Engineering Acoustics

ENGINEERING ACOUSTICS EE 363N INDEX (p,q,r) modes ....................28 2θHP half-power beamwidth .........................................16 A absorption......................27 a absorption coefficient ....21 absorption..........................27 average .........................27 measuring.....................27 absorption coefficient ..21, 28 measuring.....................21 acoustic analogies................8 acoustic impedance........3, 10 acoustic intensity ...............10 acoustic power...................10 spherical waves ............11 acoustic pressure..............5, 9 effective..........................5 adiabatic ........................7, 36 adiabatic bulk modulus........6 ambient density................2, 6 amp......................................3 amplitude.............................4 analogies..............................8 anechoic room ...................36 arbitrary direction plane wave ..................................9 architectural absorption coefficient........................28 area sphere ...........................36 average absorption.............27 average energy density ......26 axial pressure.....................19 B bulk modulus..................6 band frequency......................12 bandwidth..........................12 bass reflex..........................19 Bessel J function..........18, 34 binomial expansion............34 binomial theorem...............34 bulk modulus .......................6 C compliance......................8 c speed of sound .................3 calculus..............................34 capacitance ..........................8 center frequency ................12 characteristic impedance....10 circular source ...................15 cocktail party effect ...........30 coincidence effect..............22 complex conjugate.............33 complex numbers ..............33 compliance ..........................8 condensation................2, 6, 7 conjugate complex........................33 contiguous bands ...............12 coulomb...............................3 Cp dispersion ....................22 Cramer's rule .....................23 critical gradient..................32 Tom Penick

cross product .....................35 curl ....................................36 D(r) directivity function ...16 D(θ) directivity function..14, 15, 16 dB decibels .............2, 12, 13 dBA ...................................13 decibel .....................2, 12, 13 del......................................35 density .................................6 equilibrium.....................6 dependent variable.............36 diffuse field .......................28 diffuse field mass law........22 dipole.................................14 direct field ...................29, 30 directivity function 14, 15, 16 dispersion ..........................22 displacement particle .........................10 divergence .........................35 dot product ........................35 double walls ......................23 E energy density...............26 E(t) room energy density..26 effective acoustic pressure...5 electrical analogies ..............8 electrical impedance ..........18 electrostatic transducer ......19 energy density ...................26 direct field ....................29 reverberant field ...........30 enthalpy.............................36 entropy ..............................36 equation of state ..............6, 7 equation overview ...............6 equilibrium density..............6 Euler's equation .................34 even function.......................5 expansion chamber......24, 25 Eyring-Norris ....................28 far field..............................16 farad ....................................3 fc center frequency............12 fl lower frequency.............12 flexural wavelength ...........22 flow effects........................25 focal plane .........................16 focused source...................16 Fourier series.......................5 Fourier's law for heat conduction .......................11 frequency center............................12 frequency band ..................12 frequency band intensity level .................................13 fu upper frequency ............12 gas constant .........................7 general math ......................33 glossary .............................36 grad operator .....................35 [email protected]

gradient thermoacoustic .............32 gradient ratio .....................32 graphing terminology ........36 H enthalpy........................36 h specific enthalpy............36 half-power beamwidth ......16 harmonic wave ..................36 heat flux ............................11 Helmholtz resonator ..........25 henry ...................................3 Hooke's Law........................4 horsepower ..........................3 humidity ............................28 hyperbolic functions..........34 I acoustic intensity10, 11, 12 If spectral frequency density ........................................13 IL intensity level...............12 impedance .....................3, 10 air 10 due to air ......................18 mechanical ...................17 plane wave ...................10 radiation .......................18 spherical wave .............11 incident power...................27 independent variable .........36 inductance ...........................8 inertance..............................8 instantaneous intensity ......10 instantaneous pressure.........5 intensity.......................10, 11 intensity (dB) ..............12, 13 intensity spectrum level.....13 intervals musical.........................12 Iref reference intensity.......12 isentropic...........................36 ISL intensity spectrum level ........................................13 isothermal..........................36 isotropic.............................28 joule ....................................3 k wave number ...................2 k wave vector .....................9 kelvin ..................................3 L inertance..........................8 Laplacian...........................35 line source .........................14 linearizing an equation ......34 LM mean free path.............28 m architectural absorption coefficient........................28 magnitude..........................33 mass radiation .......................18 mass conservation ...........6, 7 material properties.............20 mean free path...................28 mechanical impedance ......17

www.teicontrols.com/notes

mechanical radiation impedance ....................... 18 modal density.................... 28 modes................................ 28 modulus of elasticity ........... 9 momentum conservation . 6, 8 monopole .......................... 13 moving coil speaker .......... 17 mr radiation mass ............. 18 mufflers....................... 24, 25 musical intervals ............... 12 N fractional octave ........... 12 n number of reflections .... 28 N(f) modal density............ 28 nabla operator ................... 35 natural angular frequency.... 4 natural frequency ................ 4 newton................................. 3 Newton's Law ..................... 4 noise.................................. 36 noise reduction .................. 30 NR noise reduction .......... 30 number of reflections ........ 28 octave bands...................... 12 odd function ........................ 5 p acoustic pressure ......... 5, 6 Pa ........................................ 3 particle displacement... 10, 22 partition............................. 21 pascal .................................. 3 paxial axial pressure ........... 19 Pe effective acoustic pressure ............................. 5 perfect adiabatic gas............ 7 phase ................................. 33 phase angle.......................... 4 phase speed ......................... 9 phasor notation.................. 33 piezoelectric transducer..... 19 pink noise.......................... 36 plane wave impedance.................... 10 velocity .......................... 9 plane waves......................... 9 polar form ........................... 4 power .......................... 10, 11 SPL .............................. 29 power absorbed ................. 27 Pref reference pressure ...... 13 pressure ........................... 6, 9 progressive plane wave ....... 9 progressive spherical wave 11 propagation ......................... 9 propagation constant ........... 2 Q quality factor ................ 29 quality factor ..................... 29 r gas constant ..................... 7 R room constant ............... 29 radiation impedance .......... 18 radiation mass ................... 18 radiation reactance ............ 18 rayleigh number ................ 16

EngineeringAcoustics.pdf

12/20/00 Page 1 of 36

rayls .....................................3 rd reverberation radius ......29 reflection............................20 reflection coefficient..........20 resonance modal............................28 reverberant field ................30 reverberation radius...........29 reverberation room ............36 reverberation time..............28 rms.................................5, 34 room acoustics...................26 room constant ....................29 room energy density ..........26 room modes .......................28 root mean square ...............34 s condensation ................2, 6 Sabin formula ....................28 sabins.................................27 series..................................34 sidebranch resonator..........26 simple harmonic motion ......4 sound ...................................3 sound decay .......................26 sound growth .....................26 sound power level..............29 sound pressure level (dB) ..13 source ..........................13, 14 space derivative .................35 space-time..........................33 speaker...............................17

specific acoustic impedance .........................................10 specific enthalpy................36 specific gas constant............7 spectral frequency density .13 speed amplitude...................4 speed of sound.....................3 sphere ................................36 spherical wave...................11 impedance ....................11 velocity ........................11 spherical wave impedance.11 SPL sound power level.....29 SPL sound pressure level..13 spring constant ....................4 standing waves ..................10 Struve function ..................18 surface density...................21 T60 reverberation time.......28 TDS ...................................36 temperature..........................3 temperature effects ............25 tesla .....................................3 thermoacoustic cycle .........31 thermoacoustic engine.......31 thermoacoustic gradient ....32 thin rod ................................9 time constant .....................26 time delay spectrometry ....36 time-average ......................33 time-averaged power .........33 TL transmission loss ...21, 22

trace wavelength ...............22 transducer electrostatic ..................19 piezoelectric.................19 transmission ......................20 transmission at oblique incidence .........................22 transmission coefficient ....20 transmission loss ...............21 composite walls............22 diffuse field..................22 expansion chamber ......25 thin partition.................21 trigonometric identities .....34 u velocity..................6, 9, 11 U volume velocity..............8 vector differential equation35 velocity................................6 plane wave .....................9 spherical wave .............11 volt ......................................3 volume sphere...........................36 volume velocity...................8 w bandwidth .....................12 Wabs power absorbed ........27 watt......................................3 wave progressive...................11 spherical.......................11 wave equation .....................6 wave number .......................2

DECIBELS [dB]

k WAVE NUMBER [rad/m]

A log based unit of energy that makes it easier to describe exponential losses, etc. The decibel means 10 bels, a unit named after Bell Laboratories.

L = 10 log

energy reference energy

[dB]

One decibel is approximately the minimum discernable amplitude difference that can be detected by the human ear over the full range of amplitude.

λ WAVELENGTH [m] Wavelength is the distance that a wave advances during one cycle. At high temperatures, the speed of sound increases so λ changes. Tk is temperature in Kelvin.

wave vector ......................... 9 wavelength .......................... 2 temperature effects....... 25 weber................................... 3 weighted sound levels ....... 13 white noise ........................ 36 Wincident incident power..... 27 Young's modulus................. 9 z acoustic impedance........ 10 z impedance................ 10, 11 z0 rayleigh number ........... 16 ZA elec. impedance due to air ........................................ 18 ZM elec. impedance due to mech. forces .................... 18 Zm mechanical impedance 17 Zr radiation impedance..... 18 Γ gradient ratio................. 32 Π acoustic power ....... 10, 11 γ ratio of specific heats....... 6 λ wavelength...................... 2 λp flexural wavelength ..... 22 λtr trace wavelength.......... 22 ρ0 equilibrium density........ 6 ρs surface density ............. 21 τ time constant ................. 26 ξ particle displacement.... 10, 22 ∇ del................................. 35 ∇× · curl............................. 36 ∇2· Laplacian.................... 35 ∇· divergence ................... 35

λ= λ=

c 2π = f k

343 Tk f 293

The wave number of propagation constant is a component of a wave function representing the wave density or wave spacing relative to distance. Sometimes represented by the letter β. See also WAVE VECTOR p9.

k=

2π ω = λ c

s CONDENSATION [no units] The ratio of the change in density to the ambient density, i.e. the degree to which the medium has condensed (or expanded) due to sound waves. For example, s = 0 means no condensation or expansion of the medium. s = -½ means the density is at one half the ambient value. s = +1 means the density is at twice the ambient value. Of course these examples are unrealistic for most sounds; the condensation will typically be close to zero.

s=

ρ − ρ0 ρ0

ρ = instantaneous density [kg/m3] ρ0 = equilibrium (ambient) density [kg/m3]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 2 of 36

UNITS

c SPEED OF SOUND [m/s] Sound travels faster in stiffer (i.e. higher B, less compressible) materials. Sound travels faster at higher temperatures.

