx

dx

dx AP r t e

L r

+

( ) ( )/ 2j j sin

/ 2

1, ,

Lt kr kx

L

A p r t e e dx

L r

=

( ) ( ) ( )j

, ,t krA

p r t e Dr

= where

( )( )12

1

2

sin sin

sin

kLD

kL

=

Directivity Function

see alsoHalf Power Beamwidth p16.


15/36


DIRECTIVITY FUNCTION

The directivity function is responsible for the lobes ofthe dispersion pattern. The function is normalized to

have a maximum value of 1 at = 0. Differentdirectivity functions are used for different elements;the following is the directivity function for the linesource.

( ) ( )

1

212

sin sin

sin

kLD

kL =

The following is the directivity function for a focusedsource.

( )( )1 2 /

/

J Gr aD r

Gr a=

J1(x) = first order Bessel J function

see alsoHalf Power Beamwidth p16.

CIRCULAR SOURCE (7.4, 7.5a)

A speaker in an enclosure may be modeled as acircular source of radius a in a rigid infinite baffle

vibrating with velocity 0ejt. For the far field pressure(r> ka

2).

( ) ( ) ( )2

j

0 0, , j2

t krkap r t c D e

r

=

where ( )( )12 sin

sin

J kaD

ka

=

Directivity function

1

null 1

3.83sin

ka

First null


r= radial distance from the source [m]

= angle with the normal from the circular source [radians]t= time [s]

0c = impedance of the medium [rayls] (415 for air)k= wave number or propagation constant [rad./m]


16/36


2HP HALF-POWER BEAMWIDTHThe angular width of the main lobe to the points wherepower drops off by 1/2; this is the point at which the

directivity function equals 1/ 2 .

For a circular source:

from the Directivity function: ( ) ( )1HP2 sin

1sin2

J kaD

ka = =

1

HP HP

1.61634sin 1.61634 2 2sinka

ka

= =

HP

3.2327 185.222 radians or degrees, for 1ka

ka ka ?


For a line source:

from the Directivity function: ( )( )12

HP 12

sin sin1

sin2

kLD

kL

= =

11HP HP2 1

2

1.391558sin 1.391558 2 2sinkL

kL

= =

FOCUSED SOURCE

The dispersion pattern of a focused source ismeasured at the focal plane, a plane passing throughthe focal point and perpendicular to the central axis.

a

= 0z

0ejt

FOCUSED SOURCE

Focal Planed

z= d

r

Focal Plane pressure ( ) ( )0p r G c D r =

where

2

2

kaG

d=

and ( ) ( )1 2 //

J Gr aD rGr a

= (Directivity function)


r= radial distance from the central axis [m]

G = constant [radians]

a = radius of the source [m]

d= focal length [m]


z0 RAYLEIGH NUMBER [rad.m]

The Rayleigh number or Rayleigh length is thedistance along the central axis from a circular pistonelement to the beginning of the far field. Beyond thispoint, complicated pressure patterns of the near fieldcan be ignored.

221

0 2

a z ka

= =


d= focal length [m]



17/36


MOVING COIL SPEAKER (14.3b, 14.5)

Model for the moving coilloudspeaker.

V u= Faraday's lawF I= Lorentz force ++ RV m

magnet

s

u

V

0 0

I

Z

R L

E

R LM

MZ

ZM M

CA u

u = velocity of the voice coil [m/s]

I= electrical current [A]

R0 = electrical resistance of the voice coil []L0 = electrical inductance of the voice coil [H]

s = spring stiffness due to flexible cone suspension material

[N/m]

Rm = mechanical resistance, a small frictional force [(N s)/m

or kg/s]

RM= effective electrical resistance due to the mechanical

resistance of the system []CM= effective electrical capacitance due to the mechanical

stiffness [F]

LM= effective electrical inductance due to the mechanical

inertia [H]

V= voltage applied to the voice coil [V]

ZE= electrical impedance due to electrical components []ZA = effective electrical impedance due to mechanical air

loading []ZM= effective electrical impedance due to the mechanical

effects of spring stiffness, mass, and (mechanical)

resistance []F= force on the voice coil [V]

=Bl coupling coefficient [N/A]B = magnetic field [Tesla (an SI unit)]

l = length of wire in the voice coil [m]

Zm MECHANICAL IMPEDANCE [(Ns)/m](1.7)

