Contains research objectives and reports on two research projects.U. S. Navy - Office of Naval Research (Contract N00014-67-A-0204-0019)Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DAAB07-71-C-030

Ingard, K. Uno

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X. PHYSICAL ACOUSTICSAcademic Research StaffProf. K. U. IngardGraduate StudentsA. A. Moretta V. K. SinghalS. R. Rogers J. A. TarvinRESEARCH OBJECTIVESOur general objective involves the study of the emission, propagation, and absorp-tion of acoustic waves in matter. Specific areas of present research include (i) theinteraction of waves with coherent light beams in fluids and solids, (ii) nonlinear acous-tics in fluids, and (iii) the generation and propagation of sound waves in moving fluids,with particular emphasis on waves propagated in ducts and generation of sound byturbulent flow.K. U. IngardA. UPSTREAM AND DOWNSTREAM SOUND RADIATION INTOA MOVING FLUIDONR (Contract N00014-67-A-0204-0019)K. U. Ingard, V. K. SinghalThe problem of sound radiation from a source in relative motion with respect to thesurrounding fluid has become of considerable interest, particularly in connection withnoise generation in aircraft and in various types of fluid machinery. Although the basiceffects, as well as many details of the influence of fluid motion on sound radiation, have1-4been identified by several investigators, these studies have been limited to mathemat-ical analysis of the idealized case of moving point sources. The problem of soundemission from real sources of finite dimensions has received much less attention. Wepresent an explicit demonstration of the influence of relative fluid motion on sound radi-ation from a stationary source, with particular attention to the relation between thesound pressure fields radiated upstream and downstream.1. Experimental ArrangementThe experimental arrangement is shown schematically in Fig. X-1. A sound sourceis mounted in one of the side walls of a rectangular duct with inner dimensionsThis work is supported by the U. S. Navy - Office of Naval Research (ContractN00014-67-A-0204-0019) and by the Joint Services Electronics Programs (U. S. Army,U.S. Navy, U.S. Air Force) under Contract DAAB07 -71-C -0300).QPR No. 108(X. PHYSICAL ACOUSTICS)3/4 X 7/8 in. The duct is connected to a steam ejector through a plenum chamber, andthe flow speed in the tube can be varied from 0 to -100 m/s, which corresponds to Machnumber ~0. 3.The loudspeaker is driven by means of a pulse generator and produces harmonicsound-pressure wave trains in the duct. The carrier frequency of these waves is chosento be considerably lower than the cutoff frequency for the first higher order acousticmode in the duct, so that at the carrier frequency only the plane-wave mode will be ableto propagate.The pressure pulses are detected by two identical pressure transducers mounted inthe side walls of the duct on the upstream and downstream sides of the sound source atequal distances from it. In the absence of flow, M = 0, the recorded pulses from thesetransducers are simultaneous and identical in shape, as demonstrated by the resultsshown in Fig. X-1. Any difference in the sensitivity of the transducers is compensatedfor by gain adjustment of the transducer amplifiers, so that in the absence of flow theamplitudes of the recorded pulses from the transducers are equal.P_M=0.45Fig. X-1.-- FLOW / Recorded pressure pulses in the upstream (p_)TRANSDUCER TRANSDUCER and the downstream (p ) directions at flow MachSOUND SOURCE +numbers I = 0 and M = 0. 3.PP,/ M=0When the air in the duct is moving, however, the pressure amplitudes areno longer equal; the amplitude in the upstream direction is larger than in thedownstream direction, and the difference increases monotonically with the flowspeed in the duct. An example of recorded upstream and downstream pressurewaves at a flow Mach number 0. 