A (amp) = C = W = J = N ·m = V ·F s V V ·s V ·s s C (coulomb) = A·s = V ·F = J = N ·m = W ·s V V V

Frequency/wavelength relation:

2 2 F (farad) = C = C = C = J = A·s V J N ·m V 2 V H (henry) = V ·s (note that H·F = s 2 ) A

In a perfect gas:

c=

γ P0 = γrTK ρ0

γ BT ρ0

where

2

C J (joule) = N ·m = V ·C = W ·s = AV · ·s = F ·V 2 = F N (newton) = J = C ·V = W ·s = kg ·m m m m s2 Pa (pascal) = N = kg = J = W ·s m 2 m·s 2 m3 m3 T (tesla) = Wb = V ·s = H · A m2 m2 m2 V (volt) = W = J = J = W ·s = N ·m = C A C A·s C C F

In liquids:

λω 2π

B = γ BT

γ = ratio of specific heats (1.4 for a diatomic gas) [no units] P0 = ambient (atmospheric) pressure ( p = P0 ). At sea level, P0 ≈ 101 kPa [Pa]

2 W (watt) = J = N ·m = C ·V = V · A = F ·V = 1 HP s s s s 746 Wb (weber) = H · A = V ·s = J A Acoustic impedance: [rayls or (Pa·s)/m]

Temperature: [°C or K] 0°C = 273.15K

c=

c = λf =

ρ0 = equilibrium (ambient) density [kg/m3] r = specific gas constant [J/(kg· K)] TK = temperature in Kelvin [K] B = ρ  ∂P  adiabatic bulk modulus [Pa] 0   ∂ρ ρ0 BT = isothermal bulk modulus, easier to measure than the adiabatic bulk modulus [Pa] Two values are given for the speed of sound in solids, Bar and Bulk. The Bar value provides for the ability of sound to distort the dimensions of solids having a small-crosssectional 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 Aluminum Brass Concrete

Tom Penick

[email protected]

www.teicontrols.com/notes

343 6300 4700 3100

Copper Glass (pyrex) Ice Steam @ 100°C

5000 5600 3200 404.8

EngineeringAcoustics.pdf

Steel Water, fresh 20°C Water, sea 13°C Wood, oak

6100 1481 1500 4000

12/20/00 Page 3 of 36

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

SIMPLE HARMONIC MOTION, POLAR FORM

s M

f s = − sx

and Newton's Law:

F = ma d 2x d 2x s yield: − sx = m and + x=0 dt 2 dt 2 m s 2 , so that the system is described by the Let ω0 = m d 2x 2 + ω0 x = 0 . equation 2 dt ω0 =

f0 =

s is the natural angular frequency in rad/s. m

ω0 2π

is the natural frequency in Hz.

x ( t ) = A cos ( ω0t + φ ) ,

where we have the new constants: amplitude:

u  A = x0 +  0   ω0 

initial phase angle:

 −u  φ = tan −1  0   ω0 x0 

2

2

Note that zero phase angle occurs at maximum positive displacement. By differentiation, it can be found that the speed of the mass is

u = −U sin ( ω0t + φ ) , where U = ω0 A

amplitude. The acceleration is

is the speed

a = −ω0U cos ( ω0t + φ ) .

Using the initial conditions, the equation can be written

The general solution takes the form

x ( t ) = A1 cos ω0t + A2 sin ω0t

Initial conditions: displacement: velocity

Solution:

The solution above can be written

x ( 0 ) = x0 , so A1 = x0 x& ( 0 ) = u0 , so

x ( t ) = x0 cos ω0 t +

u A2 = 0 ω0

u0 sin ω0t ω0

s = spring constant [no units] x = the displacement [m] m = mass [kg] u = velocity of the mass [m/s] t = time [s]

2

u   u  x ( t ) = x0 +  0  cos  ω0t − tan −1 0  ω0 x0   ω0   2

x0 = the initial position [m] u0 = the initial speed [m/s] s is the natural angular frequency in rad/s. ω0 = m It is seen that displacement lags 90° behind the speed and that the acceleration is 180° out of phase with the displacement.

SIMPLE HARMONIC MOTION, displacement – acceleration - speed Displacement, Speed, Acceleration

Acceleration

a

Displacement

Speed

x

0

π 2

u

π

3π 2

2π

ω0 t Phase Angle

Initial phase angle φ=0°

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

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 4 of 36

FOURIER SERIES

p ACOUSTIC PRESSURE [Pa]

The Fourier Series is a method of describing a complex periodic function in terms of the frequencies and amplitudes of its fundamental and harmonic frequencies.

Sound waves produce proportional changes in pressure, density, and temperature. Sound is usually measured as a change in pressure. See Plane Waves p9.

p = P − P0

Let f ( t ) = f ( t + T ) = any periodic signal where T = the period.

For a simple harmonic plane wave traveling in the x direction, p is a function of x and t: j( ωt − kx )

p ( x, t ) = Pe

f (t)

P = instantaneous pressure [Pa] 0

1T

f (t ) =

Then

where ω =

t

2T

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

∞ 1 A0 + ∑ ( An cos nωt + Bn sin nωt ) 2 n =1

Pe EFFECTIVE ACOUSTIC PRESSURE [Pa]

2π = the fundamental frequency T

A0 = the DC component and will be zero provided the function is symmetric about the t-axis. This is almost always the case in acoustics. An =

Bn =

2 T

∫

t0 +T t0

f ( t ) cos nωt dt

2 t0 +T f ( t ) sin nωt dt T ∫ t0

P0 = ambient (atmospheric) pressure ( p = P0 ). At sea level, P0 ≈ 101 kPa [Pa]

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

An is zero when f(t) is an odd function, i.e. f(t)=-f(-t), the right-hand plane is a mirror image of the lefthand plane provided one of them is first flipped about the horizontal axis, e.g. sine function.

Pe =

P 2

Pe =

Pe = P 2 = ∫ p 2 dt 2

T

P1 + P2 + P3 + L 2

2

2

P = peak acoustic pressure [Pa] p = P − P0 acoustic pressure [Pa]

Bn is zero when f(t) is an even function, i.e. f(t)=f(-t), the right-hand plane is a mirror image of the lefthand plane, e.g. cosine function.

where t0 = an arbitrary time

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 5 of 36

ρ0 EQUILIBRIUM DENSITY [kg/m3]

EQUATION OVERVIEW Equation of State (pressure)

The ambient density.

ρ0 =

B γ P0 = 2 c2 c γ BT ρ0 = 2 c

p = Bs

for ideal gases

Mass Conservation (density) 3-dimensional

for liquids

∂s v v + ∇ ⋅u = 0 ∂t

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

Pν = RT → P = ρ0 RT B = ρ0

( )

∂P ∂ρ ρ 0

∂ s ∂u + =0 ∂t ∂ x

Momentum Conservation (velocity) 3-dimensional

to the isothermal bulk modulus, 2.18×10 for water [Pa]

c = the phase speed (speed of sound) [m/s] γ = ratio of specific heats (1.4 for a diatomic gas) [no units] P0 = ambient (atmospheric) pressure ( p = P0 ). At sea

∂p ∂u + ρ0 =0 ∂x ∂t

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

∇2 p =

level, P0 ≈ 101 kPa [Pa]

P = pressure [Pa] ν = 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) Copper 8900 Glass (pyrex) 2300 Ice 920 Steam @ 100°C 0.6

1 ∂2 p c 2 ∂t 2

EQUATION OF STATE - GAS

ρ0 Equilibrium Density of Selected Materials [kg/m3] 1.21 2700 8500 2600

1-dimensional

v v ∂u ∇p + ρ 0 =0 ∂t

adiabatic bulk modulus, approximately equal 9

Air @ 20°C Aluminum Brass Concrete

1-dimensional

Steel 7700 Water, fresh 20°C 998 Water, sea 13°C 1026 Wood, oak 720

An equation of state relates the physical properties describing the thermodynamic behavior of the fluid. In acoustics, the temperature property can be ignored. In a perfect adiabatic gas, the thermal conductivity of the gas and temperature gradients due to sound waves are so small that no appreciable thermal energy transfer occurs between adjacent elements of the gas.

B ADIABATIC BULK MODULUS [Pa] B is a stiffness parameter. A larger B means the material is not as compressible and sound travels faster within the material.

Perfect adiabatic gas:

∂ P 2 B = ρ0   = ρ0 c = γ P0 ∂ρ   ρ0

Linearized:

ρ = instantaneous density [kg/m3] ρ0 = 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/m2] P0 = ambient (atmospheric) pressure ( p = P0 ). At sea level, P0 ≈ 101 kPa [Pa]

γ = ratio of specific heats (1.4 for a diatomic gas) [no units]

P ρ =  P0  ρ0 

γ

p = γ P0 s

P = instantaneous (total) pressure [Pa] P0 = ambient (atmospheric) pressure ( p = P0 ). At sea level, P0 ≈ 101 kPa [Pa] ρ = instantaneous density [kg/m3] ρ0 = equilibrium (ambient) density [kg/m3] γ = ratio of specific heats (1.4 for a diatomic gas) [no units] p = P - P0 acoustic pressure [Pa] s = ρ − ρ0 = 1 condensation [no units] ρ0

B Bulk Modulus of Selected Materials [Pa] Aluminum Brass Copper Glass (pyrex)

75×109 136×109 160×10

9

39×109

Iron (cast)

86×109

Rubber (hard)

5×109

Lead

42×109

Rubber (soft)

1×109

Quartz Steel

33×10

9

170×109

Water

*2.18×109

Water (sea)

*2.28×109

*BT, isothermal bulk modulus

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 6 of 36

EQUATION OF STATE – LIQUID

MASS CONSERVATION – one dimension

An equation of state relates the physical properties describing the thermodynamic behavior of the fluid. In acoustics, the temperature property can be ignored.

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

Adiabatic liquid:

p = Bs

x

x + dx

p = P - P0 acoustic pressure [Pa] B = ρ0

( )

∂P ∂ρ ρ 0

A = tube area

adiabatic bulk modulus, approximately equal 9

to the isothermal bulk modulus, 2.18×10 for water [Pa]

s = ρ − ρ0 = 1 condensation [no units] ρ0

( ρuA ) x+ dx

The specific gas constant r depends on the universal gas constant R and the molecular weight M of the particular gas. For air r ≈ 287 J/ ( kg·K ) .

