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
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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
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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]
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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
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343 6300 4700 3100
Copper Glass (pyrex) Ice Steam @ 100°C
5000 5600 3200 404.8
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Steel Water, fresh 20°C Water, sea 13°C Wood, oak
6100 1481 1500 4000
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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.
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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
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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
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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
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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
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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]
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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]
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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
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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
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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]
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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
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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.
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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.
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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]
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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
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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]
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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]
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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
