Engineering Acoustics

36 Pages·2000·0.547 MB·English

by  Penic T.

ENGINEERING ACOUSTICS EE 363N
INDEX
(p,q,r) modes....................28 cross product.....................35 gradient mechanical radiation
2q half-power beamwidth curl....................................36 thermoacoustic.............32 impedance.......................18
HP
.........................................16 D(r) directivity function...16 gradient ratio.....................32 modal density....................28
A absorption......................27 D(q ) directivity function..14, graphing terminology........36 modes................................28
a absorption coefficient....21 15, 16 H enthalpy........................36 modulus of elasticity...........9
absorption..........................27 dB decibels.............2, 12, 13 h specific enthalpy............36 momentum conservation.6, 8
average.........................27 dBA...................................13 half-power beamwidth......16 monopole..........................13
measuring.....................27 decibel.....................2, 12, 13 harmonic wave..................36 moving coil speaker..........17
absorption coefficient..21, 28 del......................................35 heat flux............................11 m radiation mass.............18
r
measuring.....................21 density.................................6 Helmholtz resonator..........25 mufflers.......................24, 25
acoustic analogies................8 equilibrium.....................6 henry...................................3 musical intervals...............12
acoustic impedance........3, 10 dependent variable.............36 Hooke's Law........................4 N fractional octave...........12
acoustic intensity...............10 diffuse field.......................28 horsepower..........................3 n number of reflections....28
acoustic power...................10 diffuse field mass law........22 humidity............................28 N(f) modal density............28
spherical waves............11 dipole.................................14 hyperbolic functions..........34 nabla operator...................35
acoustic pressure..............5, 9 direct field...................29, 30 I acoustic intensity10, 11, 12 natural angular frequency....4
effective..........................5 directivity function14, 15, 16 I spectral frequency density natural frequency................4
f
adiabatic........................7, 36 dispersion..........................22 ........................................13 newton.................................3
adiabatic bulk modulus........6 displacement IL intensity level...............12 Newton's Law.....................4
ambient density................2, 6 particle.........................10 impedance.....................3, 10 noise..................................36
amp......................................3 divergence.........................35 air 10 noise reduction..................30
amplitude.............................4 dot product........................35 due to air......................18 NR noise reduction..........30
analogies..............................8 double walls......................23 mechanical...................17 number of reflections........28
anechoic room...................36 E energy density...............26 plane wave...................10 octave bands......................12
arbitrary direction plane E(t) room energy density..26 radiation.......................18 odd function........................5
wave..................................9 effective acoustic pressure...5 spherical wave.............11 p acoustic pressure.........5, 6
architectural absorption electrical analogies..............8 incident power...................27 Pa........................................3
coefficient........................28 electrical impedance..........18 independent variable.........36 particle displacement...10, 22
area electrostatic transducer......19 inductance...........................8 partition.............................21
sphere...........................36 energy density...................26 inertance..............................8 pascal..................................3
average absorption.............27 direct field....................29 instantaneous intensity......10 paxial axial pressure...........19
average energy density......26 reverberant field...........30 instantaneous pressure.........5 Pe effective acoustic
axial pressure.....................19 enthalpy.............................36 intensity.......................10, 11 pressure.............................5
B bulk modulus..................6 entropy..............................36 intensity (dB)..............12, 13 perfect adiabatic gas............7
band equation of state..............6, 7 intensity spectrum level.....13 phase.................................33
frequency......................12 equation overview...............6 intervals phase angle..........................4
bandwidth..........................12 equilibrium density..............6 musical.........................12 phase speed.........................9
bass reflex..........................19 Euler's equation.................34 Iref reference intensity.......12 phasor notation..................33
Bessel J function..........18, 34 even function.......................5 isentropic...........................36 piezoelectric transducer.....19
binomial expansion............34 expansion chamber......24, 25 ISL intensity spectrum level pink noise..........................36
binomial theorem...............34 Eyring-Norris....................28 ........................................13 plane wave
bulk modulus.......................6 far field..............................16 isothermal..........................36 impedance....................10
C compliance......................8 farad....................................3 isotropic.............................28 velocity..........................9
c speed of sound.................3 f center frequency............12 joule....................................3 plane waves.........................9
c
calculus..............................34 f lower frequency.............12 k wave number...................2 polar form...........................4
l
capacitance..........................8 flexural wavelength...........22 k wave vector.....................9 power..........................10, 11