3 is shown in Fig. X-1. In addition to theobvious difference in the pressure amplitudes, a difference in the time of arrivalof the two pulses is also apparent in this figure. From this time differenceand the known distance between the sound source and the receivers, the flowMach number in the duct can be determined.The Mach number dependence of the ratio between the upstream and downstreampressure amplitudes obtained in this manner is illustrated in Fig. X-2. Measurementswere carried out at two frequencies, 800 Hz and 2000 Hz. As can be seen, there isno marked difference in the amplitude ratio at these frequencies.QPR{ No. 108(X. PHYSICAL ACOUSTICS)* /S (I+M)3.0P22.52.0/++0+0 0.1+-+++ +'t"+ 1000 Hz* 1400 Hz0.4 0.5Fig. X-2. \Mach number dependence of the measured ratio P_/P+ between the pres-sure amplitudes radiated in the upstream and downstream directions.2. Mathematical AnalysisIn the mathematical analysis of this problem we start from the wave equation forthe sound pressure field p(x, y, z, t)2  2 2 2 22 a 2p  a2p a P 2M a2p a2 p(1-M ) + + 2 2 0, (1)ax2  ay2 az2  c axatc2 a12Dx Dy Dz c atwhere c is the sound speed, and M the Mach number. The coordinates x, y, z refer toa stationary laboratory frame of reference with respect to which the unperturbed fluidis assumed to move with uniform speed Mc in the positive x direction. We wish tosolve this equation, subject to the boundary conditions peculiar to our experimentalarrangement.The duct walls, placed in the planes y = 0, y = a and z = 0, z = b, are assumed tobe rigid everywhere except in the source region, as indicated in Fig. X-3. Conse-quently, the normal components of the fluid velocity and the corresponding pressuregradients are zero at the boundaries except at the.source. If the source, located in thewall in the plane y = 0, produces a velocity perturbation u in the fluid flow in the planey = 0, the effect of the source can be expressed as the boundary conditionu = u f(x, z, t)y 0 (y= 0).QPR No. 1080 ////(I+M )2(X. PHYSICAL ACOUSTICS)SOURCE REGIONuy= uy (x, z, t)Fig. X-3. The source region is in the plane y = 0 of the duct and is definedby the perturbation of the fluid velocity u in the duct.It is important to realize, however, that this velocity perturbation of the fluid in theduct is not necessarily the same as the velocity of the oscillating air column in thethroat of the loudspeaker sourcear s(x, z, t)us t (3)5 atwhere Trs (x, z, t) is the displacement of the air column. Although u = us is valid whenthere is no mean flow in the duct, the situation is more complex when mean flow ispresent. For example, if the flow over the source region is streamlined, the trans-verse oscillatory motion of the air out of the source will result in a displacement ofthe streamlines, and this displacement gives rise to a velocity perturbation in the fluidflow given byu (x, z, t) = + Mc s(x, z, t). (4)This model of the flow perturbation produced by the source may not be quite realisticin a highly turbulent duct flow, in which case the contribution from the space derivativein Eq. 4 is expected to be reduced by the irregularities in the flow. In the absence ofthis contribution, the boundary condition in (4) reduces to u = u .y sIn our experiment, since only the plane-wave mode is transmitted along the duct,it is expedient to introduce the average sound pressure p over the duct cross section,p = f p dydz. (5)We now obtain a wave equation for p by integrating Eq. 1 over the transverse coordi-nates y, z. We make use of the fact that the duct walls are rigid, except in theQPR No. 108(X. PHYSICAL ACOUSTICS)2 2source region, and note that the average of p/a z is zero. Similarly, the average ofa2p/ay 2 is (1/A) f (-Dp/Dy) 0 dz, where (ap/ay) 0 is evaluated in the source region of thewall at y = 0. We can express (ap/ay) 0 in terms of the velocity perturbation u fromthe momentum equationP t + M  fx Uy ay"The wave equation for the average pressure p can then be expressed as2- 2- 2-p Nap a p(1-M -- = s(x, t),ax2  c axatc2 at2 ax c atwhereP t) = - ) b y(x, ,t)dz.S(X, t) + a A C a UY (x, z, t) d z.at ax oIn this inhomogeneous wave equation the right-hand side is considered to be a knownsource function s(x, t) defined in the source plane y = 0, where u is given by Eq. 4. ToYsolve this equation, we introduce the Fourier transformsikx -iwtp(x, t) = ff P(k, w) e e dkdwikx -iwts(x, t) = ff S(k, w) e e dkdw,and from Eq. 7 obtain-S(k, w)P(k, e) = -M 2w(erk)(k (+ _wherek = W+ c(kc(1+M)(10)(11)(12)k - wMc(l-M)If we let the source region be