Contains reports on three research projects.U. S. Navy (Office of Naval Research) under Contract N00014-67-A-0204-0019Joint Services Electronics Program (Contract DA 28-043-AMC-02536(E)

Galaitsis, A. G.

Ingard, K. Uno

Krischer, Charles

Mazenko, G. F.

Whitney, Jacqueline F.

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IV. PHYSICAL ACOUSTICSAcademic and Research StaffProf. K. U. IngardDr. C. KrischerA. G. GalaitsisG. F. MazenkoGraduate StudentsP. A. MontgomeryR. M. SpitzbergA. GENERALIZED SOUND MODES IN GASESWe have been investigating the existence of "generalized sound" modes outside of theregime of classical hydrodynamics, for wavelengths shorter than the mean-free path.In developing a theory to describe nonhydrodynamic behavior, traditional approachessuch as implementation of the Boltzmann equation are not valid for values of the wave-length of the order of a molecular diameter because in such short-range probing thenonlocal nature of two-particle collisions is sampled. Our work has involved first, thedevelopment of a kinetic equation that is valid for moderately dense gases and for allfrequencies and wave numbers, and second, the solution of this kinetic equation for the1.51 kr = 0.0152 kr = 0.023 kr =0.034 kr = 0.104 5 kr 5.01.0Fig. IV-1.Normalized response function for a gas.2 3X = c/kVo , where mV = KT, and nr =0.01, with m the molecular mass, K0.5s Boltzmann's constant, T temperature,co frequency, and k wave number.x = /kvThis work was supported principally by the U. S. Navy (Office of Naval Research)under Contract N00014-67-A-0204-0019; and in part by the Joint Services ElectronicsProgram (Contract DA 28-043-AMC-02536(E)).QPR No. 102 51(IV. PHYSICAL ACOUSTICS)special case of a hard-core interparticle potential. A detailed account of this work maybe found in the author's thesis and in a recent paper.1, 2In this report we merely give examples of computed response spectra such as mightbe obtained in a scattering experiment. Thus in Fig. IV-1 we have plotted the responsespectrum for a moderate density gas for various values of kr , where r is the molec-ular diameter. We see that the excitation spectrum of the system changes from the2.0A kr = 0.0015, HC = 0.001B kr -0.15, HC = 0.101.51.0A0.5 B1.0 2.0x =w /kvoFig. IV-2. Normalized response function for a gas. X = w/kV , where2 3 omV = KT and nr = 0.01, with m the molecular mass, Ko oBoltzmann's constant, T temperature, w frequency, and kwave number. Curve A gives the predictions of the Boltzmannequation, whereas curve B gives the predictions of our theory3(with kr = 0. 15, nr = 0. 10).0 otwo well-defined hydrodynamic peaks for kr = 0.015 to a free-particle Gaussianoresponse for kro = 0.10. For intermediate wave numbers we have strongly damped col-lective modes. This damping is due to a phase-mixing effect caused by the interactionQPR No. 102(IV. PHYSICAL ACOUSTICS)3of collective and free-particle modes. We have found, for densities (n) such that nr =0. 10 and wave numbers of the order of kr = 0.15, large deviations from the predictionoof the Boltzmann equation (see Fig. IV-2). Our theory predicts more highly dampedmodes. Our predictions are being tested, at present, in a series of neutron scatteringexperiments.G. F. MazenkoReferences1. G. F. Mazenko, Ph.D. Thesis, Department of Physics, M.I.T., May 1971(unpublished).2. G. F. Mazenko, "Microscopic Method for Calculating Memory Functions in Trans-port Theory," Phys. Rev. A 3, 2121-2137 (1971).B. ACOUSTICAL PROPERTIES OF LANTHANUM FLINT GLASSWe are, at present, investigating the acoustical and acousto-optical properties oflanthanum flint glass, using Bragg diffraction of light. The objective of this study isto determine the effect on the acoustical properties of introducing heavy dopants intoglass. We are also interested in determining the possibility of utilizing this high-refractive-index glass in acousto-optic modulators.Fig. IV-3.Ultrasonic attenuation as a functionof frequency for longitudinal wavesin B + L No. 8673 10 lanthanum flintat 25°C.FREQUENCY ( MHz )QPR No. 102_I i I I ll1.14±0.03I I I I i I I(IV. PHYSICAL ACOUSTICS)Measurements of the longitudinal-wave ultrasonic attenuation and velocity weremade on a sample of Bausch and Lomb No. 867310 lanthanum flint. This glass has adensity of 5.30 gm/cm 3 and a refractive index of 1. 867 for the sodium D line.1 Thesample was a rectangular parallelepiped, 5 X 2.5 X 1 cm, with 4 edges opticallypolished flat to 1/10 wave, and opposite surfaces parallel within 10 seconds of arc.The longitudinal-wave pulses were excited by a bonded X-cut quartz transducer,of 10. 1 MHz nominal fundamental-resonance frequency, driven at odd harmonics.The experimental apparatus and technique have been described, in detail, elsewhere. 2The longitudinal-wave velocity was measured at various frequencies in the range190-920 MHz. We found that vp = (5. 477 0. 