penic tom engineering acoustics 2000 doc

Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 1 of 36 ENGINEERING ACOUSTICS EE 363N INDEX (p,q,r) modes 28 2θ HP 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 C p dispersion 22 Cramer's rule 23 critical gradient 32 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 f c center frequency 12 f l 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 f u upper frequency 12 gas constant 7 general math 33 glossary 36 grad operator 35 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 I f 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 I ref 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 L M 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 mechanical radiation impedance 18 modal density 28 modes 28 modulus of elasticity 9 momentum conservation.6, 8 monopole 13 moving coil speaker 17 m r 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 p axial axial pressure 19 P e 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 P ref reference pressure 13 pressure 6, 9 progressive plane wave 9 progressive spherical wave11 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 Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 2 of 36 rayls 3 r d 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 T 60 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 W abs power absorbed 27 watt 3 wave progressive 11 spherical 11 wave equation 6 wave number 2 wave vector 9 wavelength 2 temperature effects 25 weber 3 weighted sound levels 13 white noise 36 W incident incident power 27 Young's modulus 9 z acoustic impedance 10 z impedance 10, 11 z 0 rayleigh number 16 Z A elec. impedance due to air 18 Z M elec. impedance due to mech. forces 18 Z m mechanical impedance 17 Z r radiation impedance 18 Γ gradient ratio 32 Π acoustic power 10, 11 γ ratio of specific heats 6 λ wavelength 2 λ p flexural wavelength 22 λ tr trace wavelength 22 ρ 0 equilibrium density 6 ρ s surface density 21 τ time constant 26 ξ particle displacement 10, 22 ∇ del 35 ∇×· curl 36 ∇ 2 · Laplacian 35 ∇ · divergence 35 DECIBELS [dB] 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. energy 10log reference energy L = [dB] One decibel is approximately the minimum discernable amplitude difference that can be detected by the human ear over the full range of amplitude. λλ WAVELENGTH [m] Wavelength is the distance that a wave advances during one cycle. At high temperatures, the speed of sound increases so λ changes. T k is temperature in Kelvin. 2c fk π λ== 343 293 k T f λ= k WAVE NUMBER [rad/m] 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. 2 k c πω == λ 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. 0 0 s ρ−ρ = ρ ρ = instantaneous density [kg/m 3 ] ρ 0 = equilibrium (ambient) density [kg/m 3 ] Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 3 of 36 UNITS A (amp) = ·· ·· CWJNmVF sVVsVss ==== C (coulomb) = ·· ·· JNmWs AsVF VVV ==== F (farad) = 22 2 · · CCCJAs VJNmVV ==== H (henry) = · Vs A (note that 2 H·F s= ) J (joule) = 2 2 ······ C NmVCWsAVsFV F ===== N (newton) = 2 ··· JCVWskgm mmms === Pa (pascal) = 2233 · · NkgJWs mmsmm === T (tesla) = 222 ·· WbVsHA mmm == V (volt) = ·· · WJJWsNmC ACAsCCF ===== W (watt) = 2 ···1 · 746 JNmCVFV VAHP ssss ===== Wb (weber) = ·· J HAVs A == Acoustic impedance: [rayls or (Pa·s)/m] Temperature: [°C or K] 0°C = 273.15K c SPEED OF SOUND [m/s] Sound travels faster in stiffer (i.e. higher B, less compressible) materials. Sound travels faster at higher temperatures. Frequency/wavelength relation: 2 cf λω =λ= π In a perfect gas: 0 0 K crT γ ==γ ρ P In liquids: 0 T c γ = ρ B where T =γ BB γ = ratio of specific heats (1.4 for a diatomic gas) [no units] P 0 = ambient (atmospheric) pressure ( 0 p P = ). At sea level, 0 101kPa ≈ P [Pa] ρ 0 = equilibrium (ambient) density [kg/m 3 ] r = specific gas constant [J/(kg·K)] T K = temperature in Kelvin [K] B = 0 0 ρ  ∂ ρ  ∂ρ  P adiabatic bulk modulus [Pa] B T = 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-cross- sectional area. Sound moves more slowly in Bar material. The Bulk value is used below where applicable. Speed of Sound in Selected Materials [m/s] Air @ 20°C 343 Copper 5000 Steel 6100 Aluminum 6300 Glass (pyrex) 5600 Water, fresh 20°C 1481 Brass 4700 Ice 3200 Water, sea 13°C 1500 Concrete 3100 Steam @ 100°C 404.8 Wood, oak 4000 Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 4 of 36 SIMPLE HARMONIC MOTION Restoring force on a spring (Hooke's Law): s fsx=− and Newton's Law: Fma = s M yield: 2 2 dx sxm dt −= and 2 2 0 dxs x dtm += Let 2 0 s m ω= , so that the system is described by the equation 2 2 0 2 0 dx x dt +ω= . 0 s m ω= is the natural angular frequency in rad/s. 0 0 2 f ω = π is the natural frequency in Hz. The general solution takes the form ( ) 1020 cossin xtAtAt =ω+ω Initial conditions: displacement: ( ) 0 0 xx = , so 10 Ax = velocity ( ) 0 0 xu = & , so 0 2 0 u A = ω Solution: ( ) 0 000 0 cossin u xtxtt =ω+ω ω s = spring constant [no units] x = the displacement [m] m = mass [kg] u = velocity of the mass [m/s] t = time [s] SIMPLE HARMONIC MOTION, POLAR FORM The solution above can be written ( ) ( ) 0 cos xtAt =ω+φ , where we have the new constants: amplitude: 2 2 0 0 0 u Ax  =+  ω  initial phase angle: 1 0 00 tan u x −  − φ=  ω  Note that zero phase angle occurs at maximum positive displacement. By differentiation, it can be found that the speed of the mass is ( ) 0 sin uUt =−ω+φ , where 0 UA =ω is the speed amplitude. The acceleration is ( ) 00 cos aUt =−ωω+φ . Using the initial conditions, the equation can be written ( ) 2 2 1 00 00 000 costan uu xtxt x −  =+ω−  ωω  x 0 = the initial position [m] u 0 = the initial speed [m/s] 0 s m ω= is the natural angular frequency in rad/s. 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 ω t 0 π π 2 π 2 0 π 3 2 Phase Angle Displacement x Displacement, Speed, Acceleration Speed u a Acceleration Initial phase angle φφ=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 tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 5 of 36 FOURIER SERIES 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. Let ( ) ( ) ftftT =+= any periodic signal where T = the period. 0 1T ( ) f t t 2T Then ( ) ( ) 0 1 1 cossin 2 nn n ftAAntBnt ∞ = =+ω+ω ∑ where 2 T π ω= = the fundamental frequency A 0 = the DC component and will be zero provided the function is symmetric about the t-axis. This is almost always the case in acoustics. A n = ( ) 0 0 2 cos tT t ftntdt T + ω ∫ A n 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 left- hand plane provided one of them is first flipped about the horizontal axis, e.g. sine function. B n = ( ) 0 0 2 sin tT t ftntdt T + ω ∫ B n 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 left- hand plane, e.g. cosine function. where 0 t = an arbitrary time p ACOUSTIC PRESSURE [Pa] Sound waves produce proportional changes in pressure, density, and temperature. Sound is usually measured as a change in pressure. See Plane Waves p9. 0 p =− PP For a simple harmonic plane wave traveling in the x direction, p is a function of x and t: ( ) ( ) j , tkx pxtPe ω− = P = instantaneous pressure [Pa] P 0 = ambient (atmospheric) pressure ( 0 p P = ). At sea level, 0 101kPa ≈ P [Pa] P = peak acoustic pressure [Pa] x = position along the x-axis [m] t = time [s] P e EFFECTIVE ACOUSTIC PRESSURE [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. 2 e P P = 2 22 e T PPpdt== ∫ 222 123e PPPP =+++ L P = peak acoustic pressure [Pa] p = 0 − PP acoustic pressure [Pa] Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 6 of 36 ρρ 0 EQUILIBRIUM DENSITY [kg/m 3 ] The ambient density. 