center frequency................12 flow effects........................25 kelvin..................................3 SPL..............................29
characteristic impedance....10 focal plane.........................16 L inertance..........................8 power absorbed.................27
circular source...................15 focused source...................16 Laplacian...........................35 Pref reference pressure......13
cocktail party effect...........30 Fourier series.......................5 line source.........................14 pressure...........................6, 9
coincidence effect..............22 Fourier's law for heat linearizing an equation......34 progressive plane wave.......9
complex conjugate.............33 conduction.......................11 LM mean free path.............28 progressive spherical wave11
complex numbers..............33 frequency m architectural absorption propagation.........................9
compliance..........................8 center............................12 coefficient........................28 propagation constant...........2
condensation................2, 6, 7 frequency band..................12 magnitude..........................33 Q quality factor................29
conjugate frequency band intensity mass quality factor.....................29
complex........................33 level.................................13 radiation.......................18 r gas constant.....................7
contiguous bands...............12 f upper frequency............12 mass conservation...........6, 7 R room constant...............29
u
coulomb...............................3 gas constant.........................7 material properties.............20 radiation impedance..........18
C dispersion....................22 general math......................33 mean free path...................28 radiation mass...................18
p
Cramer's rule.....................23 glossary.............................36 mechanical impedance......17 radiation reactance............18
critical gradient..................32 grad operator.....................35 rayleigh number................16
Tom Penick [email protected] www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 1 of 36
rayls.....................................3 specific acoustic impedance trace wavelength...............22 wave vector.........................9
r reverberation radius......29 .........................................10 transducer wavelength..........................2
d
reflection............................20 specific enthalpy................36 electrostatic..................19 temperature effects.......25
reflection coefficient..........20 specific gas constant............7 piezoelectric.................19 weber...................................3
resonance spectral frequency density.13 transmission......................20 weighted sound levels.......13
modal............................28 speed amplitude...................4 transmission at oblique white noise........................36
reverberant field................30 speed of sound.....................3 incidence.........................22 W incident power.....27
incident
reverberation radius...........29 sphere................................36 transmission coefficient....20 Young's modulus.................9
reverberation room............36 spherical wave...................11 transmission loss...............21 z acoustic impedance........10
reverberation time..............28 impedance....................11 composite walls............22 z impedance................10, 11
rms.................................5, 34 velocity........................11 diffuse field..................22 z rayleigh number...........16
0
room acoustics...................26 spherical wave impedance.11 expansion chamber......25 Z elec. impedance due to air
A
room constant....................29 SPL sound power level.....29 thin partition.................21 ........................................18
room energy density..........26 SPL sound pressure level..13 trigonometric identities.....34 Z elec. impedance due to
M
room modes.......................28 spring constant....................4 u velocity..................6, 9, 11 mech. forces....................18
root mean square...............34 standing waves..................10 U volume velocity..............8 Z mechanical impedance17
m
s condensation................2, 6 Struve function..................18 vector differential equation35 Z radiation impedance.....18
r
Sabin formula....................28 surface density...................21 velocity................................6 G gradient ratio.................32
sabins.................................27 T60 reverberation time.......28 plane wave.....................9 P acoustic power.......10, 11
series..................................34 TDS...................................36 spherical wave.............11 g ratio of specific heats.......6
sidebranch resonator..........26 temperature..........................3 volt......................................3 l wavelength......................2
simple harmonic motion......4 temperature effects............25 volume l flexural wavelength.....22
p
sound...................................3 tesla.....................................3 sphere...........................36 l trace wavelength..........22
sound decay.......................26 thermoacoustic cycle.........31 volume velocity...................8 r tr equilibrium density........6
sound growth.....................26 thermoacoustic engine.......31 w bandwidth.....................12 0
r surface density.............21
sound power level..............29 thermoacoustic gradient....32 W power absorbed........27 s
abs t time constant.................26
sound pressure level (dB)..13 thin rod................................9 watt......................................3
x particle displacement....10,
source..........................13, 14 time constant.....................26 wave
22
space derivative.................35 time delay spectrometry....36 progressive...................11
space-time..........................33 time-average......................33 spherical.......................11 (cid:209) del.................................35
speaker...............................17 time-averaged power.........33 wave equation.....................6 (cid:209) ×(cid:31)·c ur.l............................36
TL transmission loss...21, 22 wave number.......................2 (cid:209) 2 · Laplacian....................35
(cid:209) ·d ivergence ...................35

