High Intensitiy Sound Propagation

7/29/2019 High Intensitiy Sound Propagation

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Transfer function method of measuringacoustic intensityin a duct system with flowJ. Y. Chungand D. A. BlaserEngineeringechanicsepartment,eneralotorsesearchaboratories,, arren,ichigan8090{ReceivedMay1980;cceptedorpublication1August980)A newmethodfmeasuringhe cousticntensityfplane avesna duct ithlowsdescribed.n thismethod,he cousticransferunctionetweenwoocationsn he uctsusedodeterminehe cousticintensity.n he asefno low,heransfer-functionormulationeducesoacross-spectralelationimilarto he elationsedo measurecousticntensityn three imensions.n contrasto he elationn threedimensions,owever,he ross-spectralormulationora ductsnotimitednaccuracyy hemicrophonespacing.he ewmethodas eenerifiedxperimentallyith seriesf aboratoryests.estesultsobtainedy he ransferunctionethodevealhathenet cousticowerransmittedlonghe ipeincreasess heMachnumberncreases.heacousticoweradiatedrom hepipe pening,owever,remainsnchangedithncreasingachumber.his ifferenceetweenheransmittedndhe adiatedpowerppearsobe ueosoundbsorptionausedy orticityheddingt he ipepening.PACSnumbers:3.85.Dj, 7.60.+ i, 43.20.Mv, 3.28.Py

INTRODUCTIONThe use of the cross-spectral method of measuring

acoustic intensity '' has been shown to be effective inevaluating sound adiation (or absorption) rom a struc-tural surface. The method is applicable to generalthree-dimensional wave propagation. It is restricted,however, to a condition of no-mean flow. Munro andIngard 3 suggested that the use of the cross-spectraltechnique can be extended to a case of plane-wave pro-pagation in a uniform flow field, provided additionalterms are added to the expression for the intensity.The validity of this expression, however, has not beendemonstrated experimentally. A previous experimentalwork on the acoustic measurement in a duct system withflow was conductedby Alfredson4 who used the standing-wave-ratio method to evaluate the sound power trans-mission in a duct.

This paper describes a different method of measuringacoustic intensity in a duct system with flow. Themethod uses a transfer-function measurement to de-termine the acoustic intensity in a duct and is applica-ble to the condition of plane-wave propagation in a uni-form flow. In conjunction with the transfer functionmethod, ,Morfey's definition and formulation forsound powder n a flow field was adopted. Verificationof this new technique was made by a series of laboratorytests with flow, which include a comparison of soundpower measurements made both inside and outside of aduet system. The inside measurement was made withthe new method which is valid in the flow field while theoutside measurement was made with the cross-spectralmethod. Strictly, the cross-spectral method cannot beused in a flow field. However, all the outside measure-ments were made far away from the duet opening, hencethe effect of flow is negligible. Excellent agreementbetween the two sound power measurements was ob-tained within the frequency range where the conserva-tion of acoustical energy is known to exist.

A new expression of acoustic intensity for a plane-wave propagation with no flow is also presented. Thisexpressions significantlyifferent rom thatof thegeneral cross-spectral method. The general ease re-

quires a small microphone spacing relative to the wave-length due to the finite difference approximation, whilein the case of plane-wave propagation, no restriction isimposed on the microphone spacing.I. ACOUSTIC NTENSITY N A DUCTWITHOUT FLOWA. Formulation of acoustic intensity

Consider a stationary random acoustic propagation na duct system as depicted n Fig. 1. The acousticpres-sure p at a given location, shownas a dotted ine, canbe expressed n terms of its incidentand reflected com-ponentsPi and p,, respectively,or,P=Pi+Pr. (1)

For plane-wave propagation, the magnitude of theacoustic intensity associated with the incident wave canbe defined as,

(2)

whereSpps the autospectraldensityofp andpc isthe characteristic acoustic impedance. Likewise, themagnitude of the acoustic intensity associated with thereflected wave is,