R M

R = universal gas constant M = molecular weight

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

value due to compression) [kg/s]

r SPECIFIC GAS CONSTANT [J/(kg· K)]

r=

( ρuA) x is called the mass flux [kg/s]

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

( ρuA ) x − ( ρuA ) x+ dx = −

∂ ( ρuA ) dx ∂x

ρ dv is the mass in the center volume, so the rate at which the mass is changing can be written as

∂ ∂ ρ dv = ρA dx ∂t ∂t Equating the two expressions gives

∂ ( ρuA ) ∂ ρA dx = − dx , which can be simplified ∂t ∂x ∂ ∂ ρ + ( ρu ) = 0 ∂t ∂x u = particle velocity (due to oscillation, not flow) [m/s] ρ = instantaneous density [kg/m3] p = P - P0 acoustic pressure [Pa] A = area of the tube [m2]

MASS CONSERVATION – three dimensions v ∂ v ρ + ∇ ⋅ ( ρu ) = 0 ∂t

v ∂ ∂ ∂ where ∇ = ρ xˆ + ρ yˆ + ρ zˆ ∂x ∂y ∂z and let ρ = ρ0 (1 + s )

v v ∂ s + ∇ ⋅ u = 0 (linearized) ∂t

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 7 of 36

MOMENTUM CONSERVATION – one dimension (5.4) For the one-dimensional problem, consider sound waves traveling through a tube. Individual particles of the medium move back and forth in the x-direction. x

x + dx

( PA) x

-( PA) x+dx

A = tube area

( PA) x is the force due to sound pressure at location x in the tube [N]

( PA) x +dx

is the force due to sound pressure at location

ACOUSTIC ANALOGIES to electrical systems ACOUSTIC Impedance:

ZA =

Voltage:

∆p

Current:

U

∑ F = ( PA ) − ( PA ) x

x + dx

Z=

L INERTANCE [kg/m4] Describes the inertial properties of gas in a channel. Analogous to electrical inductance.

∂P dx = −A ∂x

Force in the tube can be written in this form, noting that this is not a partial derivative:

F = ma = ( ρA dx )

V I V =IR V I= R

p U

p = P - P0 acoustic pressure [Pa] U = volume velocity (not a vector) [m3/s] ZA = acoustic impedance [Pa·s/m3]

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:

ELECTRIC

du dt

L=

ρ0 ∆x A

ρ0 = ambient density [kg/m3] ∆x = incremental distance [m] A = cross-sectional area [m2]

For some reason, this can be written as follows:

( ρA dx )

du ∂u   ∂u = ( ρA dx )  + u  dt ∂x   ∂t

C COMPLIANCE [m6/kg]

∂u often discarded in acoustics. ∂x P = instantaneous (total) pressure [Pa or N/m2] A = area of the tube [m2] ρ = instantaneous density [kg/m3] p = P - P0 acoustic pressure [Pa] u = particle velocity (due to oscillation, not flow) [m/s] with the term u

MOMENTUM CONSERVATION – three dimensions ∂ ∂u   ∂u P + ρ + u  = 0 ∂t ∂x   ∂t v v  ∂u v v v  and ∇p + ρ  + u ⋅∇u  = 0  ∂t 

Tom Penick

[email protected]

C=

V γρ0

V = volume [m3] γ = ratio of specific heats (1.4 for a diatomic gas) [no units] ρ0 = ambient density [kg/m3]

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

U=

v v Note that uv ⋅∇uv is a quadratic term and that ρ ∂u is ∂t quadratic after multiplication

v v ∂u ∇p + ρ 0 = 0 (linearized) ∂t

The springiness of the system; a higher value means softer. Analogous to electrical capacitance.

∂V d ξ = S = uS ∂t d t

V = volume [m3] S = area [m2] u = velocity [m/s] ξ = particle displacement, the displacement of a fluid element from its equilibrium position [m]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 8 of 36

PLANE WAVES

u VELOCITY, PLANE WAVE [m/s]

PLANE WAVES (2.4, 5.7) A disturbance a great distance from the source is approximated as a plane wave. Each acoustic variable has constant amplitude and phase on any plane perpendicular to the direction of propagation. The wave equation is the same as that for a disturbance on a string under tension. There is no y or z dependence, so ∂ = ∂ = 0 . ∂ y ∂z 2 2 One-dimensional wave equation: ∂ p = 1 ∂ p ∂ x 2 c 2 ∂t 2

General Solution for the acoustic pressure of a plane wave:

The acoustic pressure divided by the impedance, also from the momentum equation:

∂p ∂u + ρ0 =0→ ∂x ∂t

propagating in the +x direction

p p = z ρ0 c

p = P - P0 acoustic pressure [Pa] z = wave impedance [rayls or (Pa· s)/m] ρ0 = equilibrium (ambient) density [kg/m3] c = dx dt is 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]

( ) ( ) p ( x, t ) = 1 Ae Be 424 3+1 424 3 j ωt − kx

u=

j ωt + kx

PROPAGATION (2.5)

propagating in the -x direction

a disturbance

F(x-c∆t)

p = P - P0 acoustic pressure [Pa] A = magnitude of the positive-traveling wave [Pa] B = magnitude of the negative-traveling wave [Pa] ω = frequency [rad/s] t = time [s] k = wave number or propagation constant [rad./m] x = position along the x-axis [m]

x c= c=

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

v k

j ωt − kx )

ARBITRARY DIRECTION PLANE WAVE

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 propagation.

v k = k x xˆ + k y yˆ + k z zˆ

The expression for an arbitrary direction plane wave contains wave numbers for the x, y, and z components.

p ( x, t ) = Ae where

(

j ωt − k x x −k y y −k z z

 ω k + k +k =  c 2 x

2 y

∆x dx → , ∆t → 0 ∆t dt

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

PROGRESSIVE PLANE WAVE (2.8) A progressive plane wave is a unidirectional plane wave—no reverse-propagating component. p ( x, t ) = Ae (

dx dt

x

∆ x = c ∆t

where

ω k +k +k =   c 2 x

)

2 y

2

2 z

2

2 z

THIN ROD PROPAGATION

A thin rod is defined as λ ? a . a

a = rod radius

c=

dx dt

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

ϒ = Young's modulus, or modulus of elasticity, a characteristic property of the material [Pa] ρ0 = equilibrium (ambient) density [kg/m3]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 9 of 36

v ξ PARTICLE DISPLACEMENT [m]

STANDING WAVES Two waves with identical frequency and phase characteristics traveling in opposite directions will cause constructive and destructive interference: ) ) p ( x, t ) = p1e ( +pe( 1424 3 12424 3 j ωt − kx

The displacement of a fluid element from its equilibrium position.

v v ∂ξ u= ∂t

j ωt + kx

moving right

v u

moving left

for p1=p2=10, k=1, t=0:

p ωρ0c

ξ=

= particle velocity [m/s]

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

v v Π = ∫ I ·d s S

S = surface surrounding the sound source, or at least the 2 surface area through which all of the sound passes [m ]

for p1=5, p2=10, k=1, t=0:

I = acoustic intensity [W/m2]

I ACOUSTIC INTENSITY [W/m2]

z SPECIFIC ACOUSTIC IMPEDANCE [rayls or (Pa· s)/m] (5.10) Specific acoustic impedance or characteristic impedance z is a property of the medium and of the type of wave being propagated. It is useful in calculations involving transmission from one medium to another. In the case of a plane wave, z is real and is independent of frequency. For spherical waves the opposite is true. In general, z is complex.

z=

p = ρ0 c (applies to progressive plane waves) u

Acoustic impedance is analogous to electrical impedance: pressure volts = impedance = velocity amps

z = r+ j x In a sense this is resistive, i.e. a loss since the wave departs from the source.

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

The time-averaged rate of energy transmission through a unit area normal to the direction of propagation; power per unit area. Note that I = 〈pu〉T is a nonlinear equation (It’s the product of two functions of space and time.) so you can't simply use jωt or take the real parts and multiply, see TimeAverage p33.

I = I (t )

T

= pu

T

1 T pu dt T ∫0

=

For a single frequency:

I=

1 Re{ p u *} 2

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

1 ∂E 1 ∂E ∂x ∂E I= = =c A ∂t A ∂x ∂t ∂V

2

,

p Pe2 I= = 2ρ0 c ρ0 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] ρ0 = equilibrium (ambient) density [kg/m3] c = dx dt is the phase speed (speed of sound) [m/s]

ρ0c Characteristic Impedance, Selected Materials (bulk) [rayls] Air @ 20°C 415 Aluminum 17×106 Brass 40×106 Concrete 8×106

Copper 44.5×106 Glass (pyrex) 12.9×106 Ice 2.95×106 Steam @ 100°C 242

Tom Penick

Steel 47×106 Water, fresh 20°C 1.48×106 Water, sea 13°C 1.54×106 Wood, oak 2.9×106

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 10 of 36

IMPEDANCE [rayls or (Pa· s)/m]

FOURIER'S LAW FOR HEAT CONDUCTION, HEAT FLUX

Spherical wave impedance is frequency dependent:

Sound waves produce proportional changes in pressure, density, and temperature. Since the periodic change in temperature is spread over the length of a wavelength, the change in temperature per unit distance is very small.

q = −K

∂T ∂x

z=

p = P - P0 acoustic pressure [Pa] u = particle velocity (due to oscillation, not flow) [m/s] ρ0 = equilibrium (ambient) density [kg/m3] is the phase speed (speed of sound) [m/s] c = dx dt k = wave number or propagation constant [rad./m]

q = heat flux [°C/m] T = temperature [°C] K = a constant x = distance [m]

r = radial distance from the center of the sphere [m]

u VELOCITY, SPHERICAL WAVE [m/s]

SPHERICAL WAVES

u=

SPHERICAL WAVES (5.11) General solution for a symmetric spherical wave:

p (r, t ) =

p ρ0 c = u 1 − j / ( kr )

p = P - P0 acoustic pressure [Pa] z = wave impedance [rayls or (Pa· s)/m] ρ0 = equilibrium (ambient) density [kg/m3] c = dx dt is the phase speed (speed of sound) [m/s]

A j( ωt − kr ) B j( ωt + kr ) e + e r424 r424 1 3 1 3 divirging from the source

p p  j  = 1 −  z ρ0 c  kr 

converging on the source

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

A j( ωt − kr ) Progressive spherical wave: p ( r , t ) = e r

Π ACOUSTIC POWER, SPHERICAL WAVES [W]

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 negativetraveling wave [Pa] ω = frequency [rad/s] t = time [s] k = wave number or propagation constant [rad./m]

r

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

v v Π = ∫ I ·d s S 2

Π = 4{ πr I

general definition

for spherical dispersion

area of a spherical surface

Π = 2{ πr 2 I

SPHERICAL WAVE BEHAVIOR Spherical wave behavior changes markedly for very small or very large radii. Since this is also a function of the wavelength, we base this on the kr product where kr ∝ r/λ.

for hemispherical dispersion

hemispherical surface

S = surface surrounding the sound source, or at least the 2 surface area through which all of the sound passes [m ] I = acoustic intensity [W/m2] r = radial distance from the center of the sphere [m]

For kr ? 1 , i.e. r ? λ (far from the source):

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

z ; ρ0 c

and with p and u in

phase.

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

z ; jωρ0 r

and p and u are 90° out of phase. The

source is not radiating power; particles are just sloshing back and forth near the source. Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 11 of 36

FREQUENCY BANDS

CONTIGUOUS BANDS

fu fl FREQUENCY BANDS The human ear perceives different frequencies at different levels. Frequencies around 3000 Hz appear loudest with a rolloff for higher and lower frequencies. Therefore in the analysis of sound levels, it is necessary to divide the frequency spectrum into segments or bands.

f u = 21/ N f l

n

300

n

fc fu

n

log

1k

f

10k

fl

n+1

n+1

fc fu

n+1

1 f cn +1 = 2N n fc

N = the bandwidth in terms of the (inverse) fractional portion of an octave, e.g. N=2 describes a ½-octave band

fc CENTER FREQUENCY [Hz]

Octave bands are the most common contiguous bands:

f cn +1 =2 f cn

fc 2

fl =

fu = f c 2

w=

fc 2

e.g. for fc = 1000 Hz, fl = 707 Hz, fu = 1414 Hz

The center frequency is the geometric mean of a frequency band.

