Contains reports on four research projects.Joint Services Electronics Programs (U.S. Army, U.S. Navy, and U.S. Air Force) under Contract DA 28-043-AMC-02536(E)U.S. Navy (Office of Naval Research) under Contract N00014-67-A-0204-001

Ingard, K. Uno

Katyl, R. H.

Krischer, Charles

Contains reports on three research projects.U. S. Navy (Office of Naval Research) under Contract Nonr-1841(42)Joint Services Electronics Programs (U. S. Army, U. S. Navy and U. S. Air Force) under Contract DA 28-043-AMC-02536(E

Schulz, H. M., III

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X. PHYSICAL ACOUSTICS*Academic and Research StaffProf. K. U. IngardProf. L. W. Dean IIIGraduate StudentsR. H. KatylC. KrischerW. M. ManheimerA. POSSIBILITY OF RADIATION-INDUCED SOUND AMPLIFICATION IN A GASThe possibility of observing acoustic-wave amplification made possible through theheating of neutral particles by electrons in a weakly ionized gas has received somerecent attention. -4 The idea is that an existing sound wave, by inducing a space-.timevariation in the electron density, would create an acoustic source, thereby generatinga wave that interferes constructively with the original 3ound wave. At moderately highsound frequencies (> 104 sec - ) the electrons behave isothermally under typical dis-charge conditions. Then, if acoustic losses resulting from viscosity and heat conductionin the neutral gas are ignored, the resulting exponential growth in the wave intensity inthe linear theory is characterized by a time constant T' = T(1+x 2). The parameter T(~1 sec under typical discharge conditions) represents a time of the order of thatrequired for the gas to absorb from the electrons an amount of energy equal to thethermal energy of the gas, while x is the tangent of the phase angle between relativedensity perturbations in the electrons and neutrals. 4 An analogous calculation for ionwaves in a fully ionized gas shows only a very slight wave amplification.5 Here, also,there is no guarantee that the gain would offset the expected losses. In fact, one of theprincipal difficulties in observing energy-transfer wave amplification is the considerableloss rate associated with the relatively low-pressure gas required to obtain an electricaldischarge.This difficulty could be overcome if the energy to be transferred into the translationalmodes of the gas could be carried internally in the gas molecules themselves. Dischargeconditions would not then be required, and one could increase the gas pressure so as toreduce ordinary acoustic losses in the completely un-ionized gas. By carrying the energyinternally, moreover, the gas molecules would eliminate the phase difference 4 betweenrelative density perturbations in the gas component that is responsible for heating andthe component that is being heated through collisions with the former. The purpose ofThis work was supported principally by the U. S. Navy (Office of Naval Research)under Contract Nonr-1841(42), and in part by the Joint Services Electronics Programs(U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E).QPR No. 86(X. PHYSICAL ACOUSTICS)the present report is to point out the existence of this kind of acoustic-instability mech-anism in a high-pressure gas, completely un-ionized, having internal modes, forexample, molecular rotational or vibrational states or atomic-electron states, excitedselectively by an external radiation field so as to be out of thermal equilibrium with thetranslational modes of the gas. The possibility of nonradiative decay through atomic ormolecular collisions in such a gas constitutes a positive-feedback condition for anacoustic wave in the gas, since the decay energy would go into the translational modesand the rate of energy transfer would be proportional to the collision frequency. Theresult could be either the spatial amplification of a wave propagating in a relatively loss-less high-pressure gas or the spontaneous oscillation of a high-Q acoustic cavity.The level separation in such a gas is large enough that at room temperature, or atleast at temperatures easily attained by cooling the gas, nearly all of the molecules arein the ground state under conditions of thermal equilibrium, that is, they are not inexcited rotational, vibrational or electronic states. Intense illumination of the gas, byelectromagnetic radiation having a narrow bandwidth about a transition frequency, forexample, a line of the rotational, vibrational or electronic