[Ilr=sp,/pc (3)where ,. s theauto pectral ensity fPt. In orderto establishherelationbetween iiandSr onecould,first of all, establish the relation between p andp by intro-ducing a convolution integral with an impulsive response7/' or

pr(t)-or(r)p,(tr)dr. (4)One has, from Eq. (4),

E{p,(t)p(t r)}= r(rl)E{p,(t)p,(t r- r/)}dr/, (5)or

1570 J.Acoust.oc. m. 8(6),Dec. 980 0001-4966/80/121570-08500.80 1980AcousticalocietyfAmerica 1570loaded 03 Aug 2012 to 217.64.162.238. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms


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SOUND SOURCE

FIG. 1. Incident and reflected pressure waves at a duct sec-tion.

(R,i(r) r(v)(Rii(r-v)dv, (6)whereE denotesheexpectedalueand Rrois thecross-correlationunction f Pi andPr while Rii s theautocorrelation function of Pi. Taking the Fouriertransform of Eq. (6), or;iR,i)ej2' r

= r(n) ( r - n)e'2''d , (7)and defining a complex reflection coefficient R as theFourier transform of the impulsive response r, one hasfrom Eq. (7)

R(f) r()e2'"=S/S . (8)

Silarly, it can be shown from Eq. (4) that

Using the relations in Eqs. (8) and (9), the relation be-tweenS and he autospectralensity of the totalacoustic pressure p can now be derived.

The autocorrelation of p can be expressed in terms ofthe incident and reflected-wave components as,

,,(r)-E{[p,(t) +p(t)][p,(t+ r)+p(t + r)]}=(r)+i(r)+(r)+(r). (10)

The Fourier transform of Eq. (10) yields,,S-S + S+ S+ , (11)

where * denotesa complex conjugate. It is clear fromEqs. (8), (), ana (11) tatS- (1+ R)(1

SOUND FLOWSOURCEI Ii li i P2ii II i

FIG. 2. Two-microphone measurement system.

or

s,,,, + I .Similarly, it can be shown that

S,W,,.=S,,/[+ l/R[ ' (14)Substitutinghe aboveexpressions f SisiandSppntoEqs. (2) and (3), one obtains he magnitudeof the acou-stic intensity components in the direction of the incidentand reflected waves, respectively, as

andIIl=s,/[oc[1 +

(15)

(16)The magnitude of the net acoustic intensity in the direc-tion of the incident wave can then be defined as,

[Io]=[I[i- [IIr, (17)where the subscript 0 indicates the no-flow condition.From Eqs. (15), (16),and (17), we obtain

[iol:(s,,/oc)[(1 Inl-)/11 (18)

B. Theoretical basis of the measurement techniqueTo evaluate he acousticntensity n a ductbasedonEq. (18) requires measurements of the autospectral

density and the complex reflection coefficient of theacoustic pressure. Measurement of the autospectraldensity is a routine matter requiring a single micro-phone. Determination of the complex reflection coef-ficient, however, cannot be achieved easily with a singlepressure measurement.

According to the theory developed in Refs. 5 and 6,the complex pressure reflection coefficient R can be.expressed in terms of the transfer functions betweenacoustic pressures at two fixed locations in the duct.As depicted in Fig. 2, let microphones be placed atlocations I and 2 along the duct wall, and let s be thespacing between these microphones. Then accordingto Refs. 5 and 6, the complex pressure reflection coef-ficient at microphone location I is

Rx= (Hx2 H )/(H r - Hx2) (19)where

Hx2=Sm/Sxx (20)is the acoustic transfer function, Sm is the cross-spec-tral density between the two microphone signals, andS is the autospectral density of the pressure at micro-phone ocation 1, also H =e 'jksandHr=e + (k=wave-number).

It can be shown from Eq. (19) that1 JR, [2 - 4 sin(ks)Im(H2)/[H. H2[ (21)

and

1571 J. Acoust.Soc. Am., Vol. 68, No. 6, December1980 J.Y. Chungand D. A. Blaser' Measuring coustic ntensity 1571

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FIG. 3. Experimental apparatus for verifying the theory ofacoustic intensity in a flow field.