The mechanical impedance is analogous to electricalimpedance but does not have the same units. Whereelectrical impedance is voltage divided by current,mechanical impedance is force divided by speed,sometimes called mechanical ohms.

m

FZ

u= mF

Rm

x

s

{ { {

Return Force due to Mass force of mechanical accelerationthe spring resistance

F sx Rmx mx =& &&

mF mx R x sx= + +&& &

let ( ) ( ) j tF t F e = % and ( ) ( ) j tX t X e = %

then j tF e% ( )2 jj tmm R s X e

= + + %

and 2 j m

FX m R s= + +

%

%

so the velocity2

j=j

jdxdt

m

FU X

m R s

= =

+ +

%% %

finally

( )

2

2

j

jj / j

mmo

m

m R sF FZ

U F m R s

+ += = =

+ +

% %

% %

{ {

{inertiadampingdue to springmass effect

j jmo ms

Z R m= +

m = mass of the speaker cone and voice coil [kg]

x = distance in the direction of motion [m]


[N/m]

Rm = mechanical resistance, a small frictional force [(Ns)/m

or kg/s]

F= force on the speaker mass [N]

= frequency in radians


18/36


ZM ELECTRICAL IMPEDANCE DUE TO

MECHANICAL FORCES []Converting mechanical impedance to electrical invertseach element.

2

M

mo

RR

= ,

2M

mC =

,

2

ML

s

=

2

M M M M

mo

Z R C LZ

= = P P

2

j j

j jmo m M

m

s Z R m Z

sR m

= + =

+

RM= effective electrical resistance due to the mechanical

resistance of the system []CM= effective electrical capacitance due to the mechanical

stiffness [F]

LM= effective electrical inductance due to the mechanical

inertia [H]

Rm = mechanical resistance, a small frictional force [(Ns)/m

or kg/s]



[N/m]

=Bl coupling coefficient [N/A]Zmo = mechanical impedance, open-circuit condition

[(Ns)/m or kg/s]

ZA ELECTRICAL IMPEDANCE DUE TO

AIR []

The factor of two in the denominator is due to loadingon both sides of the speaker cone.

2

2A

r

ZZ

=

Zr RADIATION IMPEDANCE [(Ns)/m](7.5)

This is the mechanical impedance due to airresistance. For a circular piston:

( ) ( )0 1 12 j 2r Z cS R ka X ka= +

3803

20

j , 1

, 1r

a ka

Z a c ka

=

?

The functionsR1 andX1 are defined as:

( )( ) 2 4 6 81

1

21

8 192 9216 737280

J x x x x xR x

x= + ;

( )( )

3 5 7 9

5 7

1

1

3

4, 4.32

3 45 1600 10 102H

4 8 3sin , 4.32

4

x x x x xx

xX x

xx x

x x

+ + =

+ >

;

0c = impedance of the medium [rayls] (415 for air)S = surface area of the piston [m

2]

J1 = first order Bessel J functionR1 = a function describing the real part ofZrX1 = a function describing the imaginary part ofZrx = just a placeholder here for 2ka

k= wave number or propagation constant [rad./m]


H1 = first order Struve function

= frequency in radians

mr RADIATION MASS [kg] (7.5)

The effective increase in mass due to the loading ofthe fluid (radiation impedance).

rr

Xm =

The effect of radiation mass is small for light fluidssuch as air but in a more dense fluid such as water, itcan significantly decrease the resonant frequency.

0

r

s s

m m m =

+

The functionsR1 andX1 are defined as:

Xr= radiation reactance, the imaginary part of the radiation

impedance [(Ns)/m]

= frequency in radianss = spring stiffness due to flexible cone suspension material

[N/m]



19/36


paxial AXIAL PRESSURE [Pa] (7.4a)22

20 1axial 0 2

j j ,2 2

aka p cu u r ka

r r

= = >

MA

MA E

Z Vu

Z Z=

+

2

j0axial 0j

2tMA

MA E

aZ p V e Z Z r

= +

ZMA =ZM ||ZA = effective electrical impedance due to the

mechanical components and the effect of air []ZE= electrical impedance due to the voice coil []

BASS REFLEX ENCLOSURE (14.6c)

The bass response of a speaker/cabinet system canbe improved bat the expense of an increase in lowfrequency rolloff by adding a port to the enclosure.