spectrum, would lead to anonequilibrium distribution of the gas molecules among the energy levels of the internalmode, or to an internal temperature, which we hope may be much larger than the tem-perature associated with the translational modes.The effect can be imagined as occurring in a one-dimensional electromagnetic reso-nant cavity driven externally, for instance, by a large number of parallel laser beams,so as to possess an energy-density spectrum sufficiently peaked at the transition fre-quency that a uniform nonthermal fraction f of the molecules (or atoms) are in theexcited state. For this to be true it is necessary that a random molecule (or atom) bemuch more likely to interact with a photon from the radiation field than with anothermolecule (or atom). Thus, the radiation field must have, on the average, an energy-flux density J > w nh//(, where wnn is the collision frequency, h is Planck's constant,hv is the transition energy for the internal mode, and a- is the cross section for theabsorption of a photon at that energy. Under these conditions, the time required for thetranslational modes of the gas to absorb their own thermal energy from the radiativelyexcited internal modes is of the order of T = m c /hvfwnn, where m is the mass of an n nn ngas molecule, and cn is the speed of sound. Within a factor of order unity, this timeconstant would also characterize the growth of the intensity of a sound wave in theabsence of acoustic losses.H. M. Schulz IIIReferences1. U. Ingard and K. W. Gentle, Phys. Fluids 8, 1396 (1965).2. U. Ingard, Phys. Rev. 145, 41 (1966).QPR No. 86(X. PHYSICAL ACOUSTICS)3. L. D. Tsendin, Zh. Tekh. Fiz. 35, 1972 (1965). [Translation: Soviet Phys. - Tech.Phys. 10, 1514 (1966)].4. U. Ingard and M. Schulz (to appear in Phys. Rev.).5. M. Schulz, Phys. Fluids 10, 243 (1967).B. LIGHT SCATTERING OFF SURFACE WAVE PHONONSThe dynamics of surface waves on liquids can now be measured in the wavelengthregion of ~10 - 5 cm by light-scattering techniques. In this wavelength region thedamping of the waves is quite high and first-order theories of surface wavescannot be used. We shall outline the exact solution to the surface wave prob-lem,2 using a linearized theory, and use this solution to interpret some experi-mental results.We work here with an incompressible fluid of mass density p, surface tension a, andshear viscosity L. The liquid surface is taken to be the x-y plane, with the z-axisextending out of the fluid. The velocity field is u(r,t) and the surface displacement3(x, y,t). Following Lamb, the boundary conditions required of the stress tensor Tijat the surface z = 0 are72 2Tzz = - +  (1)zz 2 ayT = T =0, (2)xz yz8u. 2 -auwhere T..ij is defined as Tij = p6.ij- + +- (u).13 3 = 8~ x 4 ax 3 ijWe now resolve the velocity field u into its independent longitudinal and transverseparts by u = -V + V x b, where V - b = 0 specifies the gauge. With this replacement foru, the linearized equation of motion, paui/8t =-aTi/ax , is separated into two equations.With the use of the incompressibility condition, V - u = 0, these areatb (4)P t = P b. (4)To simplify the problem, we look for the motion of a one-dimensional disturbance ofthe format iKx K zS= o e e e (5)QPR No. 86(X. PHYSICAL ACOUSTICS)-% -. at iKx PZb=b e e e . (6)oHere, d satisfies V 2 = 0, and (4) requires p2 = K + pa/h. For this one-dimensionalwave u = 0, and the velocity field becomes- KeKz Pez  iKxeat (7)u= = o e 0 + b e 0 e e (7)uz K JiKWith this form of u, condition (2) results inibo pa (8)pa4o 1+2K 2and upon taking the time derivative of (1) and using (3), we obtain82u 82 2ua2  a a2ucr- -p +2 z (9)z2 + 2  azat8x 8tWith the use of (7) and (8), this reduces to the following equation for a:2a + a + + +p +- = 0. (10)P ( 22 tK2)(0With a change of variables, this becomesS 2 + 2S - , +1 +2S 1 + y = 0, (11)where S = aT, T O = p/2pK = K /p, and y = (w T ) 2. This is the determiningequation for the time dependence of surface waves of a particular wavelength.Equation 11 can be squared, and the resultant quartic equation solved either exactlyor numerically. Of the four roots obtained in the solution for S(y) only two are consis-tent with the requirement p > 0. These are shown in Fig. X-1. For y -< 0. 145, bothroots are real and negative, and for y > 0. 145, the roots are complex conjugates witha negative real part. Thus, for y < 0. 145, the decay of the disturbance is nonoscillatory;there are two modes of decay with differing time constants. But for y > 0. 