007) X 105 cm/s at 25'C, where theerror brackets correspond to the ±lo- points for our data.Figure IV-3 shows the longitudinal wave attenuation as a function of frequency overthe range 190-680 MHz. The observed frequency dependence A(dB/cm) ~fl. 14 indi-cates that, in this frequency range, the dominant attenuation mechanism is neither astructural relaxation nor the interaction with thermal phonons, since both thoseprocesses give a square-law frequency dependence. We are now extending the rangeof measurements at both higher and lower frequencies, in order to obtain further infor-mation about the loss process.C. Krischer, Jacqueline F. WhitneyReferences1. "Optical Glass," Bausch and Lomb, Rochester, New York.2. C. Krischer, "Optical Measurements of Ultrasonic Attenuation and ReflectionLosses in Fused Silica,"J. Acoust. Soc. Am. 48, 1086-1092 (1970); C. Krischer,"Measurement of Sound Velocities in Crystals Using Bragg Diffraction of Light andApplications to Lanthanum Fluoride," Appl. Phys. Letters 13, 310-311 (1968).C. FREQUENCY MIXING OF SOUND IN A NONLINEAR ORIFICEExperimental data on nonlinear distortion of a pure sound tone when transmittedthrough a sharp-edged orifice have already been reported. 1In this work the spectrum of the transmitted wave is obtained when the incident waveis the superposition of two tones. A simple theoretical analysis will be used to predictthe spectrum.1. Experimental ArrangementA simple diagram of the experimental arrangement is shown in Fig. IV-4. A high-impedance piston P is fitted to the left end of a cylindrical tube, 1 3/8 in. in diameter.The diaphragm D is 0. 015 in. thick and has an orifice, 1/8 in. in diameter. The velocityQPR No. 102(IV. PHYSICAL ACOUSTICS)Fig. IV-4. Experimental arrangement. P = high-impedance piston,D = diaphragm with concentric orifice, A = hot-wireanemometer, T = pc termination, and C o = driver cavity.in the plane of the orifice is measured by the hot-wire anemometer A, and there is apc termination at the right end.If PZ and p3 are the pressures at the left and the right of the orifice, then the veloc-ity u at the orifice is in general given byp A duAp Pz - p 3  2 u + p c0 u + pt (1)3Am dtZC 11where p is the density of the air, c is the speed of sound, C 1 is the orifice coefficient,A and A are the cross sections of the orifice and the duct, respectively, 0 is a fric-o 1tion coefficient that may be expressed in terms of the orifice geometry and the viscositycoefficient, and t is the effective orifice thickness.mThe pressure p 3 represents approximately an outgoing wave given byAo (2)P 3  pci U,so that (1) may be writtenu u + pc (1+)u + du (3)2 2 A I m dtThe level L of a sinusoidal wave p cos ct is defined asPL = 20 log (4)where p = 0. 0002 dyn/cm , and L is measured in decibels.u pQPR No. 102(IV. PHYSICAL ACOUSTICS)In the low-pressure level regime (L < 130 dB) the nonlinear term of (3) may beomitted, and the orifice behaves as a linear RL circuit where the dissipative terms arepcAou/A 1 and pcAo0u/Al, that is, a friction and radiation term. The inertial termmerely introduces a phase shift between the driving force p 2 and the response u of thesystem.In the high-pressure level and low-frequency regime separation of flow appears inthe vicinity of the orifice, and the nonlinear term is much larger than both linear termswhich, consequently, are omitted.Finally, in the high-pressure levels and high-frequency regime the inertial term isno longer negligible (being roughly proportional to the frequency of the incident wave)so both the nonlinear and the inertial terms must be included in the analysis. In thisregion, however, even the complete form of (3) is only an approximation to the actualbehavior of the system.2. Calculation of the Transmitted SpectrumThe flow velocity at the orifice is calculated by assuming that the process isdescribed by the high-pressure level and low-frequency model outlined above,PPZ= u u. (5)2C 1Ifp 2 = Po (cos (x+A) + cos (8x+B)) pof(x), (6)where x = wt, w = 2Tv, v = 250 Hz, and A, B are arbitrary phase angles, thenu(x) = C 1  for f(x) = f(x) . (7)_f (x) (2)If U(x) is defined byu(x)U(x) = ± f(x) for f(x) = ± f(x) , (8)C 1then U(x) may be expressed as a Fourier seriesooU(x) = (C(N) cos Nx+S(N) sin Nx), (9)N= 1where the coefficients are given byQPR No. 102(IV. PHYSICAL ACOUSTICS)Nx)dx.Nx /(10)The integrals of (10) are carried out on a computer. The predicted spectrum wasobtained for different values of the phase angles, A, B.160 -150 -140130It was observed that C(N)3 5 7 9 11 13 15 17 19W/WlFig. IV-5. Spectrum of transmitted sound through an orifice upon whichtwo sound waves of frequencies w1 and w. = 8w 2 are incident.and S(N) are sensitive to phase changes while the total amplitude AM(N) of a partic-ular harmonic,AM(N) = [C(N)Z+S(N)21/2, (11)was virtually independent, except for N = 2, 7, 9, 14, 16, 18 which exhibit 10% maxi-mum variation of the total pressure level.The transmitted spectrum for A = 41T/5 and B = rr/20 rad is shown in Fig. IV-5.A. G. Galaitsis, U. IngardReferences1. U. Ingard, J. Acoust Soc. Am. 48, 32-33 (1970).QPR No. 102C(N)) 1S(N) / x= 0COsU ( sin

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