DECIBELS [dB] k WAVE NUMBER [rad/m]
A log based unit of energy that makes it easier to The wave number of propagation constant
describe exponential losses, etc. The decibel means is a component of a wave function
2p w
10 bels, a unit named after Bell Laboratories. representing the wave density or wave k = =
spacing relative to distance. Sometimes l c
L=10log energy [dB] represented by the le tter b . See also
reference energy WAVE VECTOR p9.

One decibel is approximately the minimum discernable
amplitude difference that can be detected by the human ear s CONDENSATION [no units]
over the full range of amplitude.
The ratio of the change in density to the ambient
density, i.e. the degree to which the medium has
ll WAVELENGTH [m] condensed (or expanded) due to sound waves. For
example, s = 0 means no condensation or expansion
Wavelength is the distance that a l= c= 2p of the medium. s = -½ means the density is at one

wave advances during one cycle. half the ambient value. s = +1 means the density is at
f k
twice the ambient value. Of course these examples
At high temperatures, the speed of 343 T are unrealistic for most sounds; the condensation will
sound increases so l changes. T is l = k typically be close to zero.
k
temperature in Kelvin. f 293 r-r
s = 0

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

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UNITS c SPEED OF SOUND [m/s]
A (amp) = CW=== JNmV= F · · Sound travels faster in stiffer (i.e. higher B, less
s VVsVs· s · compressible) materials. Sound travels faster at
higher temperatures.
JNmW·s ·
C (coulomb) = A·sV=F==· =
VV V lw
Frequency/wavelength relation: c=l =f
F (farad) = CC=C=J=A2 s =2 · 2p
V JNmV · V 2
g
H (henry) = V·s (note that H·F=s2) In a perfect gas: cr=T= gP0
A r K
0
C2
J (joule) = Nm···V··=CWsA·V==sF=V = 2
F g B
In liquids: c= T where B=g B
N (newton) = JC=VW·s·kg=m · = r T
mm m s 2 0
g = ratio of specific heats (1.4 for a diatomic gas) [no units]
NkgJW s ·
Pa (pascal) = == =
mm223sm · 3m P0 = ambient (atmospheric) pressure (p= P0 ). At sea
level, P» 101kPa [Pa]
T (tesla) = WbV=sH·A= · 0
mm22 m2 r 0 = equilibrium (ambient) density [kg/m3]
r = specific gas constant [J/(kg·K)]
WJJWsNm C · ·
V (volt) = ==== = T = temperature in Kelvin [K]
K
A CAsCC· F (cid:230) ¶ P(cid:246)
B = r (cid:231) (cid:247) adiabatic bulk modulus [Pa]
W (watt) = JN=m=C=V=···FV =1 VA·HP 2 0Ł r¶ ł r
0
sss s 746
B = isothermal bulk modulus, easier to measure than the
T
Wb (weber) = HA·V=s · = J adiabatic bulk modulus [Pa]
A
Two values are given for the speed of sound in solids, Bar
Acoustic impedance: [rayls or (Pa·s)/m] and Bulk. The Bar value provides for the ability of sound to
distort the dimensions of solids having a small-cross-
Temperature: [°C or K] 0°C = 273.15K 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 @ 20°C 343 Copper 5000 Steel 6100
Aluminum 6300 Glass (pyrex) 5600 Water, fresh 20°C 1481
Brass 4700 Ice 3200 Water, sea 13°C 1500
Concrete 3100 Steam @ 100°C 404.8 Wood, oak 4000