0 0 22 cc γ ρ== PB for ideal gases 0 2 T c γ ρ= B for liquids The equilibrium density is the inverse of the specific volume. From the ideal gas equation: 0 PRTPRT ν=→=ρ B = ( ) 0 0 ∂ ∂ρ ρ ρ P adiabatic bulk modulus, approximately equal to the isothermal bulk modulus, 2.18×10 9 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] P 0 = ambient (atmospheric) pressure ( 0 p P = ). At sea level, 0 101kPa≈P [Pa] P = pressure [Pa] ν = V/m specific volume [m 3 /kg] V = volume [m 3 ] m = mass [kg] R = gas constant (287 for air) [J/(kg·K)] T = absolute temperature [K] (°C + 273.15) ρ 0 Equilibrium Density of Selected Materials [kg/m 3 ] Air @ 20°C 1.21 Copper 8900 Steel 7700 Aluminum 2700 Glass (pyrex) 2300 Water, fresh 20°C 998 Brass 8500 Ice 920 Water, sea 13°C 1026 Concrete 2600 Steam @ 100°C 0.6 Wood, oak 720 BB 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. 0 2 000 c ρ  ∂ =ρ=ρ=γ  ∂ρ  P BP ρ = instantaneous density [kg/m 3 ] ρ 0 = equilibrium (ambient) density [kg/m 3 ] c = the phase speed (speed of sound, 343 m/s in air) [m/s] P = instantaneous (total) pressure [Pa or N/m 2 ] P 0 = ambient (atmospheric) pressure ( 0 p P = ). At sea level, 0 101kPa ≈ P [Pa] γ = ratio of specific heats (1.4 for a diatomic gas) [no units] BB Bulk Modulus of Selected Materials [Pa] Aluminum 75×10 9 Iron (cast) 86×10 9 Rubber (hard) 5×10 9 Brass 136×10 9 Lead 42×10 9 Rubber (soft) 1×10 9 Copper 160×10 9 Quartz 33×10 9 Water *2.18×10 9 Glass (pyrex) 39×10 9 Steel 170×10 9 Water (sea) *2.28×10 9 *B T , isothermal bulk modulus EQUATION OVERVIEW Equation of State (pressure) ps = B Mass Conservation (density) 3-dimensional 1-dimensional 0 s u t ∂ +∇⋅= ∂ v v 0 su tx ∂∂ += ∂∂ Momentum Conservation (velocity) 3-dimensional 1-dimensional 0 0 u p t ∂ ∇+ρ= ∂ v v 0 0 pu xt ∂∂ +ρ= ∂∂ From the above 3 equations and 3 unknowns (p, s, u) we can derive the Wave Equation 2 2 22 1 p p ct ∂ ∇= ∂ EQUATION OF STATE - GAS 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. Perfect adiabatic gas: 00 γ  ρ =  ρ  P P Linearized: 0 ps =γP P = instantaneous (total) pressure [Pa] P 0 = ambient (atmospheric) pressure ( 0 p P = ). At sea level, 0 101kPa ≈ P [Pa] ρ = instantaneous density [kg/m 3 ] ρ 0 = equilibrium (ambient) density [kg/m 3 ] γ = ratio of specific heats (1.4 for a diatomic gas) [no units] p = P - P 0 acoustic pressure [Pa] s = 0 0 1 ρ−ρ ρ = condensation [no units] Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 7 of 36 EQUATION OF STATE – LIQUID An equation of state relates the physical properties describing the thermodynamic behavior of the fluid. In acoustics, the temperature property can be ignored. Adiabatic liquid: ps = B p = P - P 0 acoustic pressure [Pa] B = ( ) 0 0 ∂ ∂ρ ρ ρ P adiabatic bulk modulus, approximately equal to the isothermal bulk modulus, 2.18×10 9 for water [Pa] s = 0 0 1 ρ−ρ ρ = condensation [no units] r SPECIFIC GAS CONSTANT [J/(kg·K)] The specific gas constant r depends on the universal gas constant R and the molecular weight M of the particular gas. For air ( ) 287J/kg·Kr ≈ . r M = R R = universal gas constant M = molecular weight MASS CONSERVATION – one dimension 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. tube area x A = x +dx ( ) x uAρ is called the mass flux [kg/s] ( ) xdx uA + ρ is what's coming out the other side (a different value due to compression) [kg/s] The difference between the rate of mass entering the 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. ( ) ( ) ( ) xxdx uA uAuAdx x + ∂ρ ρ−ρ=− ∂ dv ρ is the mass in the center volume, so the rate at which the mass is changing can be written as dvAdx tt ∂∂ ρ=ρ ∂∂ Equating the two expressions gives ( ) uA Adxdx tx ∂ρ ∂ ρ=− ∂∂ , which can be simplified ( ) 0u tx ∂∂ ρ+ρ= ∂∂ u = particle velocity (due to oscillation, not flow) [m/s] ρ = instantaneous density [kg/m 3 ] p = P - P 0 acoustic pressure [Pa] A = area of the tube [m 2 ] MASS CONSERVATION – three dimensions ( ) 0u t ∂ ρ+∇⋅ρ= ∂ v v where ˆˆ ˆ xyz xyz ∂∂∂ ∇=ρ+ρ+ρ ∂∂∂ v and let ( ) 0 1 s ρ=ρ+ 0 su t ∂ +∇⋅= ∂ v v (linearized) Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 8 