Speaker

Port

- +

+

Choose c somewhat less than /s m to add a responsepeak just below the existing damping-controlled peak.Rolloff below that point will increase from 12 dB/octave to 18dB/octave.

1c

c cL C = ,

2

c

c

Ls

= ,

2

vc

mC =

C

L

C

C

ZC

Z

RC

uI

R

V

0 0L

E

MLR

M

MZ

MC

Lc = effective electrical inductance due to the cabinet [H]

sc = cabinet stiffness [N/m]

Cc = effective electrical capacitance due to the cabinet [F]

mv = mass of the air inside the port or vent [kg]

=Bl coupling coefficient [N/A]

PIEZOELECTRIC TRANSDUCER (14.12b)

Uses a crystal (usually quartz) or a ceramic; voltage isproportional to strain. High efficiency (30% is high foracoustics.) Highly resonant. Used for microphonesand speakers.

ELECTROSTATIC TRANSDUCER(14.3a, 14.9)

A moving diaphragm of areaA is separated from astationary plate by a dielectric material (air). A biasvoltage is applied between the diaphragm and plate.Modern devices use a PVDF film for the diaphragmwhich has a permanent charge, so no bias voltage isrequired. Bias, in this case and in general, is an

attempt to linearize the output by shifting its operatingrange to a less non-linear operating region. The DCbias voltage is much greater in magnitude than thetime-variant signal voltage but is easily filtered out insignal processing.

A

dielectricV

area

oscillationx

diaphragm

0

x0

Z

V oC

I u

F

ms 2

Acoustic voltage: ( )0

1

jV I u

C= +

,

Mechanical voltage:2

1

j

ms

o

ZFI u

C= +

,

For this circuit model, there is noinverting of mechanical impedancesas in the loudspeaker circuit.

Coupling coefficient: 0 0

0

C V

x = ,

Equilibrium capacitance: 00

AC

x

=

V0 = bias voltage [V]

C0 = equilibrium capacitance due to diaphragm, back plate,

and dielectric [F]

F= force on the diaphragm [N]

I= electrical current [A]

u = electrical current due to mechanical force [A]

= coupling coefficient [N/V]u = acoustic velocity [m/s]Zms = short-circuit mechanical impedance [(N s)/m]

x0 = equilibrium position of the diaphragm [m]

A = area of the diaphragm [m2]


20/36


REFLECTION AND TRANSMISSION ATNORMAL INCIDENCE (6.2)

Incident:( )1j t k x

i i p Pe

=

Reflected:( )1j t k x

r r p P e +=

Transmitted:( )2j t k x

t t p Pe =

= 0x x

=k1 / c1 , k 2c02=1 01 1

pr

pi

=r c 2 2 2

pt

c= / , r

Boundary Conditions:

1) Pressure is equal across the boundary at x=0.

i r t i r t p p p P P P+ = + =

2) Continuity of the normal component of velocity.

i r tu u u+ =

R, T, RI, TI REFLECTION ANDTRANSMISSION COEFFICIENTS (6.2)

The ratio of reflected and transmitted magnitudes toincident magnitudes. The stiffness of the medium hasthe most effect on reflection and transmission.

i) r2 >> r1 Medium 2 is very hard compared to medium 1

and we have total reflection. 1, 2R T Note that T=2 means that the amplitude doubles, but

there is practically no energy transmitted due to highimpedance.

ii) r2 = r1 The mediums are similar and we have total

transmission. 0, 1R T= = iii) r2


21/36


a ABSORPTION COEFFICIENT

The absorption coefficient can be measured in animpedance tube by placing a sample at the end of thetube, directing an acoustic wave onto it and measuringthe standing wave ratio.

11

1

a

i

W SWRa

W SWR

= =

+

An alternative method is to place a sample in a reverberationroom and measuring the effect on the reverberation time forthe room. Difficulties with this method include variationsencountered due to the location of the sample in the roomand the presence of standing waves at various frequencies.