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SIMPLE HARMONIC MOTION SIMPLE HARMONIC MOTION,
POLAR FORM
Restoring force on a spring s
(Hooke's Lawf)s: x=- M The solution abovxet( Aca)n= bw+e fwcroittste(n ),
s 0
and Newton's Law: where we have the new constants:
Fm=a
(cid:230) (cid:246) 2
u
yield: - sx= md2x and dx2 +s x=0 amplitude: A= x02 +(cid:231)Ł w 0(cid:247)ł
dt2 dt2 m 0
s (cid:230) - u (cid:246)
Let w =2 , so that the system is described by the initial phase angle: f= tan- 1(cid:231) 0(cid:247)
0 m Ł w xł
0 0
d2x Note that zero phase angle occurs at maximum positive
equation dt2 +w 02=x 0. displacement.
By differentiation, it can be found that the speed of the mass
w = s is the natural angular frequency in rad/s. is uU=-w+f sint( ), where U =w A is the speed
0 m 0 0
w amplitude. The acceleration is aU=-ww+f ctos( ).
f = 0 is the natural frequency in Hz. 0 0
0 2p
Using the initial conditions, the equation can be written
The general solution takes the form
xt(At)A=w+10c2ostsiwn0 xt(x)=+w - 2 (cid:230)(cid:246)(cid:230)(cid:231)(cid:247)(cid:231) u0 2 ctostan (cid:246)(cid:247) - 1 u0
Initial conditions: 0 ŁłŁ w 0 ł w x
displacement: x(0)= x , so A = x 00 0
0 1 0
x = the initial position [m]
0
velocity x&(0)=u , so A = u0 u0 = the initial speed [m/s]
0 2 w
s
0 w = is the natural angular frequency in rad/s.
Solution: xt(xt)=w+ cossinwt u0 0 m
00 w 0 It is seen that displacement lags 90° behind the speed and
0
that the acceleration is 180° out of phase with the
s = spring constant [no units] displacement.
x = the displacement [m]
m = mass [kg]
SIMPLE HARMONIC MOTION,
u = velocity of the mass [m/s]
t = time [s] displacement – acceleration - speed
Displacement, Acceleration
Speed, a
Acceleration
Displacement Speed
x u
0 p p 3p 2p w 0t
2 2 Phase
Angle

Initial phase angle ff=0°
The speed of a simple oscillator leads the displacement by
90°. Acceleration and displacement are 180° out of phase
with each other.

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FOURIER SERIES p ACOUSTIC PRESSURE [Pa]
The Fourier Series is a method of describing a Sound waves produce proportional changes in
complex periodic function in terms of the frequencies pressure, density, and temperature. Sound is usually
and amplitudes of its fundamental and harmonic measured as a change in pressure. See Plane
frequencies. Waves p9.
Let ftf(t)=T+ ( = ) any periodic signal p= P- P0
where T = the period. For a simple harmonic plane wave traveling in the x
direction, p is a function of x and t:
f ( t ) px(tP,e)= j(w-tkx )
P = instantaneous pressure [Pa]
0 1T 2T t P0 = ambient (atmospheric) pressure (p= P0 ). At sea
level, P» 101kPa [Pa]
0
P = peak acoustic pressure [Pa]

x = position along the x-axis [m]
Then ftA(A)n=t+B1w+n t (cid:229) ¥ (w cossin ) t = time [s]
2 0 n n
n=1
2p Pe EFFECTIVE ACOUSTIC PRESSURE
where w= = the fundamental frequency [Pa]
T
A = the DC component and will be zero provided the The effective acoustic pressure is the rms value of the
0
function is symmetric about the t-axis. This is almost sound pressure, or the rms sum (see page 34) of the
always the case in acoustics. values of multiple acoustic sources.
An = T2 (cid:242) tt00+T ftn(td)tcos w Aothdne di sr i fgzuhentr-coht awionhnde, npi. lefa(.tn )fe (its )i s=a -anf( -t), Pe = P2 PeP2p=dt 2 = (cid:242) T 2
mirror image of the left-