] +R]2=4sin2(ks)/]H,-Hm]. (22)Substituting Eqs. (21) and (22) into Eq. (18), one has

[Io -- -(S pc)[Im(H2)/sin(ks)] (23)or

[Io -- - ( 1/pc)[ImSm)/sin(ks)], (24)where Im indicates the imaginary part. In contrast toEq. (18), Eq. (24) reveals that the net acoustic intensityin a duct for plane-wave propagation without flow can bedetermined from the cross-spectral density betweenacoustic pressures at two locations along the duct wall.Furthermore, it shows hat the intensity is directlyproportional to the imaginary part of the cross-spectraldensity. This is not surprising, however, since it wasshown previously that this proportionaiRy exists even ina general case of three-dimensional wave propagationfor which the acoustic intensity was shown to be

Iio1=- [Im(S2)/pcks]. (25)This general expression was derived using finite dif-ference approximations to evaluate the acoustic velocityfrom the acoustic pressure gradient; hence its applica-tion is restricted to small values of ks. It is interest-ing to note, however, that no such restriction is im-posed on Eq. (24) for plane-wave propagation.II. ACOUSTIC INTENSITY IN A DUCT WITH FLOWA. Formulation of acoustic intensity

The definition of acoustic intensity in a flow field is con-siderably different from that in a sound ield without flow.In this paper Morley's expression for acoustic intensityis applied to a uniform duct flow with plane-wave propa-gation. The wave propagation is assumed to be station-ary random and all results are applicable to the specialcase of a deterministic process.

According to Morley 7 the magnitude of the net acousticintensity in a uniform duct flow in the axial directioncan be expressed as

(26)where p and u are the acoustic pressure and the acous-

tic velocity, respectively, M is the mean flow Machnumber, pc is the characteristic acoustic impedance,and the brackets denote a time-averaged quantity. Fora stationary random acoustic propagation, Eq. (26) canbe rewritten as

l I= (1+ M ) Re[S,]dco(M/pc)x Sdco (MOC) S,,dco, (*7)

where co s the angular frequency and S is e cross-spectral density betweenp and u, also S and S arethe autospectral densities of p and u, respectively. A c-cordg to Eq. (27), an expression of e frequencyspectrum of acoustic intensity can also be given as[ f()]- (1+M)Re[Su]+(M/pc)S+MpcSuu.28)

Introducing the complex reflection coefficient R givenby Eqs. (8) and (9), the expression in Eq. (28) can berewritten as (see the Appendix for derivations)+hi . (29)

Usg Eqs. (13) and (14), Eq. (29) can be rewrittenas

[Ifl , - II,[r, (30)where

andI -

are e maitude of the acoustic intensity componentsin e direction of the cident and reflected waves, re-spectively, with mean flow Mach number M. It is ob-vious that Eqs. (29), (31), and (32) reduce to Eqs. (18),(2), and (3), respectively, when M=0.I. Theoreticalbasis f the measurementechnique

Just as in the case without flow using Eq. (18), to de-termine the acoustic intensity in a duct with flow usingEq. (29) requires measurementsof the quantitiesS andR. The transfer function relation given by Eq. (19) isapplicable to the case where a uniform flow exists pro-vided H i and H r are redefined as

H =exp -j[ks/(1 + M)]}, (33)and

Hr= exp{+j ks/(1 M)]}. (34)Hence, substituting Eqs. (19), (33), and (34) into Eq.(29), one has{I'l- 4pcina[ks/(1M2)]

(35)

Equation (35) shows that, due to flow, the cross-spectral density alone cannot completely describe the