0

0

0.161 1 1s

s s

Va a

S T T

= +

Wi = power incident on a surface [W]

Wa = power absorbed by ? [W]

Wr= Wi - Wa = power in the reflected sound [W]

as = absorption coefficient of the sample [no units]a0 = absorption coefficient of the empty room [no units]

V= volume of the room [m3]

Ss = surface area of the sample [no units]

Ts = reverberation time with the same in place [s]

T0 = reverberation time in the empty room [s]

a Absorption Coefficient, Selected Materials [no units]

250 Hz 1 kHz 4 kHz

acoustic tile suspended ceiling 0.50 0.75 0.60

brick 0.03 0.04 0.07

carpet 0.06 0.35 0.65

concrete 0.01 0.02 0.02

concrete block, painted 0.05 0.07 0.08

fiberglass, 1" on rigid backing 0.25 0.75 0.65

glass, heavy plate 0.06 0.03 0.02

glass, windowpane 0.25 0.12 0.04

gypsum, " on studs 0.10 0.04 0.09

floor, wooden 0.11 0.07 0.07

floor, linoleum on concrete 0.03 0.03 0.02

floor, terrazzo 0.01 0.02 0.02

upholstered seats 0.35 0.65 0.60

wood paneling, 3/8-1/2" 0.25 0.17 0.10

TRANSMISSION THROUGH PARTITIONS

TL TRANSMISSION LOSS THROUGH ATHIN PARTITION [dB] (13.15a)

For a planar, nonporous, homogeneous, flexible wall,the transmission loss is dependent on the density ofthe partition and the frequency of the noise.

0

1 j si r tP P Pc + = +

2

02Is

cT =

( ) 020log 20log 20logi st

I cTL f

I

= =

This is the Normal Incidence Mass Law. Lowfrequency roll off of the transmitted wave will be 6dB/octave. Doubling the mass of the wall will give anadditional 6 dB loss.

Loss through a thin partition in air (0c = 415):

( )020log 20log 42i

st

ITL f

I= =

Transmission loss as a function of power:

0 10logi

t

WTL

W=

s = surface density of the partition material [kg/m2]0c = impedance of the medium [rayls or (Pas)/m] (415 for

air)f= frequency [Hz]

Pi = peak acoustic pressure, incident [Pa]

Pr= peak acoustic pressure, reflected [Pa]

Pt= peak acoustic pressure, transmitted [Pa]

Ii = intensity of the incident wave [W/m2]

It= intensity of the transmitted wave [W/m2]

Wi = power of the incident wave [W/m2]

Wt= power of the transmitted wave [W/m2]

s SURFACE DENSITY [kg/m2]

The surface density affects the transmission lossthrough a material and is related to the materialdensity.

0s h =

0 = density of the material [kg/m3

]h = thickness of the material [m]


22/36


TL TRANSMISSION LOSS INCOMPOSITE WALLS AT NORMAL

INCIDENCE [dB]

For walls constructed of multiple materials, e.g. a brickwall having windows, the transmission loss is the sumof the transmission losses in the different materials.

0

110log

I

TLT=

, where 1 I i i

iT T SS= ,

for a wall in air:

2

132i

S

Tf

=

Si = area of the ith element [m

2]

Ti = TI for of the ith element (transmission intensity

coefficient [no units]

s = surface density [kg/m2]

TRANSMISSION AT OBLIQUEINCIDENCE [dB]

Waves striking a wall at an angle see less impedancethan waves at normal incidence.

pi

pr pt

( ) 2

0

1

1 cos

2

I

S

T

c

=

+

TI= transmission intensity coefficient [no units]

= angle of incidence [radians]0c = impedance of the medium [rayls or (Pas)/m] (415 for

air)

s = surface density [kg/m2]

DIFFUSE FIELD MASS LAW [dB]

In a diffuse field, sound is incident by definition at allangles with equal probability. Averaging yields anincrease in sound transmission of 5 dB over waves ofnormal incidence.

diffuse 0 5TL TL=

Loss through a thin partition in air (0c = 415):

( )diffuse 20log 47STL f=

COINCIDENCE EFFECT (13.15a)

When a plane wave strikes a thin partition at an angle,there are alternating high and low pressure zonesalong the partition that cause it to flex sinusoidally.This flexural wave propagates along the surface ofthe wall. At some frequency, there is a kind ofresonance and the wall becomes transparent to thewave. This causes a marked decrease in the

transmission loss over what is expected from themass law; it can be 10-15 dB.

Coincidence occurs

when tr = p.