300

fl

a 1 -octave band

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

fl

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

STANDARD CENTER FREQUENCIES [Hz]

fc fu

log

1k

f

16, 31.5, 63, 125, 250, 500, 1000, 2000, 4000, 8000

10k

fc =

Octave bands: 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

fu fl

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

MUSICAL INTERVALS [Hz] 1/12

w BANDWIDTH [Hz]

Each half-step is 2 times higher than the previous note. Harmonious frequency ratios:

The width of a frequency band.

fl

fu w

300

log

1k

10k

(

1

w = fu − fl = 2 2 N − 2

− 21N

)f

f

2:1 3:2 4:3 5:4

octave perfect fifth perfect fourth major third

212/12 = 2.000 27/12 = 1.489 25/16 = 1.335 24/12 = 1.260

2/1 = 2.000 3/2 = 1.500 4/3 = 1.333 5/4 = 1.200

c

IL INTENSITY LEVEL [dB] Acoustic intensity in decibels. Note that IL = SPL when IL is referenced to 10-12 and SPL is referenced to 20×10-6.

Intensity Level:

 I  IL = 10 log    I ref 

I = acoustic intensity [W/m2] Iref = the reference intensity 1×10-12 in air [W/m2]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 12 of 36

If SPECTRAL FREQUENCY DENSITY

PSL PRESSURE SPECTRUM LEVEL

[W/m ]

[dB]

2

The distribution of acoustic intensity over the frequency spectrum; the intensity at frequency f over a bandwidth of ∆f. The bandwidth ∆f is normally taken to be 1 Hz and may be suppressed.

∆I If ( f ) = ∆f

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

Iref = the reference intensity 10

-12

PSL ( f ) = ISL ( f ) = SPL in a 1 Hz band SPL = sound pressure level [dB]

dBA WEIGHTED SOUND LEVELS

ISL INTENSITY SPECTRUM LEVEL [dB]

ISL ( f ) = 10 log

Same as intensity spectrum level.

I f ( f )·1Hz I ref [Pa]

(13.2)

Since the ear doesn't perceive sound pressure levels uniformly across the frequency spectrum, several correction schemes have been devised to produce a more realistic scale. The most common is the Aweighted scale with units of dBA. From a reference point of 1000 Hz, this scale rolls off strongly for lower frequencies, has a modest gain in the 2-4 kHz region and rolls off slightly at very high frequencies. Other scales are dBB and dBC. Most standards, regulations and inexpensive sound level meters employ the Aweighted scale.

ILBAND FREQUENCY BAND INTENSITY LEVEL [dB]

ACOUSTICAL SOURCES

The sound intensity within a frequency band.

MONOPOLE (7.1)

ILBAND = ISL + 10 log w

The monopole source is a basic theoretical acoustic source consisting of a small (small ka) pulsating sphere.

ISL = intensity spectrum level [dB] w = bandwidth [Hz]

SPL SOUND PRESSURE LEVEL [dB] Acoustic pressure in decibels. Note that IL = SPL when IL is referenced to 10-12 and SPL is referenced to 20×10-6. An increase of 6 dB is equivalent to doubling the amplitude. A spherical source against a planar surface has a 3 dB advantage over a source in free space, 6 dB if it's in a corner, 9 dB in a 3-wall corner.

Sound Pressure Level:

 P  SPL = 20 log  e   Pref 

A j( ωt − kr ) e r 2 where A = jka ρ0 cu0 p (r, t ) =

a u0e jωt

p = P - P0 acoustic pressure [Pa] r = radial distance from the center of the source [m] ω = frequency [rad/s] k = wave number or propagation constant [rad./m] ρ0 = equilibrium (ambient) density [kg/m3]

c=

dx dt

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

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

for Multiple Sources:

 N  Pei  2   N SPLi /10  SPL = 10 log  ∑    = 10 log  ∑10   i =1  Pref    i =1  for Multiple Identical Sources:

SPL = SPL0 + 10 log N Pe = effective or rms acoustic pressure [Pa] Pref = the reference pressure 20×10-6 in air, 1×10-6 in water [Pa] N = the number of sources

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 13 of 36

DIPOLE

LINE SOURCE (7.3)

The dipole source is a basic theoretical acoustic source consisting of two adjacent monopoles 180° out of phase. Mathematically this approximates a single source in translational vibration, which is what we really want to model. −

+

−

=

r2

r

r1

θ

θ

x

p ( r, θ) =

z

and for

A j ωt − kr ) p ( r , θ ) = j2 e ( sin ( 12 kd sin θ ) 3 r 1424 3 144244 directivity function

p = P - P0 acoustic pressure [Pa] r = radial distance from the center of the source [m] ω = frequency [rad/s] k = wave number or propagation constant [rad./m] ρ0 = equilibrium (ambient) density [kg/m3] is the phase speed (speed of sound) [m/s]

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

[email protected]

r ? L , R ≈ r − x sin θ , L

Far from the source, the wave looks spherical:

Tom Penick

A = jka 2ρ0cu0 .

p ( r , θ ) = ∫ Pdx ( r , θ, t ) dx (abbreviated form)

A = jka ρ0 cu0 2

spherical wave

dx A j( ωt − kR ) e L R

one tiny segment of the line source,

A j( ωt − kr1 ) A j( ωt − kr2 ) e − e r1 r2

where

dx dt

θ

where Pdx is the pressure at a remote point due to

d 2

d

c=

r

Pdx ( R, θ, t ) =

Let d 2 sin

+

dx

as r→∞

r r1 θ

p (r, θ )

R

L

r2

−

x

+

p (r, θ)

θ= 0

A line source of length L is calculated as follows.

dx A j( ωt − kr + kx sin θ) e L r 1 A j( ωt − kr ) L / 2 jkx sin θ p ( r , θ, t ) = e ∫− L / 2 e dx L r A j ωt − kr ) p ( r , θ, t ) = e ( D ( θ ) where r

Pdx ( r , θ, t ) ≈

D ( θ) =

sin ( 12 kL sin θ ) 1 2

kL sin θ

Directivity Function

see also Half Power Beamwidth p16.

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 14 of 36

DIRECTIVITY FUNCTION

CIRCULAR SOURCE (7.4, 7.5a)

The directivity function is responsible for the lobes of the dispersion pattern. The function is normalized to have a maximum value of 1 at θ = 0. Different directivity functions are used for different elements; the following is the directivity function for the line source.

D ( θ) =

A speaker in an enclosure may be modeled as a circular source of radius a in a rigid infinite baffle vibrating with velocity µ0ejωt. For the far field pressure (r > ½ ka2).

p ( r , θ, t ) = j

sin ( 12 kL sin θ ) 1 2

kL sin θ where

D ( θ) =

ka 2 j ωt − kr ) ρ 0 cµ 0 D ( θ ) e ( 2r

2 J1 ( ka sin θ ) ka sin θ

Directivity function

The following is the directivity function for a focused source.

D (r) =

J1 ( 2Gr / a ) Gr / a

 3.83  θnull 1 ≈ sin −1    ka 

First null

J1(x) = first order Bessel J function r = radial distance from the source [m] θ = angle with the normal from the circular source [radians] t = time [s] ρ0c = impedance of the medium [rayls] (415 for air) k = wave number or propagation constant [rad./m]

J1(x) = first order Bessel J function see also Half Power Beamwidth p16.

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 15 of 36

2θ θHP HALF-POWER BEAMWIDTH

FOCUSED SOURCE

The angular width of the main lobe to the points where power drops off by 1/2; this is the point at which the

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

directivity function equals 1/ 2 .

FOCUSED SOURCE

z= 0

z= d

r a

µ0 e jωt

d

Focal Plane pressure

from the Directivity function: D ( θHP ) =

2 J ( ka sin θ ) 1 = 1 ka sin θ 2

 1.61634  ka sin θHP = 1.61634 ⇒ 2θHP = 2sin −1    ka  3.2327 185.22 2θHP ≈ radians or degrees, for ka ? 1 ka ka J1(x) = first order Bessel J function

p ( r ) = Gρ0c D ( r )

ka 2 2d J1 ( 2Gr / a ) and D ( r ) = (Directivity function) Gr / a where

For a circular source:

Focal Plane

G=

J1(x) = first order Bessel J function r = radial distance from the central axis [m] G = constant [radians] a = radius of the source [m] d = focal length [m] ρ0c = impedance of the medium [rayls] (415 for air) k = wave number or propagation constant [rad./m]

For a line source:

sin ( 12 kL sin θ ) from the Directivity function: D ( θ ) = 1 = HP 2

1 2

1 2

z0 RAYLEIGH NUMBER [rad.·m]

kL sin θ

 1.391558  kL sin θHP = 1.391558 ⇒ 2θHP = 2sin −1  1   2 kL 

The Rayleigh number or Rayleigh length is the distance along the central axis from a circular piston element to the beginning of the far field. Beyond this point, complicated pressure patterns of the near field can be ignored.

πa 2 1 2 z0 = = 2 ka λ a = radius of the source [m] d = focal length [m] ρ0c = impedance of the medium [rayls] (415 for air) k = wave number or propagation constant [rad./m]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 16 of 36

MOVING COIL SPEAKER (14.3b, 14.5) magnet

Model for the moving coil loudspeaker.

+

ZE R0 V

I

(1.7) The mechanical impedance is analogous to electrical impedance but does not have the same units. Where electrical impedance is voltage divided by current, mechanical impedance is force divided by speed, sometimes called mechanical ohms.

u

s

V = φu Faraday's law F = φI Lorentz force

V

−

Zm MECHANICAL IMPEDANCE [(N·s)/m]

Rm

x

ZM

L0 RM

LM

CM

ZA

φu

u = velocity of the voice coil [m/s] I = electrical current [A] R0 = electrical resistance of the voice coil [Ω] 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 [Ω] 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 [Ω] ZA = effective electrical impedance due to mechanical air loading [Ω] ZM = effective electrical impedance due to the mechanical effects of spring stiffness, mass, and (mechanical) resistance [Ω] F = force on the voice coil [V] φ = Bl coupling coefficient [N/A] B = magnetic field [Tesla (an SI unit)] l = length of wire in the voice coil [m]

Tom Penick

[email protected]

Zm =

F u

F

F − sx − Rmx { {& = Return force of the spring

Force due to mechanical resistance

s

m

mx {&&

Rm

Mass × acceleration

F = mx&& + Rm x& + sx

let

F ( t ) = F% ( ω) e jωt

and

X ( t ) = X% ( ω) e jωt

then

F% e jωt = ( −ω2 m + jωRm + s ) X% e jωt

and

X% =

F% −ω2 m + jωRm + s

so the velocity finally

Z mo =

U% =

dx dt

=jωX% =

jωF% −ω m + jωRm + s 2

F% F% −ω2 m + jωRm + s = = U% jωF% / ( −ω2 m + jωRm + s ) jω

s Z mo = Rm + { jωm − j { ω { inertia damping due to mass

spring effect

m = mass of the speaker cone and voice coil [kg] x = distance in the direction of motion [m] s = spring stiffness due to flexible cone suspension material [N/m] Rm = mechanical resistance, a small frictional force [(N·s)/m or kg/s] F = force on the speaker mass [N] ω = frequency in radians

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 17 of 36

ZM ELECTRICAL IMPEDANCE DUE TO MECHANICAL FORCES [Ω] Converting mechanical impedance to electrical inverts each element.