1572 J. Acoust.Soc.Am., Vol. 68, No. 6, December 980 J.Y. Chung nd D. A. Blaser: Measuring cousticntensity 1572



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lCALCULATOR

FLOWSOURCE

lRANDOMNOISESOURCE

DIGITALSIGNAL

LAP,;IMICROPHONEr '-JPOWER /!1SUPPLIES ! i L_J

MEASURINGAMPLIFIERSt 3 I

MICROPHONEPOWER

SUPPLIESt I'--

j12MICROPHONES(TRANSFER-FUNCTIMETHOD)

MICROPHONES(CROSS-SPECTRALMETHOD)

--''' ACOUSTICROPAGATIONFLOW-'-'"-Z

FIG. 4. Instrumentationsystem.

soundpower transmitted in a duct system (as is thecase with no flow). It should be noted, however, thatflow not only affects the incident and the reflected soundPowercomponentsy the factors of (1 +M) 2 and (1-M) 2,respectively, as shown n Eqs. (31) and (32), but alsochangeshe value of the autospectraldensitiesSii andSprPrue o changesn theacousticmpedancehrough-out the duct system.III. EXPERIMENTAL VERIFICATION OF THEORIES

To verify the transfer function method it is desirableto have a flow field with known acoustic intensity to beused as a reference to the test result. Such an acousticfield, however, s diffiuclt o find sinceno existingmethod is available for calibrating the intensity field inflow. An indirect approach was then adopted to cali-brate the intensity field. In this approach, it assumedthat the acoustic energy is conserved within the fre-quency range of interest such that the sound powertransmitted in the duct with flow is the same as thatradiated from the duct opening. The latter quantity wasdetermined experimentally using the general crossspectral method which is applicable in the region out-side the duct where the effect of flow is negligible.During the same test, the new transfer-function methodwas used inside the duct to determine the sound powertransmission in the flow. The inside and the outsidemeasurement results were then compared to determinethe validity of the new method.A. Apparatus nd instrumentation

The experimental apparatus, as shown in Fig. 3, con-sists of a pipe system, a two-microphone station formeasuring the acoustic intensity within the pipe, and atwo-microphone probe for measuring the acoustic in-

130

110

90

80

70

605040

300 I 2 3

FREQUENCY (kHz)a) M-O

130

110

90

807060504030

0 1FREQUENCY (kHz)

b) M-O. 1FIG. 5. Typical soundpressut-c pcctra in the duct with andwithout the signal from the random noise source.1573 J. Acoust.Soc. Am., Vol. 68, No. 6, December 980 J.Y. Chungand D. A. Blaser: Measuring coustic ntensity 1573



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11o

100

0

70

GO

I m G121 s-i-ksl

o 1 2 3FREQUENCY (kHz)

10

.,oni

o

-30

M=O


FIG. 6. Comparison of general and plane-wave theories fordetermining the acoustic intensity without flow.tensity outside the pipe. A 100-W, 381-mm(15-in. )-diam. speaker was used in conjunctionwith a 100-Wacoustic driver to generate a broadband noise withinthe 51-.mm(2-in. )-diam. pipe system. A compressedair system was utilized as the source of flow for thetests. Measurements were made without flow and withmean flow Mach numbers of 0.05 and 0.10 within thepipe.

Four condenser microphones with a diameter of 6.35-mm (k-in.) were used. Two of the microphones weremounted flush to the inside wall of the pipe. They wereseparated by 27 mm and located 76 and 103 mm fromthe end of the pipe. This nearness to the end of the pipeminimized viscous losses along the pipe wall. Theother two microphones spaced 16 mm apart were usedfor the probe system outside the pipe.