The wave equationfor a thin plate:

4 2

4 2 2 2

bar

120

y h c t

+ =

+

-

y

-

+

+

-

tr

h

(y, )t

Particle displacement: ( ) ( )j /

, pt y C

y t e =

Dispersion: ( ) bar3

pC f hc f

= [m/s]

Trace wavelength: trsin

=

[m]

Flexural wavelength:p

p

C

f = [m]

Coincidence frequency:

2

bar1.8

c

cf

hc

= [Hz]

TL

flog

10-15 dB

(dB)

fc

Mass law6 dB/octaveslope

Design considerations: Iff < fc, use the diffuse field masslaw to find the transmission loss. Iff > fc, redesign to avoid.Note thatfc is proportional to the inverse of the thickness.

= transverse particle displacement [m]h = panel thickness [m]cbar = bar speed for the panel material [m/s]

t= time [s]

= angle of incidence [radians]


23/36


DOUBLE WALLS

Masses in series look like series electricalconnections. We want to determine the motion of thesecond wall due to sound incident on the first.Assume that d


24/36


MUFFLERS

EXPANSION CHAMBER

When sound traveling through a pipe encounters asection with a different cross-sectional area, it sees anew impedance and some sound is reflected. Thedimensions can be chosen to optimize thetransmission loss through the exit at particular

frequencies. We assume d< .

t

pipe

pr

ip

0

d

expansionchamber

gasflow

p-

+p

l x

p

Let( )j t kx

i i p Pe

= 0

ii

pu

c=

( )j t kx

r r p P e

+

=

0

r

r

pu

c=

( )j t kxp P e

+ +=

0

pu

c

++ =

( )j t kx p P e

+ =

0

pu

c

=

( )j t kxt t p Pe

= 0

tt

pu

c=

r

i

PR

P= t

i

PT

P=

i

P

P

+ = i

P

P

=

pi = acoustic pressure, incident wave [Pa]pr= acoustic pressure, reflected wave [Pa]

p+ = pressure, expan. chamber forward-traveling wave [Pa]

p- = pressure, expan. chamber reverse-traveling wave [Pa]

pt= acoustic pressure, transmitted wave [Pa]

Pi = peak acoustic pressure, incident [Pa]

Pr= peak acoustic pressure, reflected [Pa]

P+ = peak acoustic pressure, expansion chamber forward-

traveling wave [Pa]

P- = peak acoustic pressure, expansion chamber reverse-

traveling wave [Pa]

Pt= peak acoustic pressure, transmitted [Pa]

0c = impedance of the medium [rayls or (Pas)/m] (415 for

air)R = reflection coefficient [no units]

T= transmission coefficient [no units]

= expansion exit transmission coefficient [no units] = expansion exit reflection coefficient [no units]Sp = cross-sectional area of the pipe [m

2]

Sc = cross-sectional area of the expansion chamber [m2]

Next, apply the boundary conditions:

EXPANSION CHAMBER BOUNDARYCONDITIONS

Boundary condition 1: atx = 0,

i r i r p p p p P P P P+ + + = + + = +

1i r

i i i i

P P P PR

P P P P

+ + = + + = + (i)

Boundary condition 2: atx = 0,Conservation of massby equating volumevelocities. The volumevelocity is the cross-sectional area times thenet velocity. Seep8.

Sc

Sp

Volume velocity is equalacross the boundary.

Area

Area

( ) ( ) ( ) ( ) p i r c p i r cS u u S u u S P P S P P+ + + = + =

( )1i cr

i i p i i

P SP P PR m

P P S P P

+ = =

(ii)

(i) + (ii): ( ) ( )1 1 2m m+ + = [Eqn. 1]Boundary condition 3: atx = l,

j j jkl kl kl

t tp p p P e P e Pe + + + = + =

j j jkl kl kle e Te + = [Eqn. 2]Boundary condition 4: atx = l,

( ) ( )p

c p t t

c

SS u u S u P P P

S+ + + = + =

j j j1kl kl kl

me e Te

= [Eqn. 3]m = the ratio of the cross-sectional area of the expansion

chamber to the cross-sectional area of the pipe.

