Contains reports on two research projects.U. S. Navy (Office of Naval Research) under Contract Nonr-1841(42)National Aeronautics and Space Administratio

Fleury, P. A., V

Ingard, K. Uno

Weiner, S. D.

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VIII. PHYSICAL ACOUSTICSProf. K. U. Ingard K. W. Gentle W. M. ManheimerProf. L. W. Dean III R. H. Katyvl M. A. MartinelliDr. H. L. Willke, Jr. J. L. Macon J. A. RossP. A. Fleury V A. A. Maduemezia S. D. WeinerA. SOUND WAVE GENERATION AND AMPLIFICATION IN PLASMASIn a gas discharge, the electron temperature is generally much greater than theneutral-gas temperature; consequently, energy is transferred to the neutral particlesthrough electron-neutral collisions. This energy transfer implies that any sound wavesin the neutral fluid will be nonadiabatic and the pressure must be considered as a func-tion of both density and entropy. Since energy is being delivered to the neutrals, wemay expect to observe this energy either in the steady increase of the neutral tempera-ture, or in the generation of growing waves in the neutral fluid. It is this generation ofsound waves which is of interest here.If the linearized continuity and momentum equations for a nondissipative gasap+ pV t-T = 0 (1)8tp -- + Vp = 0 (2)are combined by subtracting the time derivative of Eq. 1 from the divergence of Eq. 2,we obtain2a 2p, (3)at2where p is the mass density, and p the pressure of the neutral fluid. The density isrelated to the pressure and entropy, S, through the equation of state. If we assume aperfect gas, thendp = (1/c 2 ) dp - (p/C p) dS, (4)where c is the sound velocity, and C is the specific heat at constant pressure. Com-bining Eqs. 3 and 4 gives21  2 ap a(' a a2 2 t t at C tc at L iThis work was supported principally by the U. S. Navy (Office of Naval Research)under Contract Nonr-1841(42).QPR No. 78(VIII. PHYSICAL ACOUSTICS)Using the fact that pTn dS/dt = H, the rate of heat transfer to the neutral fluid, weobtain2where y is the ratio of specific heats. If we neglect the variations in sound velocity,Eq. 6 becomes1 p 2 y aH2 2 V p 2 at (7)a 2 It is now necessary to derive an expression for H in the case of energy transferfrom electrons to neutrals through elastic collisions. This expression is found by inte-grating the product of the energy transferred per collision and the probability of colli-sion over all possible collisions. Thus2Tr m (v 2mH =sin 0 dO 2 m (-cos0) e(ve 0) f(e) Nn (8)where 0 is the angle of scattering, a- is the differential cross section, f(-e ) is theelectron distribution function, and Nn is the neutral density. The integrals in Eq. 8depend on the energy and angular dependence of the cross section, but for our purposes,we shall assume an effective electron-neutral collision frequency, and setH = kT e(2me/mn ) Ne v e. (9)varies with the electron temperature, T . From Eqs. 7 and 9, it can be seen that thereewill be a source term in the acoustic wave equation if the electron density, the electrontemperature, or the collision frequency varies with time.In the afterglow of a gas discharge, the electron temperature decays exponentiallyand provides a source term in Eq. 7. Experimental studies1 performed in cryogenicafterglow plasmas have shown that the transverse acoustic modes of the discharge tubeare excited, and the relative amplitude of oscillation increases with decreasing neutraltemperature. These observations are in agreement with results derived from Eqs. 7and 9 for an exponentially decreasing electron temperature.If we consider variations in the collision frequency caused by variations in the neu-tral density (or pressure), thenav v e ap v apat Po at - 2 at (10)nnQPR No. 78(VIII. PHYSICAL ACOUSTICS)Inserting this into Eqs. 7 and 9 gives the homogeneous equation2a p 2 2 8ap2 V p 2 at' (11)c at cwherey-1T m NPe _e (12)2 T m N e'n n nBy substituting a plane wave, p = P exp(ikx-iwt), in Eq. 11, we obtain the dispersionrelationv = ip c 2k 2  (13)which represents growing waves. Similarly, if we consider w to be real, then thewaves