FIG. 8. Soundpower loss at the end of pipe without flow.The acoustic intensity in the duct is determined by

measurements of the transfer function and the auto-spectral density at the two-microphone station near theend of the pipe. These measurements were made usingthe instrumentation system shown in Fig. 4. In mea-suring the transfer function, errors due to both theamplitude and the phase mismatch between microphonesystems were eliminated by the sensor-switchingtechnique. ,6

Figure 5 shows typical comparisons of the measuredpressure spectra inside the duct with the noise sourceon and with the noise source off. The noise-source-offspectra represent the noise floor for these measure-ments. For M--0.1, this noise floor is seen to in-crease by nearly 40 dB over that of the case with M-0due to pressure fluctuations generated locally by flow

11o

lOO

o

70

GO

INSIDEOUTSIDE

M=O

0 1 2 3FREQUENCY (kHz)

11o

100

$0

0

70

INSIDEOUTSIDE

0.05


FIG. 7. In-pipe versus out-of-pipe sound power without flow. FIG. 9. In-pipe versus out-of-pipe sound power with M= 0.05.

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.,onIw

z

- -20w

-3own, -o

0.05


FIG. 10. Soundpower loss at the end of pipe with M = 0.05.over the microphone. For frequencies above 100 Hz,the noise-source-on signal is sufficiently above thenoise floor to provide accurate measurements.

The sound power radiated from the end of the pipewas determined from intensity measurements using thegeneral cross-spectral method made over the surfaceof a ficticious rectangular box with dimensions of 2 x 1x 1 m as illustrated in Fig. 3. The box was actuallysubdivided into 10 square surfaces, each with dimen-sions of 1 x 1 m. Using the acoustic intensity probe,each surface was manually surveyed and the space-timeaveraged cross-spectral density was measured usingthe instrumentation shown in Fig. 4. Errors in thecross-spectral density due to phase mismatch betweenmicrophone systems were again eliminated using thesensor switching technique. The total radiated soundpower was then determined by the averaged intensityand the total surface area of the rectangular box.

lO M -0.10

-10

-30


FIG. 12. Soundpower loss at the end of pipe with M = 0.10.

All measurements of autospectral densities, cross-spectral densities, and transfer functions were madewith a digitial signal analyzer. The acoustic intensi-ties and the net acoustic powers were then calculatedby a desktop calculator system.B. Tests without flow

Prior to testing with flow, a few tests were conductedwithout flow to verify the measurement theory in thelimiting case of zero Mach number. First, the netacoustic power transmitted along the pipe was deter-mined from the cross-spectral density utilizing both thegeneral three-dimensional formula of Eq. (25) and thespecial plane-wave formula of Eq. (24). As shown inFig. 6, the results using the two formulae are in closeagreement. At low frequencies, there is no observabledifference in the results. The general three-dimen-sional formula, however, slightly underestimates the

11o

100

90

0

70

60

INSIDE.... OUTSI DE

M -0.10


FIG. 11. In-pipe versus out-of-pipe sound power with M = 0.10.

11o

100

0

BO

70

M-0.... M- 0.10

60 I m I 0 I 2 3

FREQUENCY (kHz)FIG. 13. Comparison of out-of-pipe sound power with M = 0aud 0.10.

1575 J. Acoust. oc.Am.,Vol. 68, No.6, December980 J.Y. Chung ndD. A. Blaser:Measuringcousticntensity 1575



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11o

100

80

GO

EQ. (35).... EQ. (24)M: 0.05


FIG. 14. Sound power spectra for M-- 0.05 determined frommeasurement theories with and without flow effects.

sound power as the frequency increases. This error iscaused by the finite difference approximation used in thegeneral formulation and is related to the product of themicrophone spacing and the acoustic wavenumber. Forthese measurements, the microphone spacing of 27 mmgives avalue of ks=1.48 at3kHz. Hence at3kHz, theerror is

10 og[ks/sin(ks)I= 1.72 dB.It should be noted that lthough this error cn only becalculated for the case of plane-wave propagationalonga knowndirection, it represents the maximum possibleerror due to the finite difference approximation forany acoustic field.