Next, solve the three equations:


25/36


EXPANSION CHAMBER, FINAL STEPS

Solve the three equations:

( ) ( )j j j

j j j1

1 1 0 2

0

0

k l k l k l

k l k l k l

m

m m

e e e

e e e T

+

+

+ =

Cramer's Rule:( )

j

1 12

cos j sin

k l

m

T eTkl m kl

= = + +

2

1 110log 10log

I

TLT T

= =

Transmission loss in an expansion chamber:

( )2 21 1

410log 1 sin

mTL m kl = +

TL

20 kl

Design point

FLOW EFFECTS

Muffler performance is affected by flow rate, but thepreceding calculations are valid for flows up to 35 m/s.

TEMPERATURE EFFECTS

The effect of having high temperature gases in a

muffler causes the speed of sound to increase, so becomes larger.

343 273

293

T

f

+ =

= wavelength [m]f= frequency [Hz]

T= temperature [C]

HELMHOLTZ RESONATOR (10.8)

A Helmholtz resonator is a vessel having a largevolume with a relatively small neck. The gas in theneck looks like a lumped mass and the gas in thevolume looks like a spring at low frequency.

S

M

A= areal

= volumeV

Stiffness due to a gas volume:

2 2

0c A

sV

= [N/m]

Mass of the gas in neck: 0m l A= [kg]Some gas spills out of the neck, so themass plug is actually slightly longer thanthe neck. In practice, the effective lengthis:

0.8l l A +

neck

Resonance: 0 01

,2 2

S S c Af

m m l V = = =


26/36


SIDEBRANCH RESONATOR

Refer to the Helmholtz resonator above.

Helmholtzresonator

=V Volume

A = Area

tp

gasflow

ip

pr

pipe

The effect here is similar to the effect of blowing across thetop of a coke bottle. The air across the bottle creates noiseat many frequencies but the coke bottle responds only to itsresonant frequency.

2

0

0

0

TL 10log 1

f V

scff

f f

+

For a duct of impedance 0c with a Helmholtz resonatorhaving stiffness s and neck mass m, the arrangement can bemodeled as follows.

+pi pr

H

mj

s

j

u

t

t

0c p

u

0, z 0c

, z 0c

/s m , z 0

( )2 02

0j

s m cz

s m c

=

+

TL = transmission loss [dB]

f= frequency [Hz]

f0 = resonant frequency of Helmholtz resonator [Hz]V= resonator volume [m

3]

s = stiffness [m3]

ROOM ACOUSTICS

EE ENERGY DENSITY [J/m3] (5.8)

The amount of sound energy (potential and kinetic)per unit volume. In a perfectly diffuse field,Edoes notdepend on location.

2 2

rms

0 02

p P

c c= =

E

prms = acoustic pressure, rms [Pa]

P = peak acoustic pressure or pressure magnitude [Pa]

0c = impedance of the medium [rayls or (Pas)/m] (415 forair)

EE(t) ROOM ENERGY DENSITY [J/m3](12.2)

Sound growth: The following expression describesthe effect of sound energy filling a room as a source isturned on at t=0.

( ) ( )/04

1 tW

t eAc

= E

The following is the differential equation that describesthe growth of sound energy in a live room.

{{

{0

powerthe rate at which of thethe rate at whichenergy is absorbed inputenergy increasesby the surfaces sourcein the volume

4

d AcV W

dt=

E+ E

This can be rewritten to include the time constant.

04Wd

d t Ac =E+E , where

4V

Ac =

Sound decay: The following expression describesthe effect of sound dissipation as a source is turnedoff at t=0.

( ) /0tt e =E E

W0 = power of the sound source [W]

A = sound absorption, in units of metric sabinor English

sabin[m2or ft

2]

t= time [s]

= time constant [s]E0 = initial energy density [J/m

3]c = the speed of sound (343 m/s in air) [m/s]

E AVERAGE ENERGY DENSITY [J/m3](12.2)

1dV

V= E E

E= energy density [J/m3]

V= room volume [m3or ft

3]


27/36


A ABSORPTION [m2or ft2] (12.1)

The absorption or absorption areamay have units ofmetric sabins or English sabins, named for WallaceSabine (1868-1919). The absorption area can bethought of as the equivalent area to be cut out of awall in order to produce the same effect as an objectof absorptionA. See also ABSORPTIONCOEFFICIENTp21.

i A a S A= =

where1

i ia a S

S= , i i iA a S=

a = average absorption coefficient [no units]S = total surface area [m

2or ft

2]

Ai = sound absorption of a particular material in the room

[m2or ft

2]

ai = absorption coefficient of a particular material [no units]

Si = area represented by a particular material [no units]

a AVERAGE ABSORPTION [m2

or ft2

](12.3)

The average sound absorption over an area.