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 A = tube area A P ( ) -( ) x+dx A P x +x dx ( ) x AP is the force due to sound pressure at location x in the tube [N] ( ) xdx A + P is the force due to sound pressure at location x + dx in the tube (taken to be in the positive or right-hand direction) [N] The sum of the forces in the center volume is: ( ) ( ) xxdx FAAAdx x + ∂ =−=− ∂ ∑ P PP Force in the tube can be written in this form, noting that this is not a partial derivative: ( ) du FmaAdx dt ==ρ For some reason, this can be written as follows: ( ) ( ) duuu AdxAdxu dttx ∂∂  ρ=ρ+  ∂∂  with the term u u x ∂ ∂ often discarded in acoustics. P = instantaneous (total) pressure [Pa or N/m 2 ] A = area of the tube [m 2 ] ρ = instantaneous density [kg/m 3 ] p = P - P 0 acoustic pressure [Pa] u = particle velocity (due to oscillation, not flow) [m/s] MOMENTUM CONSERVATION – three dimensions 0 uu Pu ttx ∂∂∂  +ρ+=  ∂∂∂  and 0 u puu t  ∂ ∇+ρ+⋅∇=  ∂  v vv vv Note that uu ⋅∇ v vv is a quadratic term and that u t ∂ ρ ∂ v is quadratic after multiplication 0 0 u p t ∂ ∇+ρ= ∂ v v (linearized) ACOUSTIC ANALOGIES to electrical systems ACOUSTIC ELECTRIC Impedance: A p Z U = V Z I = Voltage: p ∆ VIR = Current: U V I R = p = P - P 0 acoustic pressure [Pa] U = volume velocity (not a vector) [m 3 /s] Z A = acoustic impedance [Pa·s/m 3 ] L INERTANCE [kg/m 4 ] Describes the inertial properties of gas in a channel. Analogous to electrical inductance. 0 x L A ρ∆ = ρ 0 = ambient density [kg/m 3 ] ∆x = incremental distance [m] A = cross-sectional area [m 2 ] C COMPLIANCE [m 6 /kg] The springiness of the system; a higher value means softer. Analogous to electrical capacitance. 0 V C = γρ V = volume [m 3 ] γ = ratio of specific heats (1.4 for a diatomic gas) [no units] ρ 0 = ambient density [kg/m 3 ] U VOLUME VELOCITY [m 3 /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. Vd USuS tdt ∂ξ === ∂ V = volume [m 3 ] S = area [m 2 ] u = velocity [m/s] ξ = particle displacement, the displacement of a fluid element from its equilibrium position [m] Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 9 of 36 PLANE WAVES 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 yz ∂∂ == ∂∂ . One-dimensional wave equation: 22 222 1 pp xct ∂∂ = ∂∂ General Solution for the acoustic pressure of a plane wave: ( ) ( ) ( ) jj propagating inpropagating in the +x directionthe -x direction , tkxtkx pxtAeBe ω−ω+ =+ 1424314243 p = P - P 0 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] PROGRESSIVE PLANE WAVE (2.8) A progressive plane wave is a unidirectional plane wave—no reverse-propagating component. ( ) ( ) j , tkx pxtAe ω− = ARBITRARY DIRECTION PLANE WAVE The expression for an arbitrary direction plane wave contains wave numbers for the x, y, and z components. ( ) ( ) j , xyz tkxkykz pxtAe ω−−− = where 2 222 xyz kkk c ω  ++=   u VELOCITY, PLANE WAVE [m/s] The acoustic pressure divided by the impedance, also from the momentum equation: 0 0 pu xt ∂∂ +ρ=→ ∂∂ 0 pp u zc == ρ p = P - P 0 acoustic pressure [Pa] z = wave impedance [rayls or (Pa·s)/m] ρ 0 = equilibrium (ambient) density [kg/m 3 ] 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] PROPAGATION (2.5) F( tx-c ∆ ) a disturbance xx c ∆ = x t ∆ ,0 xdx ct tdt ∆ =→∆→ ∆ c = dx dt is the phase speed (speed of sound) at which F is translated in the +x direction. [m/s] k v WAVE VECTOR [rad/m or m -1 ] The phase constant k is converted to a vector. For plane waves, the vector k v is in the direction of propagation. ˆˆ ˆ xyz kkxkykz =++ v where 2 222 xyz kkk c ω  ++=   THIN ROD PROPAGATION A thin rod is defined as a λ ? . rod radius a = a 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] ρ 0 = equilibrium (ambient) density [kg/m 3 ] Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 10 of 36 STANDING WAVES Two waves with identical frequency and phase characteristics traveling in opposite directions will cause constructive and destructive interference: ( ) ( ) ( ) 12 moving rightmoving left , jtkxjtkx pxtpepe ω−ω+ =+ 1424314243 for p 1 =p 2 =10, k=1, t=0: for p 1 =5, p 2 =10, k=1, t=0: 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. 