145, thedecay is damped oscillatory.This solution for the time dependence means that the spectrum of the scattered lightwill be split into a doublet, the so-called Brillouin doublet, for y > 0. 145. The doubletfrequency separation is (Si/rTo), and the half-intensity width of each line is (Sr /(TrT )).QPR No. 86(X. PHYSICAL ACOUSTICS)But for y < 0. 145, no splitting occurs; the spectrum consists of two unshiftedcomponents whose peak intensities are in inverse proportion to their widths.10.0-0.010.01 0.10 1.00 10.0Fig. X-1. Solution of Eq. 11 for time dependence of surface waves.iKx S(t/o) 2The wave propagates as e e , T = p/2K ,K3/p, y = (WOT o )This absence of a Brillouin splitting under certain conditions can explain theresults of an experiment that we have performed recently. Light-scattering mea-surements were made with two different liquids with conditions chosen so thata doublet was expected for the less viscous liquid (methanol), and nonoscillatorydecay was expected for the other (isopropanol). Although the doublet could notbe resolved, the spectrum for methanol was noticeably wider than for isopro-panol (see Fig. X-2).QPR No. 86METHANOLJ//RELATIVEINTENSITYFREQUENCY136 MHz(a)RELATIVEINTENSITYISOPROPANOLFREQUENCY14.1MHz(b)Fig. X-2. (a) Results for methanol. (b) Results for isopropanol.Upper solid curve: surface scattering; lower solid curve,scattering in bulk of fluid; dashed curve, instrumentresponse; points (I), theoretical curve resulting from thesolution of Eq. 11 when the effects of instrumentalbroadening and bulk scattering are included.QPR No. 86 100(X. PHYSICAL ACOUSTICS)Note that the theory that we have outlined here does not consider the effects of anychanges in structure of the liquid near the surface. Such effects can occur in polar orassociated liquids over length scales approaching a light wavelength, and would beexpected to greatly modify the scattered light spectrum. Higher resolution experiments,which are needed to examine this possibility, are now in progress.R. H. Katyl, U. IngardReferences1. R. H. Katyl and U. Ingard, Phys. Rev. Letters, 19, 64 (1967).2. V. G. Levich, Physicochemical Hydrodynamics, translated by Scripta Technica,Inc. (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962), p. 603.3. H. Lamb, Hydrodynamics (New York, Dover Publications, New York, 6th edition,1945).C. ION-DENSITY RESPONSE TO A RECTANGULAR PULSE OF SOUND IN APLASMARecent investigations of the acoustic-wave mode in a weakly ionized gas1 - 5 haveshown this to be a slightly perturbed version of ordinary sound in the neutral-gas com-ponent. The charged particles are dragged along by the neutrals, but ambipolar dif-fusion in the direction of wave propagation causes a phase and amplitude differencebetween the relative density perturbations in the neutral gas and in the electrons orions. 5 In the present report the frequency dependence of this phase and amplitudedifference is applied to the calculation of the ion-density response to a rectangularpulse of sound in a weakly ionized gas. It is shown, in particular, that the wavefrontin the ionic component is not sharp, but has a width in time inversely proportional tothe neutral-to-electron temperature ratio and the ion-neutral collision frequency, aresult qualitatively consistent with previously reported probe studies of the timedependence of the ion density during the passage of a weak shock wave.As has recently been shown, 5 the relative density perturbations (n /Nj ) (the sub-scripts j = n, i, e refer to the neutrals, ions, and electrons, respectively) in the variousfluid components are given by(nl/N) (ne/Ne) [l+i(Te/yT ~.in)] - 1 (nn/Nn) (1)for a sound wave of frequency w/2ir, where T is the temperature, w.in is (essentially) theinion-neutral collision frequency, and -n is the specific-heat ratio (1. 67 for a monatomicgas). The initial condition for a small-amplitude rectangular pulse in the three-component fluid undoubtedly depends on the method of generation, and is satisfied by aQPR No. 86 101(X. PHYSICAL ACOUSTICS)superposition of Fourier spectra of the various density and entropy waves that can existin a weakly ionized gas. Of these, however, the only mode that really propagates for< 10-6 -1 19S 10 6 sec is the acoustic mode; all the others are strongly damped. 