ENGINEERING ACOUSTICS EE 363NINDEX (p,q,r) modes pdf

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 acoustic impedance 3, 10 acoustic intensity .10 acoustic power 10 spherical waves 11 acoustic pressure 5, effective adiabatic 7, 36 adiabatic bulk modulus ambient density 2, amp amplitude analogies anechoic room 36 arbitrary direction plane wave architectural absorption coefficient 28 area sphere 36 average absorption 27 average energy density 26 axial pressure 19 B bulk modulus band frequency 12 bandwidth 12 bass reflex 19 Bessel J function 18, 34 binomial expansion 34 binomial theorem .34 bulk modulus .6 C compliance c speed of sound calculus 34 capacitance center frequency 12 characteristic impedance 10 circular source 15 cocktail party effect 30 coincidence effect 22 complex conjugate 33 complex numbers 33 compliance condensation 2, 6, 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 equilibrium 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 electrical analogies electrical impedance 18 electrostatic transducer 19 energy density 26 direct field 29 reverberant field 30 enthalpy 36 entropy 36 equation of state 6, equation overview .6 equilibrium density Euler's equation 34 even function .5 expansion chamber 24, 25 Eyring-Norris 28 far field 16 farad 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 general math 33 glossary .36 grad operator .35 tom@tomzap.com 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 Hooke's Law horsepower 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 inertance 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 k wave number k wave vector .9 kelvin L inertance 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, 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 momentum conservation 6, 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 natural frequency newton Newton's Law noise 36 noise reduction 30 NR noise reduction 30 number of reflections 28 octave bands 12 odd function p acoustic pressure 5, Pa particle displacement 10, 22 partition 21 pascal paxial axial pressure 19 Pe effective acoustic pressure perfect adiabatic gas phase 33 phase angle phase speed phasor notation 33 piezoelectric transducer 19 pink noise 36 plane wave impedance 10 velocity plane waves polar form power 10, 11 SPL 29 power absorbed 27 Pref reference pressure 13 pressure 6, progressive plane wave progressive spherical wave 11 propagation propagation constant Q quality factor 29 quality factor 29 r gas constant R room constant 29 radiation impedance 18 radiation mass 18 radiation reactance 18 rayleigh number 16 EngineeringAcoustics.pdf 12/20/00 Page 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, Sabin formula 28 sabins .27 series 34 sidebranch resonator 26 simple harmonic motion sound 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 spectral frequency density 13 speed amplitude speed of sound 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 standing waves 10 Struve function 18 surface density 21 T60 reverberation time .28 TDS 36 temperature temperature effects 25 tesla .3 thermoacoustic cycle 31 thermoacoustic engine .31 thermoacoustic gradient 32 thin rod 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 vector differential equation35 velocity plane wave .9 spherical wave .11 volt volume sphere 36 volume velocity w bandwidth .12 Wabs power absorbed 27 watt 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 wavelength temperature effects 25 weber weighted sound levels 13 white noise 36 Wincident incident power 27 Young's modulus 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 λ wavelength λp flexural wavelength 22 λtr trace wavelength 22 ρ0 equilibrium density 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 = 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 tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 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 F (farad) = C = C = C = J = A·s V J N ·m V V H (henry) = V ·s (note that H·F = s ) A In a perfect gas: c= In liquids: γB T ρ0 where B=γB T γ = ratio of specific heats (1.4 for a diatomic gas) [no units] P0 = ambient (atmospheric) pressure ( p = P ) At sea level, P ≈ 101 kPa [Pa] W (watt) = J = N ·m = C ·V = V · A = F ·V = 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= λω 2π γP = γrTK ρ0 J (joule) = N ·m = V ·C = W ·s = AV ·s = F ·V = C · F N (newton) = J = C ·V = W ·s = kg ·m m m m s2 Pa (pascal) = N = kg = J = W ·s m m·s 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 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 tom@tomzap.com 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 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 dt m s , so that the system is described by the Let ω0 = m d 2x + ω0 x = equation 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 +    ω0  initial phase angle:  −u  φ = tan −1    ω0 x0  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 ( ) = x0 , so A1 = x0 & x ( ) = u0 , so x ( t ) = x0 cos ω0 t + u A2 = 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] u   u  x ( t ) = x0 +   cos  ω0t − tan −1  ω0 x0   ω0   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 π u π 3π 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 tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 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− P 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] 1T f (t ) = Then where ω = t 2T P = peak acoustic pressure [Pa] x = position along the x-axis [m] t = time [s] ∞ A0 + ∑ ( An cos nωt + Bn sin nωt ) 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 = T ∫ t0 +T t0 f ( t ) cos nωt dt t0 +T f ( t ) sin nωt dt T ∫ t0 P0 = ambient (atmospheric) pressure ( p = P ) At sea level, P ≈ 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 Pe = Pe = P = ∫ p dt T P1 + P2 + P3 + L 2 P = peak acoustic pressure [Pa] p = P − P 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 tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page