limited to -L < x < L, so that s(x, t) is zero outsidethis region, we can express S(k, o) asS(k, w) = I +7T -Ls(x , co) exp(-ikx ) dx ,0 0QPR No. 108(13)(8)(X. PHYSICAL ACOUSTICS)where s(x, w) is the temporal Fourier transform of s(x, t). Then, from Eqs. 4 and 7,we haveS(k, o) = (iw) ()pu(1 - e (k , (14)whereb Lis (k, 1) z 0 sL zs(x,' °, co ) exp(-ikx°) dxo (15)o -Land A = b2L is the source area, uo = (-iuo) is the velocity amplitude of the source,and ro = the displacement amplitude. In the special case of a pistonlike displacementsuch that rs (xo z, C) = no in the source region, we havesin kL (16)ps(k, w) kL (16)Having obtained S(k, o), we obtain the pressure amplitude p(x, w) fromikxp(x, ) P(k, co) eikx dk = (k, dk, (17)-o00 - e (1-MA ) (k - k + )(k - k )where S(k, c) is given by Eqs. 14 and 15.The poles k = k and k = k , although located on the k axis in the present analysis,would contain a small positive and negative part, respectively, if some damping mech-anism, such as viscosity or heat conduction, had been included in the analysis. Eval-uating the integral by contour integration in the complex k plane, we can complete thecontour in the upper k plane for x > L, and thus include the pole at k = k . The corre-sponding solution is thenp= A (pcu 2 s(k+, ~ ) exp x . (18)+0 (1+ M) c(1+ M)Similarly, by closing the path in the lower half plane, we find, for x < -L,s 1 1 .(oP A (pcuo 2 s(k c ) exp 1 x . (19)(A 21-) c(1-M7)These solutions represent the waves transmitted in the downstream and upstream direc-tions traveling with speeds c(1+M) and c(l-M), respectively, as expected. It is inter-esting to note that the amplitudes of these waves are different in the presence of flow inQPR No. 108(X. PHYSICAL ACOUSTICS)the duct. If the source region is acoustically compact, that is, if L is much smallerthan the acoustic wavelength X, the value of the source function s is close to unity, ascan be seen in the special example given in Eq. 16, and the ratio of the upstream anddownstream pressure amplitudes becomesp_ (l+M) 2(20)p (1-M )2 "3. DiscussionIt should be emphasized that the Mach number dependence of the wave amplitudes,expressed by the factors (1+M)-2 and (I-M)-2 in Eqs. 18 and 19 and leading to theamplitude ratio in Eq. 20, depends intimately on the nature of the source and the velocityperturbation that it produces in the fluid. In the analysis carried out here the relation-ship between the (known) displacement of the air column in the loudspeaker throat andthe corresponding velocity perturbation produced in the duct flow has been assumed tobe described by Eq. 4, a relation based on the model of an oscillatory displacement ofstreamlined flow over the source region. In highly turbulent flow it may be more real-istic to use as a boundary condition u = u = as /at (obtained by neglecting a/ax iny sEq. 4), which means that the velocity perturbation in the duct flow equals the velocityin the loudspeaker throat. The amplitude ratio p /p obtained in this case isIn Fig. X-2, which shows the measured amplitude ratio Ip_ /p+ as a function of theflow Mach number, we have also plotted the functions F 1 (M) = (1+M)2/(I-M)2 andF 2 (M) = (1+M)/(1-M), which represent the theoretical results obtained on the basis ofthe two different boundary conditions that were considered. It is interesting to find thatthe experimental results fall between these theoretical curves. At Mach numbers lessthan -~0. 1, the data are in good agreement with the function F 2, which indicates thatthe streamline model of the flow is meaningful. As the Mach number is increased, how-ever, the data show a trend toward the function F 1 , which favors the boundary conditionU =u.y sThe experiments were carried out at the M. I. T. Gas Turbine Laboratory, and wewish to thank Angelo Moretta for assistance in setting up the flow facility for the exper-iments.References1. H. Billing, Trans. Max Planck Institute fiir Stromungsforschung, No. 26, p. 231,1948.2. D. I. Blokhintsev, "Acoustics of a Nonhomogeneous Moving Medium," NACA TM1399, February 1956 (translation).QPR No. 108(X. PHYSICAL ACOUSTICS)3. M. J. Lighthill, Proc. Roy. Soc. (London) A 222, 1-32 (1954).4. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill Book Company,New York, 1968), pp. 682-689.B. SOUND ABSORPTION BY A SINGLE RESONATOR IN A DUCTONR (Contract N00014-67-A-0204-0019)A. G. Galaitsis1. IntroductionThe absorption characteristics of an acoustic resonator as a duct termination havebeen studied in detail.1, 2 Here we consider the influence of a nonlinearly respondingresonator attached to a side wall of a duct on a sound wave propagating along the duct.The basic features of the problem are depicted in Fig. X-4. A plane soundwave pi propagates to the left along the duct D and excites the resonator C which com-municates with the duct through the orifice O. Consequently, a fraction of the incidentwave is dissipated in the resonator, another part, pr, is reflected back toward the soundsource and the rest, pt, propagates freely beyond the resonator.We shall first derive expressions for the transmitted and reflected fractions of theincident energy and compare them with measured values.2. TheoryConsider the system in Fig. X-4. The radius and cross section area of the orificeare r0 and A , the width and cross section of the duct are d and A, and the wavelengthof the sound wave is X. We shall assume that d, ro << , so that we deal only with planeCO/o D+ oUL uR PtPrFig. X-4. Acoustic resonator connected in parallel to a duct.waves. In the absence of reflection at the far end of the duct the total fields pL and PHto the left and right of the resonator areQPh No. 108(X. PHYSICAL ACOUSTICS)pL o(eikx + R e-ikx) -iwtPL po(e +l e ) eikx-ictP = poT eand the corresponding acoustic velocities arePo (eikx -ikx -iwtu c(e -Re )eL pcPo ikx-iwtUR =-TeR pcThe coefficients R and T are obtained by combining the boundary conditionsu - [UL-UR(x= 0)PL = PRwith the equation giving the response of the resonatorPR = pcSoUo (x = 0),where u is the acoustic velocity at the orifice, a = Ao/A, and 0 (the impedance of theorifice and resonator combination) is given byuo= e - ix = 11 - + eo0 c o - i(X + ,).o rExpression (3) contains the linear contributions 0o' X0,linear contribution proportional toU1 o'"We substitute (1) and (2) in (3)-(5), and get2o + o3and Xr, as well as the non-20T-2{ + OHere R and T depend on u 1 through o .Therefore ul must be calculated before (6) canbe evaluated. We calculate u l by observing thatSpR (x= 0)pc o01QPR No. 108-LSPL too;d130 7N".t40--KFREQUENCY (Hz)1.04,OS ATRANSMtTTED. , .- ..S1FREQUENCY (Hz)Fig. X-5. Predicted reflected (RE) and transmitted (TR) fractions of theincident power.QPR No. 108till___ __I .2T AN MITTED.- : , TTr:FREQUENCY (Hz)Fig. X-6. Measured reflected (RE) andincident power.transmitted (TR) fractions of theQPR No. 1089 80 0FREQUENCY (Hz)L.L"L 1, 14 1:-":~ c_, - ,. =-=i=,i; < : < TU< :r::.2: : : : :--:1.i, : :T:.,4.1 10'1 - iIm-1 I: 1ki.(X. PHYSICAL ACOUSTICS)which leads to24 3 2 22 2q 1 l + 2ql ou + (0 + X )u 1 2 = 0,with q1 = 1. 1/c and q 2 = p 2 /pc and p, = pR(x= 0). The roots of (8) are obtained by com-puter and R and T are subsequently evaluated. The results are shown in Fig. X-5,where RE = R and TR = Ti for different values of p 2 .3. ExperimentThe experimental arrangement for measuring RE and TR has been described in aprevious report. 4 The experimental values for TR and RE were obtained by measuringthe maximum and minimum values of the standing waves to the left (H x , Hn) and right(h x , hn) of the acoustic resonator and then takingh +hnTR = h x nH +H2RE= Hx + Hnx nfor different values of p2 . The results are shown in Fig. X-6. The agreement is sat-isfactory and the nonlinear nature of the interaction is obvious in both cases. As theamplitude of the driving field increases, the impedance of the resonator increasesbecause of the presence of the 1. 1 ul/c term, and the absorption decreases. Conse-quently a greater fraction of the power is transmitted.References1. U. Ingard and H. Ising, J. Acoust. Soc. Am. 42, 6-17 (1967).2. U. Ingard, J. Acoust. Soc. Am. 44, 1155-1156 (1968).3. A. G. Galaitsis, Ph. D. Thesis, Department of Physics, M. I. T., August 1972, seeSec. 2.1.4. A. G. Galaitsis, Quarterly Progress Report No. 106, 1Research Laboratory of Elec-tronics, AI.I.T., July 15, 1972, pp. 44-47.QPR No. 108

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