Zr RADIATION IMPEDANCE [(N·s)/m] (7.5) This is the mechanical impedance due to air resistance. For a circular piston:

Z r = ρ0 cS  R1 ( 2ka ) + j X 1 ( 2ka ) 

m φ2 φ2 RM = , CM = 2 , LM = φ Rmo s ZM =

 j 8 ρ a 3ω, ka = 1 Zr ≈  3 2 0 ka ? 1  πa ρ0 c,

φ2 = RM P CM P LM Z mo

Z mo = Rm + jωm − j

The functions R1 and X1 are defined as:

s φ → ZM = s ω Rm + jωm − j ω 2

RM = effective electrical resistance due to the mechanical resistance of the system [Ω] 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 [(N·s)/m or kg/s] m = mass of the speaker cone and voice coil [kg] s = spring stiffness due to flexible cone suspension material [N/m] φ = Bl coupling coefficient [N/A] Zmo = mechanical impedance, open-circuit condition [(N·s)/m or kg/s]

R1 ( x ) = 1 −

2 J1 ( x ) x

;

x2 x4 x6 x8 − + − 8 192 9216 737280

 4  x x3 x5 x7 x9 − 5+ 7   − + 2H1 ( x )  π  3 45 1600 10 10 ; X1 ( x ) = x 8 3π   4  πx + πx 3 sin  x − 4  , 

AIR [Ω] φ2 2Z r

(7.5)

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

mr =

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

ZA =

x > 4.32

ρ0c = impedance of the medium [rayls] (415 for air) S = surface area of the piston [m2] J1 = first order Bessel J function R1 = a function describing the real part of Zr X1 = a function describing the imaginary part of Zr x = just a placeholder here for 2ka k = wave number or propagation constant [rad./m] a = radius of the source [m] H1 = first order Struve function ω = frequency in radians

mr RADIATION MASS [kg] ZA ELECTRICAL IMPEDANCE DUE TO

  , x ≤ 4.32 

Xr ω

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

ω0 =

s → m

s m + mr

The functions R1 and X1 are defined as: Xr = radiation reactance, the imaginary part of the radiation impedance [(N·s)/m] ω = frequency in radians s = spring stiffness due to flexible cone suspension material [N/m] m = mass of the speaker cone and voice coil [kg]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 18 of 36

paxial AXIAL PRESSURE [Pa] paxial

ka 2 ωρ a 2 =j ρ0 c u = j 0 u , 2r 2r Z MA V u= Z MA + Z E φ paxial

ELECTROSTATIC TRANSDUCER

(7.4a)

r > ka 1 2

2

Z MA jωρ0 a 2 = V0 e jωt Z MA + Z E 2φr

ZMA = ZM || ZA = effective electrical impedance due to the mechanical components and the effect of air [Ω] ZE = electrical impedance due to the voice coil [Ω]

(14.3a, 14.9) A moving diaphragm of area A is separated from a stationary plate by a dielectric material (air). A bias voltage is applied between the diaphragm and plate. Modern devices use a PVDF film for the diaphragm which has a permanent charge, so no bias voltage is required. Bias, in this case and in general, is an attempt to linearize the output by shifting its operating range to a less non-linear operating region. The DC bias voltage is much greater in magnitude than the time-variant signal voltage but is easily filtered out in signal processing. oscillation area

A

diaphragm

BASS REFLEX ENCLOSURE (14.6c) V

The bass response of a speaker/cabinet system can be improved bat the expense of an increase in low frequency rolloff by adding a port to the enclosure.

dielectric

x x0

ε

0

φu

I

Speaker

Zms φ2

+

V -

+

Port

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

1 Lc Cc

ωc =

φ2 Lc = sc

,

ZE R0 V I

,

Cc =

RM

LM

Mechanical voltage:

mv φ2 Coupling coefficient:

RC CM

φu

F 1 Z = I + ms2 φu , φ jωCo φ

φ=

LC CC

PIEZOELECTRIC TRANSDUCER (14.12b) Uses a crystal (usually quartz) or a ceramic; voltage is proportional to strain. High efficiency (30% is high for acoustics.) Highly resonant. Used for microphones and speakers.

[email protected]

1 ( I + φu ) , jωC0

For this circuit model, there is no inverting of mechanical impedances as in the loudspeaker circuit.

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] φ = Bl coupling coefficient [N/A]

Tom Penick

Acoustic voltage: V =

ZC

ZM

L0

F φ

Co

Equilibrium capacitance:

C0V0 , x0 C0 =

εA x0

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] φu = electrical current due to mechanical force [A] φ = 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]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 19 of 36

REFLECTION AND TRANSMISSION AT NORMAL INCIDENCE (6.2) Incident:

( pi = Pe i

j ωt − k1 x )

Reflected:

pr = Pr e (

Transmitted:

( pt = Pe t

j ωt + k1 x )

j ωt − k2 x )

k1 = ω /c1 , r1 = ρ01 c1

k2 = ω /c2 , r2 = ρ02 c2

USING REFLECTION TO DETERMINE MATERIAL PROPERTIES (6.1) The impedance of a material (and thereby its density) can be determined by bouncing a plane wave off of the material at normal incidence and measuring the relative sound pressure levels. However, there are two possible results since we don't know the phase of the reflected wave, i.e. Pr can be positive or negative. r1 = ( ρ 0 c) 1

pi

Pi

pt pr

r2 = ( ρ 0 c) 2 test material

Pr

x

x=0

SPL difference = 20 log

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

pi + pr = pt → Pi + Pr = Pt 2) Continuity of the normal component of velocity.

ui + ur = ut

R, T, RI, TI REFLECTION AND TRANSMISSION COEFFICIENTS (6.2)

± Pr r −r Pi and =R= 2 1 ± Pr Pi r2 + r1

Pi = peak acoustic pressure, incident [Pa] Pr = peak acoustic pressure, reflected [Pa] r1 = characteristic acoustic impedances of the known material (ρ0c)1 [rayls or (Pa·s)/m] r2 = characteristic acoustic impedance of the unknown material (ρ0c)2 [rayls or (Pa·s)/m] ρ0 = equilibrium (ambient) density [kg/m3] c = the phase speed (speed of sound, 343 m/s in air) [m/s]

The ratio of reflected and transmitted magnitudes to incident magnitudes. The stiffness of the medium has the most effect on reflection and transmission. i) r2 >> r1 Medium 2 is very hard compared to medium 1 and we have total reflection. R ≈ 1, T ≈ 2 Note that T=2 means that the amplitude doubles, but there is practically no energy transmitted due to high impedance. ii) r2 = r1 The mediums are similar and we have total transmission. R = 0, T = 1 iii) r2 << r1 Medium 2 is very soft and we have total reflection with the waveform inverted. R ≈ −1, T

P r −r R= r = 2 1 Pi r2 + r1 T=

Pt 2r2 = Pi r2 + r1

 r −r  RI =  2 1   r2 + r1  TI =

≈0

2

4r2 r1

( r2 + r1 )

2

Pi, Pr, Pt = peak acoustic pressure (or magnitude) of incident, reflected, and transmitted waves [Pa] r1, r2 = characteristic acoustic impedances of the materials (ρ0c)1, (ρ0c)2 [rayls or (Pa·s)/m] ρ0 = equilibrium (ambient) density [kg/m3] c = the phase speed (speed of sound, 343 m/s in air) [m/s] RI = reflection intensity coefficient [no units] TI = transmission intensity coefficient [no units]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 20 of 36

TRANSMISSION THROUGH PARTITIONS

a ABSORPTION COEFFICIENT The absorption coefficient can be measured in an impedance tube by placing a sample at the end of the tube, directing an acoustic wave onto it and measuring the standing wave ratio.

a=

Wa SWR − 1 = 1− Wi SWR + 1

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

0.161V as = a0 + Ss

a Absorption Coefficient, Selected Materials [no units] 250 Hz 0.50 0.03 0.06 0.01 0.05 0.25 0.06 0.25 0.10 0.11 0.03 0.01 0.35 0.25

1 kHz 0.75 0.04 0.35 0.02 0.07 0.75 0.03 0.12 0.04 0.07 0.03 0.02 0.65 0.17

THIN PARTITION [dB] (13.15a) For a planar, nonporous, homogeneous, flexible wall, the transmission loss is dependent on the density of the partition and the frequency of the noise.

  2ρ c  ωρ  Pi + Pr = 1 + j s  Pt TI =  0  ρ0 c    ωρ s  TL = 20 log

2

Ii ρc = 20 log ( f ρ s ) − 20 log 0 π It

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

1 1  −   Ts T0 

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]

acoustic tile suspended ceiling brick carpet concrete concrete block, painted fiberglass, 1" on rigid backing glass, heavy plate glass, windowpane gypsum, ½ " on studs floor, wooden floor, linoleum on concrete floor, terrazzo upholstered seats wood paneling, 3/8-1/2"

TL TRANSMISSION LOSS THROUGH A

4 kHz 0.60 0.07 0.65 0.02 0.08 0.65 0.02 0.04 0.09 0.07 0.02 0.02 0.60 0.10

Loss through a thin partition in air (ρ0c = 415):

TL0 = 20 log

Ii = 20 log ( f ρ s ) − 42 It

Transmission loss as a function of power:

TL0 = 10log

Wi Wt

ρs = surface density of the partition material [kg/m2] ρ0c = impedance of the medium [rayls or (Pa·s)/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]

ρs SURFACE DENSITY [kg/m2] The surface density affects the transmission loss through a material and is related to the material density.

ρ s = ρ0 h ρ0 = density of the material [kg/m3] h = thickness of the material [m]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 21 of 36

TL TRANSMISSION LOSS IN

COINCIDENCE EFFECT (13.15a)

COMPOSITE WALLS AT NORMAL INCIDENCE [dB]

When a plane wave strikes a thin partition at an angle, there are alternating high and low pressure zones along the partition that cause it to flex sinusoidally. This flexural wave propagates along the surface of the wall. At some frequency, there is a kind of resonance and the wall becomes transparent to the wave. This causes a marked decrease in the transmission loss over what is expected from the mass law; it can be 10-15 dB.

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

TL0 = 10 log

1 , where TI = 1 ∑ Ti Si , TI S i

  for a wall in air: Ti =  132   f ρS 

2

Coincidence occurs when λtr = λp.