will grow in space. Equation 11 has also been studied for waves in cylindricaland spherical discharge tubes, and in both cases, yields self-excited sound waves. Inthese problems, the fact that the electron density is not uniform leads to couplingbetween the various acoustic modes, but this does not alter the fact that the waves areself-excited.In an experimental study of ionic sound waves, Alexeff and Neidigh 2 also observedordinary sound waves in a spherical discharge tube. They found that these ordinary soundwaves did not occur at low pressures, but as the pressure increased (above -1 mm),sound waves were spontaneously generated. From Eq. 12, we see that the growthrate, P, is proportional to pressure for constant degree of ionization. Dissipativeeffects in the bulk of the gas and at the boundaries, resulting from heat conduction andviscosity, decrease with increasing pressure. Thus as the pressure increases, thegrowth mechanism will dominate the dissipative mechanisms as observed experimen-tally.In the experiments mentioned above, these sound waves have been detected throughtheir effect on the electron fluid. The waves in the neutral fluid tend to produce wavesin the ion fluid through ion-neutral collisions. These ion waves then tend to produceelectron waves through the electric fields that they establish. It is these electron wavesthat may be detected by microwave interactions or with probes.As seen in Eq. 9, any electron waves that are in phase with the sound waves, willincrease the source term in Eq. 7. If we assume that the electron waves have the samerelative amplitude as the neutral wavesne/Ne = nn/Nn,then the growth rate, P, has twice the value given in Eq. 12.QPR No. 78(VIII. PHYSICAL ACOUSTICS)The important factors in maximizing the growth rate for these acoustic wavesare: decreasing the neutral temperature (as has been done by Berlande, Goldan, andGoldstein), and increasing the pressure and degree of ionization (as has been done byAlexeff and Neidigh). If the features of both these experiments were combined, it shouldbe possible to detect the amplification of a propagating sound wave fairly easily.S. D. Weiner, K. U. IngardReferences1. J. Berlande, P. D. Goldan, and L. Goldstein, Appl. Phys. Letters 5, 51-2 (1964).2. I. Alexeff and R. V. Neidigh, Phys. Rev. 129, 516-27 (1963).B. HYPERSONIC RELAXATION EFFECTS AND BRILLOUIN SCATTERINGIn a previous report,l some preliminary results of hypersonic velocity and absorp-tion measurements in acetic acid were presented. These measurements were made byobserving the frequency shifts and broadening in light from an He3-Ne laser scatteredby the liquid's thermal phonons. The sound propagation characteristics are inferredfrom the scattered light through the equationsvv = + 2v 1 c sin 0/2s o ca vS S6vs 2r 'where v and vs are frequencies of incident light and hypersound; 0 is the scatteringangle; as is the sound absorption coefficient; ri is the liquid's refractive index, and cis the velocity of light in vacuo; and 6vs is the broadening in frequency of the scatteredlight.In the previous report1 the possibility of resonant acoustic damping in acetic acid,suggested by small oscillations observed in the velocity with frequency and by someroughly corresponding peaks in the total scattered linewidth, was expressed. Sincethese initial measurements were taken, sufficient improvement has been made in theaccuracy of the experiment to rule out any acoustic resonance in acetic acid between1 kMc and 6 kMc of sufficient strength to cause a velocity change Avs/Vs > 0. 1%. Thepreviously observed peaks in linewidth were attributable to fluctuations in the maserpower.As Fig. VIII-1 shows, we have now observed a new relaxation in acetic acid. Otherliquids that were studied either showed no dispersion from the ultrasonic (~1 Mc) to thehypersonic (~6 kMc) ranges or exhibited relaxations. Table VIII-1 summarizes the datafor these liquids. Those liquids exhibiting an increase in sound velocity from theQPR No. 78(VIII. PHYSICAL ACOUSTICS)ultrasonic to hypersonic range, evidence acoustic relaxation processes that are activein the kilomegacycle frequency range. Thus far, no resonances have been observed inthe damping of hypersonic waves.11801170E>- 1160o1150-j._ ULTRASONIC> 1140 - VALUEI 2 3 4 5/s N kMcFig. VIII-1. Hypersonic velocity in acetic acid values normalized toAVT = 21. 