hthaenmd pisl afinrset pflriopvpiedde da boonuet of Pe =+ P12 + P22 + P32 L
the horizontal axis, e.g.
P = peak acoustic pressure [Pa]
sine function.
p = P- Pacoustic pressure [Pa]
0
B = 2 (cid:242) t0+T ftn(td)stin w Bn is zero when f(t) is an
n even function, i.e. f(t)=f(-t),
T t0 the right-hand plane is a
mirror image of the left-
hand plane, e.g. cosine
function.
where t = an arbitrary time
0

Tom Penick [email protected] www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 5 of 36
rr EQUILIBRIUM DENSITY [kg/m3] EQUATION OVERVIEW
0
The ambient density. Equation of State (pressure)
r= =B g P0 for ideal gases p= Bs
0 c2 c2
Mass Conservation (density)
g
B
r = T 3-dimensional 1-dimensional
for liquids
0 c2 ¶ s+(cid:209)(cid:215)v =uv 0 ¶ s+¶ u =0
The equilibrium density is the inverse of the specific volume. ¶ t ¶ t ¶ x
From the ideal gas equation:
Pn=Rfi=TPr RT Momentum Conservation (velocity)
0
( ) 3-dimensional 1-dimensional
B = r 0 ¶r¶ P r 0adiabatic bulk modulus, approximately equal (cid:209)+vrp =¶ uv 0 ¶ p+r ¶ u= 0
to the isothermal bulk modulus, 2.18×109 for water [Pa] 0 ¶ t ¶ x 0¶ t
c = the phase speed (speed of sound) [m/s]
From the above 3 equations and 3 unknowns (p, s, u)
g = ratio of specific heats (1.4 for a diatomic gas) [no units]
we can derive the Wave Equation
P0 = ambient (atmospheric) pressure (p= P0 ). At sea 1 ¶ 2p
P = plerveesls, urP0e »[P1a0]1 kPa [Pa] (cid:209) 2p= c2 ¶ t2
n = V/m specific volume [m3/kg]

V = volume [m3]
m = mass [kg] EQUATION OF STATE - GAS
R = gas constant (287 for air) [J/(kg·K) ]
An equation of state relates the physical properties
T = absolute temperature [K] (°C + 273.15)
describing the thermodynamic behavior of the fluid. In
r 0 Equilibrium Density of Selected Materials [kg/m3] acoustics, the temperature property can be ignored.
Air @ 20°C 1.21 Copper 8900 Steel 7700
In a perfect adiabatic gas, the thermal conductivity of
Aluminum 2700 Glass (pyrex) 2300 Water, fresh 20°C 998
the gas and temperature gradients due to sound
Brass 8500 Ice 920 Water, sea 13°C 1026
waves are so small that no appreciable thermal
Concrete 2600 Steam @ 100°C 0.6 Wood, oak 720
energy transfer occurs between adjacent elements of

the gas.
BB ADIABATIC BULK MODULUS [Pa] g
(cid:230) r (cid:246)
P
B is a stiffness parameter. A larger B means the Perfect adiabatic gas: =(cid:231) (cid:247)
material is not as compressible and sound travels P Ł r ł
0 0
faster within the material.
p=g Ps
(cid:230) ¶ (cid:246) Linearized: 0
P
B=r=r=(cid:231)g (cid:247) c2 P
00Ł ¶r ł 0 P = instantaneous (total) pressure [Pa]
r 0 P0 = ambient (atmospheric) pressure (p= P0 ). At sea
r = instantaneous density [kg/m3] level, P» 101kPa [Pa]
r 0 = equilibrium (ambient) density [kg/m3] r = instantan0eous density [kg/m3]
c = the phase speed (speed of sound, 343 m/s in air) [m/s] r = equilibrium (ambient) density [kg/m3]
P = instantaneous (total) pressure [Pa or N/m2] g 0= ratio of specific heats (1.4 for a diatomic gas) [no units]
P0 = ambient (atmospheric) pressure (p= P0 ). At sea p = P - P0 acoustic pressure [Pa]
level, P0 » 101kPa [Pa] s = r-r 0 1condensation [no units]
g = ratio of specific heats (1.4 for a diatomic gas) [no units] r =
0
BB Bulk Modulus of Selected Materials [Pa]
Aluminum 75×109 Iron (cast) 86×109 Rubber (hard) 5×109
Brass 136×109 Lead 42×109 Rubber (soft) 1×109
Copper 160×109 Quartz 33×109 Water *2.18×109
Glass (pyrex) 39×109 Steel 170×109 Water (sea) *2.28×109
*B , isothermal bulk modulus
T