of 36 ρ0 EQUILIBRIUM DENSITY [kg/m3] EQUATION OVERVIEW Equation of State (pressure) The ambient density ρ0 = B γP = 20 c c γB T ρ0 = c p = Bs for ideal gases Mass Conservation (density) 3-dimensional for liquids ∂s v v + ∇ ⋅u = ∂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 ∂ 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 = P ) At sea ∂p ∂u + ρ0 =0 ∂x ∂t From the above equations and unknowns (p, s, u) we can derive the Wave Equation ∇2 p = level, P ≈ 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 ∂2 p c ∂t 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 ∂t adiabatic bulk modulus, approximately equal 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 B = ρ0   = ρ0 c = γ P ∂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 = P ) At sea level, P ≈ 101 kPa [Pa] γ = ratio of specific heats (1.4 for a diatomic gas) [no units] P ρ =  P  ρ0  γ p = γ Ps P = instantaneous (total) pressure [Pa] P0 = ambient (atmospheric) pressure ( p = P ) At sea level, P ≈ 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 = condensation [no units] ρ0 B Bulk Modulus of Selected Materials [Pa] Aluminum Brass Copper Glass (pyrex) 75×109 136×109 160×10 39×109 Iron (cast) 86×109 Rubber (hard) 5×109 Lead 42×109 Rubber (soft) 1×109 Quartz Steel 33×10 170×109 Water *2.18×109 Water (sea) *2.28×109 *BT, isothermal bulk modulus Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 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 A = tube area adiabatic bulk modulus, approximately equal to the isothermal bulk modulus, 2.18×10 for water [Pa] s = ρ − ρ0 = 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 ) = ∂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 ) = ∂t v where ∇ = ρ x ∂ + ρ y ∂ + ρ z ∂ ˆ ˆ ˆ ∂x ∂y ∂z and let ρ = ρ0 (1 + s ) v v ∂ s + ∇ ⋅ u = (linearized) ∂t Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 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 -( 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 ACOUSTIC is the force due to sound pressure at location Impedance: ZA = Voltage: x + dx ( PA) x ACOUSTIC ANALOGIES to electrical systems ∆p Current: ∑ F = ( PA ) − ( PA ) x x + dx Z= p = P - P0 acoustic pressure [Pa] U = volume velocity (not a vector) [m3/s] ZA = acoustic impedance [Pa·s/m3] 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 U 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  = ∂t ∂x   ∂t v v  ∂u v v v  and ∇p + ρ  + u ⋅∇u  =  ∂t  Tom Penick tom@tomzap.com 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 vv Note that u ⋅∇u is a quadratic term and that ρ ∂u is ∂t quadratic after multiplication v v ∂u ∇p + ρ = (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 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 ∂ = ∂ = ∂ y ∂z 2 One-dimensional wave equation: ∂ p = ∂ p ∂ x c ∂t 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 is the phase speed (speed of sound) [m/s] dt k = wave number or propagation constant [rad./m] r = radial distance from the center of the sphere [m] ) ) p ( x, t ) = ( + ( Ae 424 Be 424 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 ˆ ˆ ˆ k = kx x + k y y + k z z 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 x y ∆x dx → , ∆t → ∆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 x ) y 2 z 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 tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 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 ( + p2 e ( 24 24 4 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 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 T pu dt T ∫0 = For a single frequency: I= Re{ p u *} For a plane harmonic wave traveling in the +z direction: ∂E ∂E ∂x ∂E I= = =c A ∂t A ∂x ∂t ∂V , 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 is the phase speed (speed of sound) [m/s] dt 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 tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 10 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 , where TI = ∑ Ti Si , TI S i   for a wall in air: Ti =  132   f ρS  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  ωρS  1+  cos θ   2ρ0 c  + ∂ 4ξ 12 ∂ 2ξ + 2 =0 ∂ y h cbar ∂ t 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] 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 - 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 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 dB over waves of normal incidence TLdiffuse = TL0 − Loss through a thin partition in air (ρ0c = 415): TLdiffuse = 20 log ( f ρ S ) − 47 Tom Penick tom@tomzap.com 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 f0 Double walls are most effective Z w ; − jω3 x2 d From Newton's Law F=ma: Mass 1: Mass 2: s ( x1 − x2 ) = m2 &&2 x , xi = X i ( ω) e ρ s1 ρ s d ρ0 c TL ; 20log ω3 (ρ s1 ρ s ) ; 18dB/octave 2ρ0 c At very high frequencies f 1: Thermoacoustic heat engine Heat flow generates sound (does work) QH + V out QC Γ < 1: Thermoacoustic refrigerator Acoustic energy pumps heat from cold end to hot end of stack QH + V in QC Tom Penick tom@tomzap.com 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 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 + y 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 ˆ ˆ E ( r , t ) = cos ( ωt + z ) x + 4sin ( ωt + z ) y π j 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 ) ˆ ˆ = Re{2e j3 z e jωt x + ( − j) 4e j3 z e jωt y} v v ˆ ˆ E ( r ) = 2e j3 z x − j4e j3 z y 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 ) = T ∫ v ( t ) i ( t ) dt watts T For a single frequency: P (t ) = Re{V I * } watts T = period [s] V = voltage in the phasor domain [s] I* = complex conjugate of the phasor domain current [A] Tom Penick tom@tomzap.com 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 ) x , x =1 x 3x x 35 x ; 1− + − + −L, − < x < 2 16 128 1+ x ; + x + x4 + x6 + L , − < x < 2 1− x ; + x + 3x + x + L , − < x < 2 (1 − x ) ; − x + x2 − x3 + L , − < x < 2 1+ x ; + x + x2 + x3 + L , − < x < 2 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