6°C by using AT -4. 8 m/sec/C.Previous work in the hypersonic range has suggested a possible resonance in H20,2attributable to reported decreases in the sound velocity with increasing frequency.These results, however, were reported only for single hypersonic frequencies. There-fore, we measured sound velocity in H 2 0 at a variety of frequencies (1-7 kMc) and tem-peratures (10 0 -50'C). We found no departure from the 5-Me values reported byGreenspan.3The high accuracy (<0. 1%o) of our velocity measurements for maximum scatteringangle (0=1800) was achieved largely through a greatly improved signal-to-noise ratioresulting from the arrangement shown in Fig. VIII-2. This arrangement increases theintensity of the light scattered into the receiving system by a factor of approximately100, as compared with that viewed in the variable angle arrangement. 4 Also, a newHe 3 -Ne laser, capable of a continuous output of 65 mW at 6328 A, was constructed andused for these experiments. A much improved Fabry-Perot interferometer, capableof resolving 150 Mc, was also used.Because of the different line shapes of the maser, the Fabry-Perot interferometer,and the phonon broadening, the extraction of 6vs from the total observed scattered line-width is cumbersome. A direct subtraction of the instrumental linewidth (maser plusFabry-Perot) from the total linewidth is presented in Table VIII-1. Such subtraction isonly valid when all line shapes involved are Lorentzian. In the present situation thisprocedure gives phonon broadenings 10-30% too small. A numerical procedure to correctQPR No. 78LiquidAcetic AcidBenzeneTolueneNitrobenzenebCC14CS2H2ocAcetoneEthyl EtherTable VIII- 1.v (~1 Mc)a1144 (m/sec)13241324146292011581488. 611901000Comparison of ultrasonic and hypersonic velocity data inliquids with and without relaxation effects.v (0= 180 0)1180 (m/sec)150113761558104612501488. 01190999T oC vs (0=180 )22. 00 5. 11 (kMc)21. 90 7. 1021.80 6.4922. 50 7. 6222. 50 4. 8221. 50 6.4022. 00 6. 2621.60 5. 0821.40 4.28(6vs)sub370 (Mc)24548067043055200175180aK. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press,New York, 1959), pp. 506, 419, 357. The temperatures of the ultrasonic measurements were notspecified beyond identifying them as "room temperature." (6vs)ext is the linewidth predicted by ultra-2sonic absorption data, when scaled up as v to kMc frequencies.I, sUJ. R. Pellam and K. K. Galt, J. Chem. Phys. 14, 608, 1946.CSee M. Greenspan and T. Tschiegg, J. Res. Natl. Bur. Std. 59, 249 (1957).(6vs)ext550 (Mc)8, 4007351,2001,28543, 900220155(VIII. PHYSICAL ACOUSTICS)PRESSURE 0.3-mm- SCANNING PINHOLEFABRY-PEROT7326Fig. VIII-2.REFLECTOR VARIAN G-14CONSTANT RECORDERFLOW- RATEREGULATORARGONExperimental arrangement for observation of Brillouinscattering at 1800. The maser is linearly polarizedperpendicular to the scattering area.this error is now being worked out. The present results on linewidths are only semi-quantitative.These experiments were done in collaboration with R. Y. Chiao of the Optical MaserGroup, M. I. T., which is directed by Professor C. H. Townes and Professor A. Javan.P. A. Fleury VReferences1. P. A. Fleury V, Quarterly Progress Report No. 76, Research Laboratory of Elec-tronics, M.I.T., January 15, 1965, p. 67.Y. Chiao andB. Lastovka,B. P. Stoicheff, J. Opt. Soc. Am. 54, 1286 (1964); G. B. Benedek,K. Fritsch, and T. Greytak, J. Opt. Soc. Am. 54, 1284 (1964).3. M. Greenspan and C. Tschiegg, J. Res. Natl. Bur. Stds. (U. S.) 59, 249 (1957).4. P. A. Fleury V, op. cit., see Fig. XII-1.The research of this group is supported in part by the National Aeronautics andSpace Administration.QPR No. 78LIQUID CELL2. R.J.

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