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EQUATION OF STATE – LIQUID MASS CONSERVATION – one dimension
An equation of state relates the physical properties For the one-dimensional problem, consider sound
describing the thermodynamic behavior of the fluid. In waves traveling through a tube. Individual particles of
acoustics, the temperature property can be ignored. the medium move back and forth in the x-direction.
Adiabatic liquid: p= Bs x x+dx
p = P - P acoustic pressure [Pa]
(0 ) A= tube area
B = r 0 ¶r¶ P r adiabatic bulk modulus, approximately equal
0
to the isothermal bulk modulus, 2.18×109 for water [Pa] (r uA) is called the mass flux [kg/s]
r-r x
s = r 0 =1condensation [no units] (r uA)xd+x is what's coming out the other side (a different
0
value due to compression) [kg/s]

The difference between the rate of mass entering the center
r SPECIFIC GAS CONSTANT [J/(kg·K) ] volume (A dx) and the rate at which it leaves the center
volume is the rate at which the mass is changing in the
The specific gas constant r depends on the universal center volume.
gpaasrt iccounlasrt agnats R. Faonrd a tihr er m» o2l8e7cJu/kla(gr· Kwei)g.h t M of the (r-r=u-Au)Adx( ) ¶ r( uA)
xxdx + ¶ x
r = R r dv is the mass in the center volume, so the rate at which
M the mass is changing can be written as
¶ ¶
R = universal gas constant r dv=Adxr
M = molecular weight ¶ t ¶ t
Equating the two expressions gives
¶ ¶ r( uA)
r=A- dxdx , which can be simplified
¶ t ¶ x
¶ ¶
r+r = ( u) 0
¶ t ¶ x
u = particle velocity (due to oscillation, not flow) [m/s]
r = instantaneous density [kg/m3]
p = P - P acoustic pressure [Pa]
0
A = area of the tube [m2]

MASS CONSERVATION –
three dimensions
¶
r+(cid:209)(cid:215)r v=( uv) 0
¶ t
¶¶ ¶
where (cid:209)v=r+r+r xˆ yˆ zˆ
¶ xy ¶ z ¶
and let r=r +(1 s)
0
¶
s+(cid:209)(cid:215)v =uv 0 (linearized)
¶ t

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MOMENTUM CONSERVATION – one ACOUSTIC ANALOGIES
dimension (5.4) to electrical systems
For the one-dimensional problem, consider sound ACOUSTIC ELECTRIC
waves traveling through a tube. Individual particles of p V
the medium move back and forth in the x-direction. Impedance: Z = Z =
A U I
x x+dx Voltage: D p VI=R
( P A )x -( P A )x+dx A= tube area Current: U I =V
R
(PA) is the force due to sound pressure at location x in p = P - P0 acoustic pressure [Pa]
x the tube [N] U = volume velocity (not a vector) [m3/s]
(PA) is the force due to sound pressure at location ZA = acoustic impedance [Pa·s/m3]
xd+x
x + dx in the tube (taken to be in the positive or
right-hand direction) [N] L INERTANCE [kg/m4]
The sum of the forces in the center volume is:
Describes the inertial properties of gas in a channel.
(cid:229) FA=A-=(A-Pdx ) ( P ) ¶ P Analogous to electrical inductance.
xxdx + ¶ x r D x
L= 0
Force in the tube can be written in this form, noting that this A
is not a partial derivative:
r = ambient density [kg/m3]
0
Fm=a=Adr x ( )du D x = incremental distance [m]
dt A = cross-sectional area [m2]
For some reason, this can be written as follows:
(r=Ar dxA)ddxuu+( uu )(cid:230)(cid:231) ¶ ¶ (cid:246)(cid:247) C COMPLIANCE [m6/kg]
dtt x Ł ¶ ¶ ł
The springiness of the system; a higher value means
with the term u¶ u often discarded in acoustics. softer. Analogous to electrical capacitance.
¶ x
V
P = instantaneous (total) pressure [Pa or N/m2] C = gr
A = area of the tube [m2] 0
r = instantaneous density [kg/m3] V = volume [m3]
p = P - P acoustic pressure [Pa] g = ratio of specific heats (1.4 for a diatomic gas) [no units]
0
u = particle velocity (due to oscillation, not flow) [m/s] r = ambient density [kg/m3]
0