1 2 Since theFourier spectrum of a rectangular pulse is dominated by the low-frequency components,an accurate description of the pulse for times t 10 sec after generation is achievedby considering the acoustic Fourier components only.Under the simplifying assumptions that the plane pulse generated at z = 0 and travelingin the +z direction is of small amplitude and substantially free from ordinary acousticlosses such as viscosity and heat conduction, the rectangular pulse shape thereby beingmaintained for all times of experimental interest, the relative density perturbation inthe neutral gas can be represented by (nn/Nn) = e(t-z/c), where 0(x) = 1 for x > 0, and0(x) = 0 for x < 0. The corresponding relative ion-density perturbation for times ofexperimental interest is given byn (z, t) E +oo + exp(-iot'+iwz/c) dt'dwNi 2 7-oo 0 1 + iw(Te/yYn .in)xp[(Tn win /Te )(t-z/c)], t < z/cE (2)1 t > z/cThus, the leading edge of the pulse traveling in the neutral gas is preceded by a distur-bance in the charged particles. This disturbance, caused by ambipolar diffusion ofcharged particles ahead of the wavefront in the neutral gas, is characterized by a time-4 -37 = Te/Y-nTn in 10- 10 sec, and so the wavefront associated with the ion densityis not at all sharply defined, even for a perfectly sharp wavefront in the neutral gas.This lack of sharpness in the ion-density wavefront is consistent with a waveformobtained in recent experiments designed to measure the speed of a weak, spark-generatedshock wave in an argon discharge.6 The structure of the wavefront was not then of inter-est, and no comment was made on it, although the experimental conditions should haveproduced a sharp wavefront in the neutral gas. In particular, the previous experimentsdid not firmly establish whether the disturbance in the charged particles trails the wave-front in the neutral gas or leads it, as predicted here. This matter can be decided onlyby future experiments specifically designed to resolve the question.U. Ingard, H. M. Schulz IIIFootnotes and References1. G. M. Sessler, Phys. Fluids 7, 90 (1964).2. U. Ingard and K. W. Gentle, Phys. Fluids 8, 1396 (1965).QPR No. 86 102(X. PHYSICAL ACOUSTICS)3. U. Ingard, Phys. Rev. 145, 41 (1966).4. L. D. Tsendin, Zh. Tekh. Fiz. 35, 1972 (1965) [Translation: Soviet Phys. - Tech.Phys. 10, 1514 (1966)].5. U. Ingard and M. Schulz (to appear in Phys. Rev.).6. K. W. Gentle and U. Ingard, Appl. Phys. Letters 5, 105 (1964).7. It might be mentioned parenthetically that the total number of ions and electrons isconserved by an appropriate disturbance preceding the trailing edge of a real pulse.This is not of essence in the present discussion, since one may imagine a pulseof duration much longer than T.QPR No. 86 103

Physical Acoustics

VII. PHYSICAL ACOUSTICS*Academic and Research StaffProf. K. U. IngardProf. L. W. Dean IIIGraduate StudentsR. H. KatylC. KrischerW. M. ManheimerA. THERMALLY EXCITED SURFACE WAVES: OPTICAL HETERODYNESCATTERING EXPERIMENTAs we have reported elsewherel the resolution of classical interferometric experi-ments has been insufficient to provide an unambiguous determination of the spectrum oflight scattered by thermal fluctuations of a liquid surface. To overcome this difficulty,we have examined the possibility of using recently developed optical heterodyne spec-troscopy2 in surface scattering experiments.In this technique, the laser and scattered light beams are combined at the surfaceof a square-law photodetector. If conditions of beam alignment and spatial coherencyare met, temporal interference occurs, and components of current at the differencefrequencies between the various Fourier frequency components will be produced. Thusthe surface fluctuation spectra can be directly observed by radiofrequency spectrumanalysis of the phototube current.The application of this technique is particularly attractive at small scattering angles,because of the A2 enhancement of the scattered intensity, and the increase of spectralpower density due to lower damping. In a typical experiment, the expected Brillouinshift is 0. 38 MHz, and the linewidth 0. 