Next, the net acoustic power was determined fromcross-spectral density measurements made inside thepipe using Eq. (24) and from cross-spectral densitymeasurements made outside the pipe using Eq. (25)

11o

lOO

so

70

GO

EQ. (3.5).... EQ. (24)M :0.10

FIG. 15. Sound ower spectra for M = 0.10 determined rommeasurement theories with and without flow effects.

with a microphone spacing of 16 ram. As shown inFig. 7, these two measurements agree quite closely,hence confirming the accuracy of the two different mea-surement procedures. It should be noted that the maxi-mum error in the outside measurement at 3 kHz corre-sponding to the microphone spacing of 16 mm is 0.6 dB,which could account for some of the deviation betweenthe two results at high frequencies. Viscous lossesalong the wall and at the end of the pipe would also con-tribute to the deviation that increases with frequency.This difference in the two measurements is plotted inFig. 8 as the ratio of radiated-to-transmitted acousticpower. The sharp decrease in the power ratio is indi-cative of a high loss of acoustic power between the twomeasurement surfaces. The power loss is directly re-lated to the vorticity shed at the pipe exit. The acousticdisturbance creates the vorticity which, due to fluidviscosity, dissipates into heat and shows up as an energydeficit in the acoustic radiation process. This pheno-menon will be discussed further in the following section.C. Tests with flow

With mean flow Mach numbers of 0.05 and 0.1, theacoustic intensity inside the duct was evaluated basedon Eq. (35). The soundpower radiation from the ductopening under the same flow conditions was also evalu-ated based on Eq. (25).

Comparisons of the net acoustic power transmittedalong the pipe and radiated from the end of the pipe areshown in Figs. 9 and 10 for M=0.05, and in Figs. 11and 12 for M =0.10. Above I kHz, there is good agree-ment between measurements inside and outside of thepipe. This confirms the theory of the transfer-functionmethod.

Close examination of Figs. 9 and 11 reveals that, asthe Mach number increases, the net acoustic powertransmitted along the pipe increases while the net ra-diated acoustic power appears to remain nearly fixed.In fact, the net radiated acoustic ,power at these flowMach numbers is essentially the same in both amplitudeand spectral distribution as the measured data withoutflow (Fig. 13). Thus, although ccording o Morley'sdefinition 7 the uniform mean flow increases the net flowof acoustic energy propagating along the pipe, this in-creased flow of energy is not radiated acoustically fromthe end of the pipe, but instead it is absorbed due to in-teractions with the flow field. Morfey also indicatedthat when vorticity is present, acoustic energy need notbe conserved. More recently, Bechert et al., s'9 havestudied specific examples where vorticity is present inlet flows through nozzles and orifices. They found thatif the flow at a nozzle exit is slightly unsteady, i.e., ifsound s present, fluctuating vorticity is shed due tothe existence of a Kutta conditionat the edge of a rigidsurface. At low frequencies, this process extracts en-ergy from the sound ield. This conclusion s in agree-ment with the current experimental finding of the dif-ference in transmitted versus radiated acoustic powershown in Figs. 9-12.

Bechert also presented an approximate model for the1576 J. Acoust.Soc. Am., Vol. 68, No. 6, December 980 J.Y. Chung nd D. A. Blaser: Measuring coustic ntensity 1576



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absorption of sound due to vorticity shedding at a noz-zle exit. 9 The following expression was given based onhis analysis'Ws/Wr=[(1 + {M")/4M](ka)" , (36)

where W/Wr is the ratio of radiated-to-transmittedacoustic power, M is the Mach number, and ka is thedimensionless wavenumber based on the radius of thepipe a. It must be noted that Eq. (36) is only applica-ble to very low values of the dimensionless wavenumber

The ratio of radiated-to-transmitted acoustic powerfrom Figs. 9 and 11 were plotted in Figs. 10 and 12,respectively, and compared to the absorption model ofEq. (36). Although the magnitude of the sound absorp-tion does not fit the model well, the increase in absorp-tion ith decreasing frequency is essentially correct.Also, the increase in absorption with increased Machnumber is well predicted by the model. All of thesefindings tend to support the verification of the transfer-function method of measuring acoustic intensity in aduct system with flow.