MOMENTUM CONSERVATION – U VOLUME VELOCITY [m3/s]
three dimensions Although termed a velocity, volume velocity is not a
¶¶ ¶ (cid:230) u u(cid:246) vector. Volume velocity in a (uniform flow) duct is the
P+r+(cid:231) =u (cid:247) 0 product of the cross-sectional area and the velocity.
¶¶ tt ¶ Łx ł
¶ V dx
and (cid:209)+vr+p(cid:215)(cid:209)u =(cid:230)(cid:231)u¶ uv v vv(cid:246)(cid:247) 0 US=u=S¶ tdt =
Ł ¶ t ł
Note that uv(cid:215)(cid:209) vuv is a quadratic term and that r ¶ uv is VS == avroeluam [me 2[]m 3]
¶ t
u = velocity [m/s]
quadratic after multiplication
x = particle displacement, the displacement of a fluid
(cid:209)+vrp =¶ uv 0 (linearized) element from its equilibrium position [m]
0 ¶ t

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PLANE WAVES
u VELOCITY, PLANE WAVE [m/s]
PLANE WAVES (2.4, 5.7) The acoustic pressure divided by the impedance, also
A disturbance a great distance from the source is from the momentum equation:
approximated as a plane wave. Each acoustic
variable has constant amplitude and phase on any ¶ p+r=¶ fi u 0 u = p = p
plane perpendicular to the direction of propagation. ¶ x 0¶ t z r c
0
The wave equation is the same as that for a
disturbance on a string under tension. p = P - P acoustic pressure [Pa]
0
¶ ¶ z = wave impedance [rayls or (Pa·s)/m ]
There is no y or z dependence, so ¶ y =¶ z =0. r 0 = equilibrium (ambient) density [kg/m3]
c = dx is the phase speed (speed of sound) [m/s]
One-dimensional wave equation: ¶ 2p = 1¶ 2p k = wdatve number or propagation constant [rad./m]
¶ xc22 t2¶ r = radial distance from the center of the sphere [m]
General Solution
for the acoustic px(tA,eB)e= 14j(w-w+2tkxtk4x) +31j(42)43 PROPAGATION (2.5)
pressure of a propagating inpropagating in
plane wave: the +x directionthe -x direction a disturbance
F(x-cD t)
p = P - P acoustic pressure [Pa]
0
A = magnitude of the positive-traveling wave [Pa]
x x
B = magnitude of the negative-traveling wave [Pa]
w = frequency [rad/s] D x=cD t
t = time [s] D xdx
k = wave number or propagation constant [rad./m] c=fiD fi , t 0
x = position along the x-axis [m] D tdt
c = dx is the phase speed (speed of sound) at which F is
dt
PROGRESSIVE PLANE WAVE (2.8) translated in the +x direction. [m/s]
A progressive plane wave is a unidirectional plane
wave—no reverse-propagating component. v
px(tA,e)= j(w-tkx ) k WAVE VECTOR [rad/m or m-1]
The phase constant k is converted to a vector. For
v
plane waves, the vector k is in the direction of
ARBITRARY DIRECTION PLANE WAVE propagation.
v
The expression for an arbitrary direction plane wave kk=xkykˆz+ ˆ + ˆ where
x y z
contains wave numbers for the x, y, and z
(cid:230) w (cid:246) 2
components. k2 +k2 +k2 =(cid:231) (cid:247)
px(tA,e)= j(w- tkx- xk- ykyz z ) x y z Ł cł