13 MHz, a good example of a spectrum unobserv-able with classical interferometry, but easily observed with radiofrequency spectrumanalysis.To observe the spectrum, we have constructed a multi-mode Argon-ion laser andhave assembled the optics required to align the two beams involved in the heterodynescattering experiment. In this experiment, the scattered light is chopped, mixed withthe laser beam, and the r. f. spectrum analyzed with a commercially available spectrumanalyzer, using a synchronous detector.Calculation shows that 10 mw of incident light should produce an r. f. signal whosepeak power is about 10% of the local oscillator shot noise power. At this low powerlevel (10 mw), the laser is relatively quiet. The amplitude noise spectra of the lightThis work was supported principally by the U.S. Navy (Office of Naval Research) underContract N00014-67-A-0204-0019 and in part by the Joint Services Electronics Programs(U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E).QPR No. 87(VII. PHYSICAL ACOUSTICS)has a half-power width of less than 20 kc, thus permitting observation of the scatteredspectrum.Construction of the apparatus is virtually complete, and scattering experiments arecurrently in progress.R. H. Katyl, U. IngardReferences1. R. H. Katyl and U. Ingard, Phys. Rev. Letters 19, 64 (1967).2. H. Z. Cummins, N. Knable, and Y. Yeh, Phys. Rev. Letters 12, 150 (1964).B. SURFACE PHONON LIFETIME BY CONDITIONAL PROBABILITYMEASUREMENTWe are attempting to study the statistics of the fluctuations of light scattered by aliquid surface to determine the surface phonon lifetime in a wavelength region whereclassical spectroscopy cannot resolve the spectrum. Using a technique proposed byMandel, 1,2 and used by Scarl, 3 we are attempting to measure the conditional prob-ability pc(T) for the arrival of a second photoelectron at time T removed from thearrival of the first photoelectron. For thermal, spatially coherent light, this prob-ability ispc(T) = a<I> [I+ly(1 , 1) (T)I2]where a is a proportionality constant, <I> the average light intensity, and y (T) thenormalized self-coherence function. For surface scattered light, y Z has the formexp [-2T/T ] , where Ts is the surface phonon lifetime. For scattering angles such that(nA/ko) ~ 20, Tf ~ 0. 8 microsecond, and the electronics required to see the correla-tion are not too demanding.Using a single-mode Helium-Neon laser with the beam focused by a 10-cm lens to asmall spot on a methanol surface (under conditions of internal reflection), we have seenthe correlation in arrival of photoelectrons, provided the collecting angle was ~10mrad.For larger collecting angles the correlation disappeared, indicating that the collectedlight was spatially incoherent. (10 mrad corresponds roughly to the area of coherent-3illumination produced by a source of diameter 10-3 cm, the approximate size of sourceused.)The digital electronics used were fabricated mostly from Fairchild Integrated Cir-cuit 4L-914. An ITT FW-130 photomultiplier was used as photodetector. Severe prob-lems of trigger level drift and dead-time corrections were encountered; also, becauseof the long counting times (10-20 min) mechanical stability was a problem.QPR No. 87(VII. PHYSICAL ACOUSTICS)The setup will be improved, and a more detailed report made later.R. H. KatylReferences1. L. Mandel, in Progress in Optics, edited by E. Wolf (North-Holland Publishing Co.,Amsterdam, and John Wiley and Sons, Inc., New York, 1963), Vol. 2, p. 181.2. L. Mandel, Electromagnetic Theory and Antennas, edited by E. C. Jordan(Macmillan Co., New York, 1963), Part 2, p. 181.3. D. B. Scarl, Phys. Rev. Letters 17, 663 (1966).C. DIFFRACTION OF LIGHT BY ULTRASONIC WAVES IN SINGLE CRYSTALSIn our investigation of the diffraction of light by ultrasonic waves in single crystals,we are presently studying the photoelastic and acoustic properties of 2 crystals havinghexagonal symmetry: LaF 3 and CdS.The hexagonal crystal groups possess acoustic isotropy over the entire basal plane,since the elastic constant tensor components are invariant for rotations about thec-axis (c). For any wave vector k lying in the basal plane the three correspondingacoustic waves are pure modes (with polarization vectors u II k, u II k X c, andu II c, respectively). No other crystal group possesses this property. A hexagonalcrystal having low ultrasonic attenuation and relatively high index of refraction andphotoelastic constants (diffracting power) would be of considerable interest as a mate-rial for elasto-optic modulators.In our experiments we observe Bragg-type diffraction of a 6328 A laser beam byultrasonic wave pulses (generated by a quartz transducer bonded to the crystal speci-men) in the frequency range of 0. 2 Gc to 1. 