Si = area of the ith element [m2] Ti = TI for of the ith element (transmission intensity coefficient [no units] ρs = surface density [kg/m2]

The wave equation for a thin plate:

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

θ

1  ωρS  1+  cos θ   2ρ0 c 

+

∂ 4ξ 12 ∂ 2ξ + =0 2 ∂ y 4 h 2cbar ∂ t2

ξ ( y, t )

-

θ

+

h Particle displacement: Dispersion:

pi

TI ( θ ) =

λ tr

-

ξ ( y, t ) = e

(

jω t − y / C p

)

π hcbar f [m/s] 3 λ Trace wavelength: λ tr = [m] sin θ Cp [m] Flexural wavelength: λ p = f c2 Coincidence frequency: f c = [Hz] 1.8hcbar

pt θ θ

y

+

TRANSMISSION AT OBLIQUE INCIDENCE [dB]

pr

-

2

Cp ( f ) =

TL

TI = transmission intensity coefficient [no units] θ = angle of incidence [radians] ρ0c = impedance of the medium [rayls or (Pa·s)/m] (415 for

(dB)

Mass law 6 dB/octave slope

air)

10-15 dB

ρs = surface density [kg/m2]

fc

DIFFUSE FIELD MASS LAW [dB] In a diffuse field, sound is incident by definition at all angles with equal probability. Averaging yields an increase in sound transmission of 5 dB over waves of normal incidence.

TLdiffuse = TL0 − 5 Loss through a thin partition in air (ρ0c = 415):

TLdiffuse = 20 log ( f ρ S ) − 47

Tom Penick

[email protected]

log f

Design considerations: If f < fc, use the diffuse field mass law to find the transmission loss. If f > fc, redesign to avoid. Note that fc is proportional to the inverse of the thickness.

ξ = transverse particle displacement [m] h = panel thickness [m] cbar = bar speed for the panel material [m/s] t = time [s] θ = angle of incidence [radians]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 22 of 36

DOUBLE WALLS

DOUBLE WALL result

Masses in series look like series electrical connections. We want to determine the motion of the second wall due to sound incident on the first. Assume that d << λ and let:

mi = ρS i

th

mass per unit area of i wall

γP0 ρ0c = stiffness per unit area of air d d f1 = pi + pr force per unit area on wall 1

f1

m1

m2 x1

Double walls are most effective.

ρs2

Z w ; − jω3

From Newton's Law F=ma:

Mass 2:

s ( x1 − x2 ) = m2 && x2

Let

f1 = F1e

jωt

,

xi = X i ( ω) e

ω ( ρ s1 ρ s 2 ) ; 6dB/octave 2ρ0c

ρs1

d

f1 − s ( x1 − x2 ) = m1 && x1

Z w ; j ω ( ρ s1 + ρ s 2 )

At high frequencies f > f0

x2

Mass 1:

Both walls move together (in phase) like one wall of twice the mass. So the mass law is recovered.

TL ; 20log

s=

s

At low frequencies f < f0

ρ s1 ρ s 2 d ρ0 c 2

TL ; 20log

ω3 (ρ s1 ρ s 2 ) ; 18dB/octave 2ρ02c 3

At very high frequencies f << f0 The walls decouple. The transmission loss is the sum of the losses of the two walls; there is no interaction.

TL ; TL1 + TL2 ; 12dB/octave

jωt

A x b 644474448 } } 2  s − m1ω − s   X 1   F1    =  s − m2 ω2   X 2   0   −s

Apply Cramer's rule,

Xi =

∆i ∆

∆ = det A and ∆ i = det A, with b in the i th column

where

The wall impedance is

ZW =

F1 ∆ ∆ mm   = = j ( m1 + m2 ) ω − 1 2 ω3  jω ∆ 2 jωs s  

ρ S 1ρ S 2 d 3   → ZW =j ( ρ S 1 + ρS 2 ) ω − ω 2 ρ c   0 Resonance occurs at ZW=0:

f0 =

1 ρ0 c 2  1 1  +   2π d  ρ S 1 ρ S 2 

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 23 of 36

MUFFLERS

EXPANSION CHAMBER BOUNDARY CONDITIONS

EXPANSION CHAMBER When sound traveling through a pipe encounters a section with a different cross-sectional area, it sees a new impedance and some sound is reflected. The dimensions can be chosen to optimize the transmission loss through the exit at particular frequencies. We assume d < λ. expansion chamber

gas flow

pi pr

p+ p-

d

pt

pipe 0

Let

pi = Pe i

l

j( ωt − kx )

pr = Pr e

j( ωt + kx )

j ωt − kx ) p+ = P+ e ( j ωt + kx ) p− = P− e (

( pt = Pe t

j ωt − kx )

R=

Pr Pi

T=

Pt Pi

x

p ui = i ρ0 c p ur = − r ρ0 c p u+ = + ρ0 c p u− = − − ρ0 c p ut = t ρ0 c P P α= + β= − Pi Pi

Boundary condition 1: at x = 0, pi + pr = p+ + p− → Pi + Pr = P+ + P− →

Pi Pr P+ P− + = + → 1 + R = α + β (i) Pi Pi Pi Pi Boundary condition 2: at x = 0, Conservation of mass by equating volume velocities. The volume velocity is the crosssectional area times the net velocity. See p8.

Volume velocity is equal across the boundary. Area

Sc Area Sp

S p ( ui + ur ) = S c ( u+ + u− ) → S p ( Pi − Pr ) = S c ( P+ − P− ) →

Pi Pr Sc  P+ P−  (ii) − =  −  → 1 − R = m (α − β) Pi Pi S p  Pi Pi 

(i) + (ii):

(1 + m) α + (1 − m) β = 2

[Eqn. 1]

Boundary condition 3: at x = l, − jkl p+ + p− = pt → P+ e − jkl + P− e jkl = Pe → t

αe− jkl + β e jkl = Te− jkl

[Eqn. 2]

Boundary condition 4: at x = l, Sc ( u+ + u− ) = S p ut →

( P+ + P− ) =

αe− jkl − β e jkl = m1 Te− jkl

Sp Sc

Pt [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:

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 forwardtraveling wave [Pa] P- = peak acoustic pressure, expansion chamber reversetraveling wave [Pa] Pt = peak acoustic pressure, transmitted [Pa] ρ0c = impedance of the medium [rayls or (Pa·s)/m] (415 for air)

R = reflection coefficient [no units] T = transmission coefficient [no units] α = expansion exit transmission coefficient [no units] β = expansion exit reflection coefficient [no units] Sp = cross-sectional area of the pipe [m2] Sc = cross-sectional area of the expansion chamber [m2] Next, apply the boundary conditions:

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 24 of 36

EXPANSION CHAMBER, FINAL STEPS

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

Solve the three equations:

(1 + m )  − jk l  e  e− jk l

(1 − m ) + jk l

e −e + jk l

Cramer's Rule:

TL = 10 log

HELMHOLTZ RESONATOR (10.8)

0  α   2  −e− jk l   β  =  0  − m1 e− jk l  T   0 

l

S M

e jk l ∆T T= = ∆ cos kl + j 12 ( m + m1 ) sin kl 1 1 = 10 log 2 TI T

V = volume

Stiffness due to a gas volume: Mass of the gas in neck:

Transmission loss in an expansion chamber:

π

2π

m = ρ0 l ′ A

ρ 0 c 2 A2 V

[N/m]

[kg] neck

l ′ ≈ l + 0.8 A

Design point

0

s=

Some gas spills out of the neck, so the mass plug is actually slightly longer than the neck. In practice, the effective length is:

2 TL = 10 log 1 + 14 ( m − m1 ) sin 2 kl   

TL

A= area

Resonance:

kl

ω0 =

S , m

f0 =

1 S c A = 2π m 2π l ′V

FLOW EFFECTS Muffler performance is affected by flow rate, but the preceding 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 λ becomes larger.

λ=

343 T + 273 f 293

λ = wavelength [m] f = frequency [Hz] T = temperature [°C]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 25 of 36

E(t) ROOM ENERGY DENSITY [J/m3]

SIDEBRANCH RESONATOR

(12.2) Sound growth: The following expression describes the effect of sound energy filling a room as a source is turned on at t=0.

Refer to the Helmholtz resonator above.

V = Volume A = Area

Helmholtz resonator pipe

gas flow

pi pr

E (t ) =

pt

The effect here is similar to the effect of blowing across the top of a coke bottle. The air across the bottle creates noise at many frequencies but the coke bottle responds only to its resonant frequency.

  π f V 2  0     sc   TL ≈ 10 log 1 +  f0     f   f − f       0

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

dE Ac V + E = W0 { d t 4 { power { the rate at which

the rate at which energy increases in the volume

pi + pr

z → ρ0c

Ø ω→ ∞,

z → ρ0c

ρ0 c

s/m , z → 0

pt

τ

(s − ω m)ρ c 0

s − ω m + jωρ0c 2

TL = transmission loss [dB] f = frequency [Hz] f0 = resonant frequency of Helmholtz resonator [Hz] V = resonator volume [m3] s = stiffness [m3]

E

τ=

4V Ac

AVERAGE ENERGY DENSITY [J/m3] (12.2)

E = (5.8)

The amount of sound energy (potential and kinetic) per unit volume. In a perfectly diffuse field, E does not depend on location.

E=

, where

W0 = power of the sound source [W] A = sound absorption, in units of metric sabin or English 2 2 sabin [m or ft ] t = time [s] τ = time constant [s] E0 = initial energy density [J/m3] c = the speed of sound (343 m/s in air) [m/s]

ROOM ACOUSTICS

E ENERGY DENSITY [J/m3]

dE 4W0 +E = dt Ac

E ( t ) = E0 e− t / τ

2

z=

of the input source

Sound decay: The following expression describes the effect of sound dissipation as a source is turned off at t=0.

ut

s jω

Ø ω→ 0, Ø ω→

jωm

energy is absorbed by the surfaces

This can be rewritten to include the time constant.

For a duct of impedance ρ0c with a Helmholtz resonator having stiffness s and neck mass m, the arrangement can be modeled as follows.

uH

4W0 1 − e−t / τ ) ( Ac

1 E dV V∫

E = energy density [J/m3] V = room volume [m3 or ft3]

2 prms P2 = ρ0 c 2ρ0 c

prms = acoustic pressure, rms [Pa] P = peak acoustic pressure or pressure magnitude [Pa] ρ0c = impedance of the medium [rayls or (Pa·s)/m] (415 for air)

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 26 of 36

A ABSORPTION [m2 or ft2]

The absorption or absorption area may have units of metric sabins or English sabins, named for Wallace Sabine (1868-1919). The absorption area can be thought of as the equivalent area to be cut out of a wall in order to produce the same effect as an object of absorption A. See also ABSORPTION COEFFICIENT p21.