(cid:230) w (cid:246) 2
where k2+k2 +k2 =(cid:231) (cid:247) THIN ROD PROPAGATION
xy z Ł cł A thin rod is defined as l ?a.

a
a= rod radius
c = dx is the phase speed (speed of sound) [m/s]
dt
¡ = Young's modulus, or modulus of elasticity, a
characteristic property of the material [Pa]
r = equilibrium (ambient) density [kg/m3]
0

Tom Penick [email protected] www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 9 of 36
STANDING WAVES xv PARTICLE DISPLACEMENT [m]
Two waves with identical frequency and phase The displacement of a fluid element from its
characteristics traveling in opposite directions will equilibrium position.
cause constructive and destructive interference: ¶x v
p
px(tp,ep)=e 1 jt(kw-wx+jtkx )+ 2 ( ) uv = ¶ t x= rw c
1424314243 0
moving rightmoving left
for p =p =10, k=1, t=0: uv = particle velocity [m/s]
1 2

PP ACOUSTIC POWER [W]
Acoustic power is usually small compared to the
power required to produce it.
P = (cid:242) Ivd·sv
S
S = surface surrounding the sound source, or at least the
for p =5, p =10, k=1, t=0: surface area through which all of the sound passes [m2]
1 2
I = acoustic intensity [W/m2]

I ACOUSTIC INTENSITY [W/m2]
The time-averaged rate of energy transmission
through a unit area normal to the direction of
propagation; power per unit area. Note that I = Æ puæ
T
is a nonlinear equation (It’s the product of two

functions of space and time.) so you can't simply use
z SPECIFIC ACOUSTIC IMPEDANCE jw t or take the real parts and multiply, see Time-
[rayls or (Pa·s)/m ] (5.10) Average p33.
Simppeecidfiacn acceo uz sisti ca imprpoepdeartnyc oef othr ec hmaerdaicutmer aisntdic o f the IIt=p=up(ud)t = 1 (cid:242) T
T T T 0
type of wave being propagated. It is useful in
calculations involving transmission from one medium For a single frequency:
tiso ianndoetpheenr.d eInn tt hoef fcreaqsuee onfc ay .p lFaonre swpahveeri,c za li sw raevael sa nthde Ip=u1 Re{ *}
2
opposite is true. In general, z is complex.
p For a plane harmonic wave traveling in the +z direction:
z ==ur 0c (applies to progressive plane waves) 1 ¶¶¶ EEx¶1 E p2 P2
I == = c , I = = e
AtA¶¶¶ xt ¶ V 2r c r c
Acoustic impedance is analogous to electrical 0 0
impedance: T = period [s]
I(t) = instantaneous intensity [W/m2]
pressurev=olitmspedance= p = P - P acoustic pressure [Pa]
velocityamps 0
|p| = peak acoustic pressure [Pa]
z = r+ j x u = particle velocity (due to oscillation, not flow) [m/s]
P = effective or rms acoustic pressure [Pa]
e
In a sense this is resistive, In a sense this is reactive, r 0 = equilibrium (ambient) density [kg/m3]
id.eep. aar ltos sfsr osmin cteh eth seo uwracvee. iann tihmatp ethdiism veanlut eto r epprorepsaegnattsion. c = ddxt is the phase speed (speed of sound) [m/s]
r 0c Characteristic Impedance, Selected Materials (bulk) [rayls]
Air @ 20°C 415 Copper 44.5×106 Steel 47×106
Aluminum 17×106 Glass (pyrex) 12.9×106 Water, fresh 20°C 1.48×106
Brass 40×106 Ice 2.95×106 Water, sea 13°C 1.54×106
Concrete 8×106 Steam @ 100°C 242 Wood, oak 2.9×106

Tom Penick [email protected] www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 10 of 36