0 Gc. The deflected light pulses aredetected by a photomultiplier tube, further amplified, and displayed on an oscilloscope.From measurements of light pulse amplitude and spacing, diffraction angle, input r-fpower, etc., we can obtain values for the photoelastic constants, elastic constants, and5, 2ultrasonic attenuation to better than 5% accuracy. In particular, this method ofmeasuring ultrasonic attenuation is much more accurate than the conventional pulse-echo technique (utilizing a transducer as the detector of the acoustic power) since theproblem of interference effects (arising from oblique incidence of the ultrasonic waveon the transducer) are eliminated. Initial measurements indicate that neither LaF 3nor CdS compare very favorably with the many low acoustic-loss crystals presentlyavailable (e. g., a (k = < 1120>, u = < 1010 >, f= 0. 25 Gc) = 0. 85 db/sec for CdS; a (k=< 11Z0>, u= < 1120 >, f= 0. 25 Gc) = 0. 58 db/Lsec for LaF 3 ; as compared to a (k= < 0001 >,u =< 1010>, f= 0. 25 Gc) = 0. 05 db/isec for TiO). 3 It is interesting to note that thegroup of known extremely low acoustic-loss materials include crystals of cubicQPR No. 87(VII. PHYSICAL ACOUSTICS)(Bi 1 2 GeOZ 0 , YIG, YAG), trigonal (LiNbO 3, A12 0 3, LiTaO3 ) and tetragonal (TiOZ) sym-metry - but none of hexagonal symmetry.X2  X 2  X 2X2 D. X1 D.Dd N X Dd a a I X11= 45 # = 1/2 tan-1 (P / P66 16X3  X3(a) HEXAGONAL SYMMETRY, ( b ) TETRAGONAL SYMMETRY ( T II)(HI OR H II)Fig. VII-1. Acoustic diffraction rotator for (a) crystal with hexagonalsymmetry, and (b) crystal with tetragonal symmetry.In the diffraction of linearly polarized light by transverse acoustic waves, it ispossible to obtain polarization rotation of the diffracted light, analogous to an optical1/2-wave plate. For this to occur, the light must be incident along an optic axis, andmust also be perpendicular to the acoustic wave polarization vector. 4 This effect isshown, in Fig. VII-1, for crystals of hexagonal and tetragonal symmetry. In both cases,the light is incident along the x3-axis; the acoustic wave propagates in the x l-directionand is polarized along the x2-axis. If the incident beam displacement vector, D i , formsan angle a with respect to the rotated axis, x2, then the diffracted beam displacementvector Dd forms an angle -a with that axis. The angle 4 depends on the symmetry andphotoelastic constants of the crystal in question. We constructed such a polarizationrotator using fused quartz (isotropic). For a total acoustic power of 10 watts, weobserved that Id/I = -14 db, in good agreement with the theoretical predictions of the62diffraction power (n p ; n = index of refraction, p = photoelastic constant). Using amaterial of higher diffracting power (e. g., CdS) it should be possible to construct anelasto-optic polarization rotator having better than 50% efficiency.C. KrischerReferences1. C. F. Quate, C. D. W. Wilkinson, and D. K. Winslow, Proc. IEEE 53, 1604 (1965).2. R. W. Dixon and M. G. Cohen, Appl. Phys. Letters 8, 205 (1966).3. T. A. Midford and S. Wanuga, J. Appl. Phys. 36, 3362 (1965).4. H. Mueller and Z. Krist. A99, 122 (1938).QPR No. 87(VII. PHYSICAL ACOUSTICS)D. NONLINEAR ACOUSTICS: SOUND PROPAGATION IN A DUCTIn many problems involving sound transmission in ducts the sound-pressure levelsinvolved are frequently so high that nonlinear effects play a role in the attenuation ofthe sound. There are two different types of nonlinear effects to be considered. Thenonlinearity of the gas itself leads to formation of a "shocked" sound wave ("saw-tooth"wave). This effect, which is not peculiar to ducted waves and occurs also in free waves,is significant at quite high sound-pressure levels, say, in excess of 160 dB (pure tone).1The attenuation of the shocked sound wave in a rigid tube has been studied by Rudnick.The second type of nonlinearity, which is peculiar to the ducted waves, results fromthe interaction of the sound wave with the duct boundaries. For many acoustical duct-lining materials a marked nonlinearity in the acoustic response sets in at comparativelylow levels, where the effect of shock formation is insignificant. As a result of this non-linearity, the sound attenuation per unit length will vary with position, and the sound-pressure level will deviate from the usual linear decrease with distance along the duct.We have calculated the pressure variation along a duct with a particular type of bound-ary lining. The results are given in a paper scheduled for publication in the Novemberissue (1967) of the I.A. S.A.U. IngardReferences1. I. Rudnick, J. Acoust. Soc. Am. 25, 1012 (1953).I r51 X --QPR No. 87

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