A = a S = ∑ Ai where a =

Wabs POWER ABSORBED [W]

(12.1)

1 ∑ ai Si , Ai = ai Si S

Wabs = aWincident

a

AVERAGE ABSORPTION [m2 or ft2]

1 ∑ ai Si S

Wincident = power incident on the surface [W] a = average absorption coefficient [no units] S = total surface area [m2 or ft2] Ai = sound absorption of a particular material in the room [m2 or ft2] ai = absorption coefficient of a particular material [no units] Si = area represented by a particular material [no units]

Wincident INCIDENT POWER [W]

a

= average absorption coefficient [no units] S = total surface area [m2 or ft2] Ai = sound absorption of a particular material in the room [m2 or ft2] ai = absorption coefficient of a particular material [no units] Si = area represented by a particular material [no units]

where a =

(12.2)

(12.2)

The total power incident on the walls of a room.

Wincident =

1 Sc E 4

S = total surface area [m2 or ft2] c = the speed of sound (343 m/s in air) [m/s]

E

3

= average energy density [J/m ]

(12.3) The average sound absorption over an area.

a=

1 Wabs A ai Si , a = = ∑ S Wincident S

A = total absorption area [m2 or ft2] S = total surface area [m2 or ft2] ai = absorption coefficient of a particular material [no units] Si = area represented by a particular material [no units] Wabs = power absorbed by the surfaces [W] Wincident = power incident on the surfaces [W]

MEASURING ABSORPTION [m2 or ft2] The absorption of a sample can be measured by placing the sample in a reverberation chamber and measuring the effect it has on reverberation time. The absorption value for a person @ 1kHz is about 0.95 m2, for a piece of furniture about 0.08 m2. See also ABSORPTION COEFFICIENT p21.

1 1 As = 0.161V  −   Ts T0  As = sound absorption of the sample [m2 or ft2] V = volume of the room [m3] Ts = reverberation time with the sample in place [s] T0 = reverberation time in the empty room [s]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 27 of 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 is linear when viewed on a log scale. SPL [dB] 60 dB

t =0

−60 =

(metric),

Including air absorption (for

0.161V T= A + 4mV

ARCHITECTURAL [no units] (12.3) I = I 0e − mx = I 0e − mct

(English)

a ≤ 0.2 ):

For most architectural applications, the air absorption coefficient can be approximated as: 1.7

(metric)

0.161V 4mV − S ln (1 − a )

(metric)

τ = time constant [s] V = room volume [m3 or ft3] A = sound absorption, in units of metric sabin or English 2 2 sabin [m or ft ] a = average absorption coefficient [no units] m = air absorption coefficient [no units] S = total surface area [m2 or ft2]

I = acoustic intensity [W/m2] I0 = initial acoustic intensity [W/m2] h = relative humidity (limited to the range 20 to 70%) [%] f = frequency (limited to the range 1.5 to 10 kHz) [Hz] c = the speed of sound (343 m/s in air) [m/s] α = air absorption coefficient due to combined factors [no

units]

LM MEAN FREE PATH [m] The average distance between reflections in a rectangular room. This works out to 2L/3 for a cubic room and 2d/3 for a sphere.

LM =

(p,q,r) MODES, rectangular cavity

(9.1)

The modes of a volume are the frequencies at which resonances occur, and are a function of the room dimensions. For example, the lowest mode will be the frequency for which the longest dimension equals ½ -wavelength and is represented by (1,0,0). 2

m = 2α

m = 5.5 × 10 −4 ( 50 / h )( f /1000 )

More accurate, Eyring-Norris reverberation formula:

T=

4π 2 Vf c3

m AIR ABSORPTION COEFFICIENT,

Sabin formula:

0.049V T= A

N( f )≈

f = frequency [Hz] V = room volume [m3] c = the speed of sound (343 m/s in air) [m/s]

T 4V 10 log e ⇒ T = 13.816τ , τ = τ ac

0.161V T= A

(9.2)

The number of modes (resonant frequencies) per unit hertz. The modal density increases with frequency until it becomes a diffuse field. In a diffuse field, the modal structure is obscured and the sound field seems isotropic, i.e. the SPL is equal everywhere. Rectangular room:

t

T60

N(f) MODAL DENSITY [Hz-1]

2

c  p  q   r  f ( p, q, r ) =   +  +  2  L  W   H 

2

p, q, and r form the mode numbers. They are integers representing the number of half-wavelengths in the length, width, and height respectively. To avoid having more than one mode at the same frequency, the ratio of any two room dimensions should not be a whole number. Some recommended room dimension ratios are 1.6:1.25:1.0 for small rooms and 2.4:1.5:1.0 or 3.2:1.3:1.0 for large rooms.

4V S

V = room volume [m3 or ft3] S = total surface area [m2 or ft2]

n NUMBER OF REFLECTIONS [no units] The number of acoustic reflections in a room in time t.

n=

ct ctS = LM 4V

c = the speed of sound (343 m/s in air) [m/s] t = time [s] LM = mean free path [m] V = room volume [m3 or ft3] S = total surface area [m2 or ft2]

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] Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 28 of 36

R ROOM CONSTANT [m2] Note that

DIRECT FIELD

R ; A when a is very small. A aS R= = 1− a 1− a

The direct field is that part of a room in which the dominant sound comes directly (unreflected) from the source. SPL [dB]

A = sound absorption, in units of metric sabin or English 2 2 sabin [m or ft ] a = average absorption coefficient [no units] S = total surface area [m2 or ft2]

Direct field (slope is 6 dB per doubling of distance)

Reverberant field

SPLrev

rd

SPL SOUND POWER LEVEL [dB] W Source: SPL = Lw = 10 log Wref

Energy density (direct):

 ρ cW  4  Q +  +10 log  0 2 ref  SPL = Lw + 10 log  2 R  4πr  Pref 3 144 42444

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

4  Q +  SPL = Lw + 10 log  2 R  4πr

Edir =

log r

I Q W = c c 4 πr 2

[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] W = sound power level of the source [W] r = distance from the source to the observation point [m]

rd REVERBERATION RADIUS [m]

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 20×10-6 in air, 1×10-6 in water [Pa]

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

rd =

QR 16π

Q = quality factor (Q=1 when source is remote from all surfaces) [no units] R = room constant [m2]

Q QUALITY FACTOR [no units] A factor that is dependent on the location of a source relative to reflective surfaces. The source strength or amplitude 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. r = distance from the source to the observation point [m] R = room constant [m2]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 29 of 36

REVERBERANT FIELD

COCKTAIL PARTY EFFECT

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

Consider a room of a given volume V and reverberation time T, and assume a fixed distance d among speakers and listeners in M small conversational groups. In each group, only one person is speaking at a time. There is a theoretical maximum number of groups that can exist before the onset of instability and loss of intelligibility. That is, as more conversations are added, one must speak louder in order to be heard. But with everyone speaking louder, the background noise increases, hence the instability.

Direct field

SPL [dB]

(slope is 6 dB per doubling of distance)

Reverberant field

SPLrev B

rd

log r d

Sound power level:

A

SPL rev = Lw + 6 − 10 log R [dB]

Energy density (reverberant):

Erev =

4Wrev 4W [J/m3] = Ac Rc

SPLrev = sound power level in the reverberant field [m] r = distance from the source to the observation point [m] rd = distance from the source to the reverberant field boundary [m] Lw = sound power level of the source [dB] R = room constant [m2] Wrev = reverberant sound power level of the room [W] W = sound power level of the source [W]

NR NOISE REDUCTION [dB]

Energy density at B due to speaker A:

Reverberant energy density due to other M-1 conversations:

Erev = ( M − 1)

SNR =

S TL = NR + 10log w R2

, provided

E1 Erev

=

1  R  + 1  2 M − 1  16πd 

Notice that the power W drops out of the equation. Now if we require that this signal to noise ratio be some minimum required in order for the listener to be able to understand the speaker, the expression can be written:

NR = SPL1 − SPL2

M < 1+

For measuring transmission loss:

4W Rc

Signal to noise ratio:

(13.13)

The noise reduction from one room to an adjoining room is the difference between the sound power levels in the two rooms. The value is used in the measurement of transmission loss for various partition materials and construction.

4 W 1 +   2 c  4 πd R

E1 =

1  R  + 1  2 SNR min  16πd 

Assume R ; A so that T = 0.161V / A ; 0.161V / R , then we can rewrite the expression in terms of the room volume V and the reverberation constant T.

R2 ; A2

Sw = surface area of the wall [m2] R2 = room constant of the receiving room [m2] A2 = sound absorption or absorption area of the receiving 2 room [m ]

M < 1+

1  V  + 1  2 SNR min  312d T 

[mks units]

If we further assume that to the listener, the speaker must be as loud as the background noise, then the maximum number of speakers (conversations) in the room is

M max ; 2 +

V 312d 2T

W = power output of a speaker [W] M = the number of speakers (or groups) R = room constant [m2] d = distance between speakers in the same group [m] E = energy density [J/m3]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 30 of 36

THERMOACOUSTIC CYCLE

THERMOACOUSTIC ENGINE

Consider a small volume of air in acoustic oscillation at t0 and ambient pressure p0. As it moves toward the source, pressure and temperature increase while volume decreases. The volume of air slows and reverses direction at t1 and transfers heat to the metal plate. As the volume of air moves away from the source, pressure and temperature decrease. At the volume reaches t3, it slows and again reverses direction. The cooler volume absorbs heat from the metal place. This action takes place all along the length of the metal plate, forming a bucket brigade of heat transfer.

A transducer in one end of a half-wavelength chamber creates a high power standing wave. Thin metal plates are positioned ¼ of the way from one end so that velocity, displacement and pressure amplitudes will all be high.

Thermoacoustic Engine λ 2

Transducer

Hot

Displacement

ξ

Cold

Velocity

Pressure

time

p t0

t1

t2

u

t

Displacement

t3

ξ

t0 Pressure

Source

p

Metal plate

t1 Source

ξ0

0

λ 8

p = A cos kx sin ωt , k =

Q

Metal plate

t2 Source

u=−

A sin kx cos ωt ρ0 c

ξ=−

A sin kx cos ωt ωρ0c

Metal plate

t3 Source

λ 4

3λ 8

λ 2

length

π πc , ω= λ/2 λ/2

ξ0 Q

Metal plate

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 31 of 36

THERMOACOUSTIC GRADIENT Also called critical gradient. The oscillatory motion and oscillatory temperature of gas particles along the metal plates establishes a temperature gradient along the plates. The parallel stacking of plates increases the power of the engine but does not affect the gradient.

T δT

x

2ξ 0

Maximum temperature gradient:

dT dx

= ( γ − 1) kT0 critical

γ = ratio of specific heats (1.4 for a diatomic gas) [no units] T0 = ambient temperature [K] k = wave number or propagation constant [rad./m]

Γ GRADIENT RATIO The ratio of the operating temperature gradient to the critical gradient.