1i ia a S

S= , abs

incident

W Aa

W S= =

A = total absorption area [m2or ft

2]

S = total surface area [m2or ft

2]

ai = absorption coefficient of a particular material [no units]


Wabs = power absorbed by the surfaces [W]

Wincident = power incident on the surfaces [W]

MEASURING ABSORPTION [m2or ft2]

The absorption of a sample can be measured byplacing the sample in a reverberation chamber andmeasuring the effect it has on reverberation time. Theabsorption value for a person @ 1kHz is about0.95 m

2, for a piece of furniture about 0.08 m2. Seealso ABSORPTION COEFFICIENTp21.

0

1 10.161s

s

A VT T

=

As = sound absorption of the sample [m2or ft

2]

V= volume of the room [m3

]Ts = reverberation time with the sample in place [s]

T0 = reverberation time in the empty room [s]

Wabs POWER ABSORBED [W] (12.2)

abs incidentW aW= where1

i ia a SS

= Wincident = power incident on the surface [W]

a = average absorption coefficient [no units]S = total surface area [m

2or ft

2]

Ai = sound absorption of a particular material in the room

[m2

or ft2

]ai = absorption coefficient of a particular material [no units]


Wincident INCIDENT POWER [W] (12.2)

The total power incident on the walls of a room.

incident

1

4W Sc= E


2]

c = the speed of sound (343 m/s in air) [m/s]

E = average energy density [J/m3]


28/36


T60 REVERBERATION TIME [s] (12.3)

The time required for a sound to decay by 60 dB, i.e.to one millionth of its previous value. Sound decay islinear when viewed on a log scale.

=0tT

60

SPL[dB]

t

60 dB

60 10log 13.816T

e T = =

,4V

ac =

Sabin formula:

0.161VT

A= (metric),

0.049VT

A= (English)

Including air absorption (for 0.2a ):0.161

4

VT

A mV =

+(metric)

More accurate, Eyring-Norris reverberation formula:

( )

0.161

4 ln 1

VT

mV S a=

(metric)

= time constant [s]V= room volume [m

3or ft3]


sabin[m2or ft

2]

a = average absorption coefficient [no units]

m = air absorption coefficient [no units]S = total surface area [m

2or ft

2]

(p,q,r) MODES, rectangular cavity (9.1)

The modes of a volume are the frequencies at whichresonances occur, and are a function of the roomdimensions. For example, the lowest mode will be thefrequency for which the longest dimension equals -wavelength and is represented by (1,0,0).

( )2 2 2

, ,2

c p q r f p q r

L W H = + +

p, q, and rform the mode numbers. They are integersrepresenting the number of half-wavelengths in the length,width, and height respectively. To avoid having more thanone mode at the same frequency, the ratio of any two roomdimensions should not be a whole number. Somerecommended room dimension ratios are 1.6:1.25:1.0 forsmall rooms and 2.4:1.5:1.0 or 3.2:1.3:1.0 for large rooms.

f= frequency [Hz]

c = the speed of sound (343 m/s in air) [m/s]L, W, H= room length, width, and height respectively [m]

N(f) MODAL DENSITY [Hz-1] (9.2)

The number of modes (resonant frequencies) per unithertz. The modal density increases with frequencyuntil it becomes a diffuse field. In a diffuse field, themodal structure is obscured and the sound fieldseems isotropic, i.e. the SPL is equal everywhere.

Rectangular room: ( ) 234

N f Vf

c

f= frequency [Hz]

V= room volume [m3]


m AIR ABSORPTION COEFFICIENT,ARCHITECTURAL [no units] (12.3)

0 0

mx mct I I e I e

= = 2m = For most architectural applications, the air absorptioncoefficient can be approximated as:

( )( )1.74

5.5 10 50 / /1000m h f

= I= acoustic intensity [W/m2]I0 = initial acoustic intensity [W/m

2]

h = relative humidity (limited to the range 20 to 70%) [%]

f= frequency (limited to the range 1.5 to 10 kHz) [Hz]


= air absorption coefficient due to combined factors [nounits]

LM MEAN FREE PATH [m]

The average distance between reflections in arectangular room. This works out to 2L/3 for a cubicroom and 2d/3 for a sphere.