As a matter of interest, the experimental data withinthe pipe system with flow were reprocessed using theno-flow cross-spectral formula. Shown in Figs. 14 and15 are comparisons of the net acoustic power deter-mined from the formula without flow, i.e., Eq. (24),and the formula with flow, i.e., Eq. (35), for the twoflow conditions. At most frequencies the no-flow theoryunderestimates the actual acoustic power; however, ina low-frequency region where the pressure-reflectioncoefficient exceeds unity, the no-flow theory overesti-mates the actual acoustic power. It is seen that evenfor mean-flow Mach numbers below 0.1, considerableerror in the net acoustic power flow would be inducedby neglecting the effect of flow.APPENDIX

In a duct system with uniform mean flow the acousticpressure p and the acoustic velocity u can be expressedin terms of the incident and reflected acoustic pressuresPi and Pt, respectively, as

P=P +Pt, (A1)u= (p - pr)/pc . (A2)

Introducing correlation functions of p and u as' (R"(r)=lim p(t)u(tr)dt, (A3)

(R(r) im7 p(t)p(tr)dt, (A4)--. o(R"(r)=lim u(t)u(t+)dt, (AS)

and using the relations in Eqs. (A1) and (A2), one has6t,(r)-lim[fo_oo p(t)p(tr)dt- p(t)p(t+ r)dt+ p(t)p,(t r)dt- p(t)p(t r)d , (A6)

or

(AT)Taking Fourier transform of Eq. (A?) one obtains aspectral relation of

$,.= S,,S,,,+ ***S,,,]/pcor

S.= S - S r - 2 m(S,)]/pc.A similar derivation ields

(A8)

(A9)

S..= S,,S,,,- 2Re(S,)]/(pc) (A10)Substituting these expressions into Eq. (28) we have

1 +M 2pc

M

or1 +M 2

pcM

M+ pS,,,,[1IRI - 2Re(R)], (A12)whereR and Ri are definedn Eqs. (8) and 9), re-spectively. Using Eq. (13) in Eq. (A12) one obtains

+n]*. (A13)

Ij. y. Chung, Cross-SpectralMethod f MeasuringAcousticIntensity Without Error Caused by Instrument Phase Mis-match," J. Acoust. Soc. Am. 64, 1613-1616 (1978).2j. y. Chung, . Pope, andD. A. Feldmaier,"ApplicationfAcoustic Intensity Measurement to Engine Noise Evaluation,"SAE Paper No. 790502 (1979).3D. H. MunroandK. U. Ir/gard,"On AcousticntensityMea-surements in the Presence of Mean Flow," J. Acoust. Soc.Am. 65, 1402-1406 (1979).4R. J. Alfredson, The Design ndOptimization f ExhaustSilencers," Ph.D. thesis, U. Southampton, England, July1970.5D. A. Blaser andJ. Y. Chung, A Transfer-Function ech-nique for Measuring the Acoustic Characteristics of DuctSystems with Flow," Proc. INTER NOISE '78 (1978).6j. y. Chung ndD. A. Blaser, "Transfer functionmethod fmeasuring in-duct acoustic properties, I. Theory and II.Experiment," J. Acoust. Soc. Am. 68, 907-921 (1980).?C.L. Morfey, "Acoustic nergy n Nonuniformlows,"J.Sound Vib. 14, 159-170 (1971).SD.W. Bechert,U. Michel, andE. Pfizenmaier, Experimentson the Transmission of SoundThrough Jets," paper presentedat AIAA 4th Aeroacoustics Conference, Atlanta, Georgia,October 1977, AIAA paper 77-1278.SD.W. Bechert, Sound bsorptionausedy VorticityShed-ding, Demonstrated with a Jet Flow," paper presented atAIAA 5th Aeroacoustics Conference, Seattle, Washington,March 1979, AIAA paper 79-0575.

1577 J. Acoust.Soc. Am., Vol. 68, No. 6, December1980 J.Y. Chungand D. A. Blaser: Measuring coustic ntensity 1577

High Intensitiy Sound Propagation

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