0 p zc u ==ρ (applies to progressive plane waves) Acoustic impedance is analogous to electrical impedance: pressurevolts impedance velocityamps == In a sense this is reactive, in that this value represents an impediment to propagation. In a sense this is resistive, i.e. a loss since the wave departs from the source. z = r+ xj ρ 0 c Characteristic Impedance, Selected Materials (bulk) [rayls] Air @ 20°C 415 Copper 44.5×10 6 Steel 47×10 6 Aluminum 17×10 6 Glass (pyrex) 12.9×10 6 Water, fresh 20°C 1.48×10 6 Brass 40×10 6 Ice 2.95×10 6 Water, sea 13°C 1.54×10 6 Concrete 8×10 6 Steam @ 100°C 242 Wood, oak 2.9×10 6 ξ v PARTICLE DISPLACEMENT [m] The displacement of a fluid element from its equilibrium position. u t ∂ξ = ∂ v v 0 p c ξ= ωρ u v = particle velocity [m/s] ΠΠ ACOUSTIC POWER [W] Acoustic power is usually small compared to the power required to produce it. · S IdsΠ= ∫ v v S = surface surrounding the sound source, or at least the surface area through which all of the sound passes [m 2 ] I = acoustic intensity [W/m 2 ] I ACOUSTIC INTENSITY [W/m 2 ] 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 Time- Average p33. ( ) 0 1 T T T IItpupudt T === ∫ For a single frequency: { } 1 * 2 Ipu= Re For a plane harmonic wave traveling in the +z direction: 11 EExE Ic AtAxtV ∂∂∂∂ === ∂∂∂∂ , 2 2 00 2 e p P I cc == ρρ T = period [s] I(t) = instantaneous intensity [W/m 2 ] p = P - P 0 acoustic pressure [Pa] |p| = peak acoustic pressure [Pa] u = particle velocity (due to oscillation, not flow) [m/s] P e = effective or rms acoustic pressure [Pa] ρ 0 = equilibrium (ambient) density [kg/m 3 ] c = dx dt is the phase speed (speed of sound) [m/s] [...]... 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 tom@ tomzap.com 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... 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 tom@ tomzap.com 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;... 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 tom@ tomzap.com 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... = 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 tom@ tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 15 of 36 2θHP HALF-POWER BEAMWIDTH θ FOCUSED SOURCE The angular width of the main lobe to the points where power drops off by 1/2; this is the... 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 tom@ tomzap.com 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... 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 tom@ tomzap.com 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)... 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 tom@ tomzap.com 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... 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 tom@ tomzap.com 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... ω − 1 2 ω3  jω ∆ 2 jωs s   ρ S 1ρ S 2 d 3   → ZW =j ( ρ S 1 + ρS 2 ) ω − ω ρ0 c 2   Resonance occurs at ZW=0: f0 = 1 ρ0 c 2  1 1  +   2π d  ρS 1 ρS 2    Tom Penick tom@ tomzap.com 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... 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 tom@ tomzap.com 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 . Tom Penick tom@ tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 1 of 36 ENGINEERING ACOUSTICS EE 363N INDEX (p,q,r) modes. (1.4 for a diatomic gas) [no units] p = P - P 0 acoustic pressure [Pa] s = 0 0 1 ρ−ρ ρ = condensation [no units] Tom Penick tom@ tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf. density [kg/m 3 ] ρ 0 = equilibrium (ambient) density [kg/m 3 ] Tom Penick tom@ tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 3 of 36 UNITS A (amp) = ·· ·· CWJNmVF sVVsVss ====