4M

VL

S=


3]


2]

n NUMBER OF REFLECTIONS [no units]

The number of acoustic reflections in a room in time t.

4M

ct ctSn

L V

= =

c = the speed of sound (343 m/s in air) [m/s]t= time [s]

LM= mean free path [m]


3]

S = total surface area [m2or ft2]


29/36


R ROOM CONSTANT [m2]

Note that R A; when a is very small.

1 1

A a SR

a a= =


sabin[m2or ft

2]

a = average absorption coefficient [no units]S = total surface area [m2or ft2]

SPL SOUND POWER LEVEL [dB]

Source:

ref

SPL 10logwW

LW

= =

0 ref

2 2

ref

In units, this is 10 log 1.04,which is small enough to be ignored.

4SPL 10log 10log

4w

cWQL

r R P

= + + + mks

144424443

24SPL 10log

4w

QLr R

= + +

Lw = sound power level of the source [dB]

W= sound power level of the source [W]

Wref= reference power level, 10-12[W]

Q = quality factor [no units]

r= distance from the source to the observation point [m]

R = room constant [m2]

Pref= the reference pressure 2010-6 in air, 110-6 in water

[Pa]

Q QUALITY FACTOR [no units]A factor that is dependent on the location of a sourcerelative to reflective surfaces. The source strength oramplitude of volume velocity.

Q = 1 The source is located away from surfaces.

Q = 2 The source is located on a hard surface.

Q = 4 The source is located in a 2-way corner.



DIRECT FIELD

The direct field is that part of a room in which thedominant sound comes directly (unreflected) from thesource.

SPL[dB]

SPLrev

dr

(slope is 6 dB perdoubling of distance)

Reverberantfield

rlog

Direct field

Energy density (direct): dir 24

I Q W

c c r= =

E [J/m3]

I= acoustic intensity [W/m2]

c = the speed of sound (343 m/s in air) [m/s]Q = quality factor (Q=1 when source is remote from all

surfaces) [no units]



rd REVERBERATION RADIUS [m]

The distance from the source at which the SPL due tothe source falls to the level of the reverberant field.

16d

QRr =

Q = quality factor (Q=1 when source is remote from all

surfaces) [no units]



30/36


REVERBERANT FIELD

The area in a room that is remote enough from thesound source that movement within the field does notcause appreciable change in sound level. The area ofthe room not in the direct field.

SPL[dB]

SPLrev

dr

(slope is 6 dB perdoubling of distance)

Reverberantfield

rlog

Direct field

Sound power level: revSPL 6 10logwL R= + [dB]

Energy density (reverberant): revrev

4 4W W

Ac Rc= =E [J/m3]

SPLrev = sound power level in the reverberant field [m]


rd= distance from the source to the reverberant field

boundary [m]Lw = sound power level of the source [dB]


Wrev = reverberant sound power level of the room [W]


NR NOISE REDUCTION [dB] (13.13)The noise reduction from one room to an adjoining room isthe difference between the sound power levels in the tworooms. The value is used in the measurement oftransmission loss for various partition materials andconstruction.

1 2NR SPL SPL=

For measuring transmission loss:

2

TL NR 10log wS

R= + , provided 2 2R A;

Sw = surface area of the wall [m2]

R2 = room constant of the receiving room [m2]

A2 = sound absorption or absorption areaof the receiving

room [m2]

COCKTAIL PARTY EFFECTConsider a room of a given volume Vand reverberation timeT, and assume a fixed distance damong speakers andlisteners inMsmall conversational groups. In each group,only one person is speaking at a time. There is a theoreticalmaximum number of groups that can exist before the onsetof instability and loss of intelligibility. That is, as moreconversations are added, one must speak louder in order tobe heard. But with everyone speaking louder, the

background noise increases, hence the instability.

B

d

A

Energy density at B due to speaker A:

1 2

1 4

4

W

c d R

= + E

Reverberant energy density due to other M-1

conversations:

( )rev4

1W

MRc

= E

Signal to noise ratio:

1

2

rev

1SNR 1

1 16

R

M d

= = +

E

E

Notice that the power Wdrops out of the equation.

Now if we require that this signal t

penic, tom - engineering acoustics [www.teicontrols.com 2000]

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