Γ=

dT / d x dT / d x critical

Γ > 1: Thermoacoustic heat engine Heat flow generates sound (does work)

QH + Vout QC Γ < 1: Thermoacoustic refrigerator Acoustic energy pumps heat from cold end to hot end of stack

QH + Vin QC

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 32 of 36

GENERAL MATHEMATICAL PHASOR NOTATION

x + j y COMPLEX NUMBERS Im y

When the excitation is sinusoidal and under steadystate conditions, we can express a partial derivative in phasor notation, by replacing ∂ with jω . For

A

∂t

example, the Telegrapher's equation ∂V = − L ∂I ∂z ∂t ∂ V becomes = − LjωI . Note that V ( z , t ) and ∂z I ( z , t ) are functions of position and time (space-time

θ x

0

Re

jθ

x + jy = Ae = A cos θ + jA sin θ Re{ x + jy} = x = A cos θ

functions) and V ( z ) and I ( z ) are functions of position

Im{ x + jy} = y = A sin θ

only.

Magnitude { x + jy} = A = x 2 + y 2 Phase { x + jy} = θ = tan −1 j=e

Sine and cosine functions are converted to exponentials in the phasor domain.

y x

Example:

v v E ( r , t ) = 2 cos ( ωt + 3 z ) xˆ + 4sin ( ωt + 3 z ) yˆ

π j 2

The magnitude of a complex number may be written as the absolute value.

Magnitude {x + jy} = x + jy The square of the magnitude of a complex number is the product of the complex number and its complex conjugate. The complex conjugate is the expression formed by reversing the signs of the imaginary terms.

x + jy = ( x + jy )( x + jy ) * = ( x + jy )( x − jy ) 2

= Re{2e j3 z e jωt xˆ + ( − j) 4e j3 z e jωt yˆ }

v v E ( r ) = 2e j3 z xˆ − j4e j3 z yˆ

TIME-AVERAGE When two functions are multiplied, they cannot be converted to the phasor domain and multiplied. Instead, we convert each function to the phasor domain and multiply one by the complex conjugate of the other and divide the result by two. The complex conjugate is the expression formed by reversing the signs of the imaginary terms. For example, the function for power is:

P (t ) = v ( t ) i (t )

watts

Time-averaged power is:

P (t ) =

1 T

∫ v ( t ) i ( t ) dt watts T

0

For a single frequency:

P (t ) =

1 Re{V I * } watts 2

T = period [s] V = voltage in the phasor domain [s] I* = complex conjugate of the phasor domain current [A]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 33 of 36

RMS

SERIES

rms stands for root mean square. root

mean

square

〈 p 2〉

f ( t )rms =

1+ x ; 1+ f (t )

2

1 x , x =1 2

1 x 3x 2 5 x 3 35 x 4 ; 1− + − + − L , − 12 < x < 12 2 8 16 128 1+ x

1 ; 1 + x 2 + x 4 + x 6 + L , − 12 < x < 12 2 1− x 1 ; 1 + 2 x + 3x 2 + 4 x 3 + L , − 12 < x < 12 2 (1 − x ) 1 ; 1 − x + x 2 − x 3 + L , − 12 < x < 12 1+ x 1 ; 1 + x + x 2 + x 3 + L , − 12 < x < 12 1− x

The plot below shows a sine wave and its rms value, along with the intermediate steps of squaring the sine function and taking the mean value of the square. Notice that for this type of function, the mean value of the square is ½ the peak value of the square.

BINOMIAL THEOREM Also called binomial expansion. When m is a positive integer, this is a finite series of m+1 terms. When m is not a positive integer, the series converges for -1<x<1.

(1 + x )

EULER'S EQUATION e jφ = cos φ + j sin φ TRIGONOMETRIC IDENTITIES e+ jθ + e− jθ = 2 cos θ

J1:

m

HYPERBOLIC FUNCTIONS j sin θ = sinh ( jθ ) j cos θ = cosh ( jθ )

CALCULUS ∫ sin u du = 12 u − 14 sin 2u + C

j tan θ = tanh ( jθ )

2

2

u du = 12 u + 14 sin 2u + C

m ( m − 1) 2 m ( m − 1)( m − 2 )L ( m − n + 1) n x +L + x +L 2! n!

BESSEL FUNCTION EXPANSION z 2z3 3z 5 + 2 − 2 2 +L , z = 1 2 2·4 2·4 ·6

e+ jθ − e− jθ = j 2sin θ e ± jθ = cos θ ± j sin θ

∫ cos

= 1 + mx +

LINEARIZING AN EQUATION Small nonlinear terms are removed. Nonlinear terms include: • variables raised to a power • variables multiplied by other variables ∆ values are considered variables, e.g. ∆t.

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 34 of 36

∇ NABLA, DEL OR GRAD OPERATOR

DOT PRODUCT The dot product is a scalar value.

A • B = (xˆ Ax + yˆ Ay + zˆ Az ) • (xˆ B x + yˆ B y + zˆ B z ) = Ax B x + Ay B y + Az B z

A • B = A B cos ψ AB

B ψ

xˆ • yˆ = 0 , xˆ • xˆ = 1 B • yˆ = (xˆ Bx + yˆ By + zˆ Bz ) • yˆ = By Projection of B along â:

A A•B

B

B

(B • aˆ )aˆ

ψ

â

ψ

â

The dot product of 90° vectors is zero. The dot product is commutative and distributive:

A•B = B•A

A • (B + C) = A • B + A • C

Compare the ∇ operation to taking the time derivative. Where ∂/∂t means to take the derivative with respect to time and introduces a s-1 component to the units of the result, the ∇ operation means to take the derivative with respect to distance (in 3 dimensions) and introduces a m-1 component to the units of the result. ∇ terms may be called space derivatives and an equation which contains the ∇ operator may be called a vector differential equation. In other words ∇A is how fast A changes as you move through space. ∂A ∂A ∂A in rectangular ∇A = xˆ + yˆ + zˆ coordinates: ∂x ∂y ∂z in cylindrical ∂A ˆ 1 ∂A ∂A ∇A = rˆ +φ + zˆ coordinates:

CROSS PRODUCT

A × B = (xˆ Ax + yˆ Ay + zˆ Az ) × (xˆ Bx + yˆ B y + zˆ B z )

= xˆ (Ay Bz − Az B y ) + yˆ ( Az Bx − Ax B z ) + zˆ (Ax B y − Ay Bx )

A × B = nˆ A B sin ψ AB ˆ is the unit vector normal to both A where n and B (thumb of right-hand rule).

B × A = −A × B x×y = z y × x = −z φ× z = r φ× r = −z

x×x = 0

The cross product is distributive:

A × (B + C) = A × B + A × C

A×B n A

∂r r ∂φ ∂z ∂A ˆ 1 ∂A ˆ 1 ∂A ∇A = rˆ +θ +φ ∂r r ∂θ r sin θ ∂φ

in spherical coordinates:

ψ B

∇2 THE LAPLACIAN The divergence of a gradient Laplacian of a scalar in rectangular coordinates: Laplacian of a vector in rectangular coordinates: In spherical and cylindrical coordinates:

∇2 A =

∂2 A ∂2 A ∂2 A + + ∂x 2 ∂y 2 ∂z 2

v ∂ 2 Ay ∂ 2 Ax ∂ 2 Az ˆ ˆ ∇ A = xˆ + y + z ∂x 2 ∂y 2 ∂z 2 2

∇ 2 A ≡ ∇ (∇·A ) − ∇ × ∇ × A = grad (div A ) − curl(curl A )

Also, we have:

A × ( B × C) = ( A ⋅ C) B − ( A ⋅ B ) C

∇⋅ DIVERGENCE The del operator followed by the dot product operator is read as "the divergence of" and is an operation performed on a vector. In rectangular coordinates, ∇⋅ means the sum of the partial derivatives of the magnitudes in the x, y, and z directions with respect to the x, y, and z variables. The result is a scalar, and a factor of m-1 is contributed to the units of the result. For example, in this form of Gauss' law, where D is a density per unit area, ∇⋅D becomes a density per unit volume.

div D = ∇ ⋅ D =

∂ Dx ∂ Dy ∂ Dz + + =ρ ∂x ∂y ∂z

D = electric flux density vector D = εE [C/m2] ρ = source charge density [C/m3]

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 35 of 36

CURL curl B = ∇ × B

GLOSSARY

The circulation around an enclosed area. The curl of vector B is in rectangular coordinates:

curl B = ∇ × B =

 ∂B ∂B   ∂B ∂B   ∂B ∂B  xˆ  z − y  + yˆ  x − z  + zˆ  y − x  ∂z  ∂x   ∂x ∂y   ∂z  ∂y in cylindrical coordinates: curl B = ∇ × B =

 1 ∂Bz ∂Bφ  ˆ  ∂Br ∂Bz  1  ∂ ( rBφ ) ∂Br  ˆ − +φ  − + − rˆ  z    r  ∂r ∂z  ∂r  ∂φ   ∂z  r ∂φ in spherical coordinates:

curl B = ∇ × B = rˆ

1  ∂ ( Bφ sin θ ) ∂Bθ  −  + r sin θ  ∂θ ∂φ 

1  1 ∂Br ∂ ( rBφ )  ˆ 1  ∂ ( rBθ ) ∂Br  θˆ − −  +φ   r  sin θ ∂φ ∂r  r  ∂r ∂θ  The divergence of a curl is always zero:

∇·(∇ × H ) = 0

SPHERE Area = πd = 4πr Volume = 16 πd 3 = 43 πr 3 2

2

adiabatic Occurring without loss or gain or heat. anechoic room Highly absorptive room. a ≈ 1. enthalpy (H) A thermodynamic property. The sum of the internal energy U and the volume-pressure product PV. If a body is heated without changing its volume or pressure, then the change in enthalpy will equal the heat transfer. Units of kJ. Enthalpy also refers to the more commonly used specific enthalpy or enthalpy per unit mass h, which has units of kJ/kg. entropy A measure of the unavailable energy in a closed thermodynamic system, varies in direct proportion to temperature change of the system. The thermal charge. harmonic wave A waveform that is sinusoidal in time. isentropic Having constant entropy, no change in thermal charge. However there could be heat flow in and out, analogous to current flow. isothermal Having constant temperature, no heat flow to/from the surroundings. Analogous to voltage. pink noise Noise composed of all audible frequencies with a 3 dB per octave attenuation with frequency increase. The attenuation is based on a per Hz value; the SPLs for each octave are equal. reverberation room Characterized by long decay time. a0 << 1, large T0.. TDS time delay spectrometry. A sophisticated method for obtaining anechoic results in echoic spaces. white noise Noise composed of all audible frequencies at equal amplitude per Hz. For a more comprehensive glossary, see the file DictionaryOfAcousticTerms.PDF.

GRAPHING TERMINOLOGY With x being the horizontal axis and y the vertical, we have a graph of y versus x or y as a function of x. The x-axis represents the independent variable and the y-axis represents the dependent variable, so that when a graph is used to illustrate data, the data of regular interval (often this is time) is plotted on the x-axis and the corresponding data is dependent on those values and is plotted on the yaxis.

Tom Penick

[email protected]

www.teicontrols.com/notes

EngineeringAcoustics.pdf

12/20/00 Page 36 of 36