Contains reports on four research projects.U. S. Navy (Office of Naval Research) under Contract Nonr- 1841(42

Chrisman, B. L.

Ingard, K. Uno

Irwin, G. M.

Maling, G. C., Jr.

Willke, H. L., Jr.

Witting, J. M.

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VIII. PHYSICAL ACOUSTICSProf. K. U. IngardDr. L. W. Dean IIIDr. G. C. Maling, Jr.Dr. H. L. Willke, Jr.B. L.P. A.K. W.G. M.A. A.ChrismanFleury VGentleIrwinMaduemeziaW. M. ManheimerJ. H. TurnerS. D. WeinerJ. M. WittingA. SOUND EMISSION FROM AN AMPLITUDE-MODULATED RADIOFREQUENCYDISCHARGEThe generation of sound waves by means of a modulated high-frequency dischargedescribed in the last report has been further investigated by measurements of the soundpressure generated as a function of the gas pressure. The modulated discharge islocated at one end of a tube, and the sound is received at the other end. To preventstanding waves in the tube, the tube is terminated by absorptive material which prevents2"(b)8. 518.54 8 12 16 20 24 28PRESSURE (in. Hg)(e)2.5(c)0 4 8 12 16 20 24 28PRESSURE (in. Hg)(f)0 4 8 12 16 20 24 28PRESSURE (in Hg)(g)Fig. VIII-1.Pressure dependence of the sound inten-sity produced by amplitude-modulated RFdischarge in Helium. Frequency: (a) 10 kc;(b) 15 kc; (c) 20 kc; (d) 25 kc; (e) 30 kc;(f) 35 kc; (g) 60 kc.This work was supported in part by the U. S. Navy (Office of Naval Research) underContract Nonr- 1841(42).QPR No. 74(VIII. PHYSICAL ACOUSTICS)sound reflection. A typical sequence of curves illustrating the pressure dependence onthe acoustic intensity is given in Fig. VIII-1. These curves refer to generation intohelium. It is interesting to note that a "resonance" occurs at a certain pressure, andthe pressure at this resonance increases as the frequency increases. Similar resultshave been obtained for argon. This effect is not yet understood and further work is inprogress.B. L. Chrisman, G. M. Irwin, U. IngardReferences1. B. L. Chrisman and G. M. Irwin, Investigation of ultrasonic coupling and relax-ation effects at a corona-neutral gas interface, Quarterly Progress Report No. 73,Research Laboratory of Electronics, M.I.T., April 15, 1964, pp. 23-24.B. FLOW STABILITY AND THE ENERGY IDENTITYDifficulty with a class of perturbed flows of irregular time dependence, whose insta-bility would seem to indicate creation of energy, has led to an investigation of the energysources for unstable flows. It will be seen that, in an exact analysis, the total kineticenergy of perturbation tends to zero with time for perturbations not affecting the strengthof the energy sources for the basic flow.The energy identity for the Navier-Stokes equations is obtained formally as follows:0 = d3 y Vi(ati- vi i j j i+iV=~ e d32 i i jd= 3yt -vviAv + v a j(2PVClearly, if v 0 0, and if the boundary conditions on P and the vi are so prescribedthat the surface integrals in (1) vanish, then the v. will tend to zero for large t. In par-2 1ticular, if v is integrable, we have' v2 3 3t 2 d3y = -v (v ) 2 d3y < 0,so that v2 d3 y must continue to decrease in time as long as one of theQPR No. 74(VIII. PHYSICAL ACOUSTICS)v. is nonzero somewhere.1When applied to the stability problem for parallel flows, the energy identity demon-strates for us the nonexistence of spatially localized instabilities: plausibly enough, theperturbation must encompass an energy source for the basic flow.If we suppose the basic parallel flow to be directed along the x 3 coordinate, we maywrite for the perturbed flow v i = u i + W(x 1 , x Z ) i3 P = p + x3Po, with -vAW6i 3 +a.x P = 0.13OIn order to avoid difficulties associated with an infinite total energy5 dx 1 dx2 5 dx 3 w 2for the basic flow, we consider the term-by-term limit of the difference between theenergy identities associated with the perturbed and unperturbed flows as we extend thedomain of integration to the whole space. Supposing that the u i are integrable andvanish at rigid boundaries and, for simplicity, that W is bounded, and suppressing inadvance terms that vanish in the limit, we get&s5 d 3 y at + v(a jv) 2 - v(2jWi) + 5 dS[-vWanu 3 + 6 n 3 Wp+ "Vd3 t - (jui )2 - v(W) u3 + dS[-vW u3 +n 3Wp+. ..V V Vd3 y ' + v( ui 2 + dS[-vW U 3E 3  Boundaries+ lim 5 dx 1 5 dx 2 W[p(x 3 )-p(-x 3 )]. (2)x 3-If there are no sliding boundaries, then WB 0 so that (2) gives us whatwe want: If p(x 3 ) - p(-x 3 ) = o(1), then all surface contributions vanish, so that, withv * 0, d 3 v is strictly negative as long as any of the u. are nonzero. Underthese conditions, v2 cannot remain everywhere positive unless the u. tend to zero almosteverywhere in time. If sliding boundaries are admitted, these considerations hold onlyif anu 3 vanishes at such boundaries. In general, perturbations must couple to the energysources for the basic flow if they are not to decay in time; that is, the surface termsmust not vanish.In the special case v = 0, the basic flow maintains itself without energy sources,QPR No. 74(VIII. PHYSICAL ACOUSTICS)so Po = 0, and it is no longer possible to extract energy from sliding boundaries. Onthe other hand, since no sources of dissipation exist, we haved3y = 0,even when the remaining surface term vanishes, so that the perturbed flow maintainsitself in the mean.H. L. Willke, Jr.C. ACOUSTIC POWER RADIATED BY A MONOPOLE SOUND SOURCE IN A DUCTThe acoustic power (W) radiated by a sound source located near boundaries may bequite different from the power output in free space (W ).2 Ingard and Lamb andWaterhouse2 have dealt with sources near rigid surfaces. Solutions for W/W whenthe surface is partially reflecting have been presented by Grine,3 Maling,4-6 andWaterhouse7 for various boundary conditions. For a constant-strength (that is, con-stant mass flow) monopole the expression 6 for W/W° isW 1=1 +- Im (pr), (1)o kAwhere Im (pr) is the imaginary part of the reflected pressure at the source. The sourcepressure is ps = (A/r) exp(ikr), and k is the wave number, w/c. The solutions forW/W 0 that have been worked out thus far deal with sources between parallel walls, ina corner, and over an infinite plane.One interesting extension of this work is a calculation of W/W for a point-monopolesource in a duct. If the source is located at position (x , Yo, z ) in a duct that hasdimensions a and b in the x-y plane, and is infinite in the z-direction, the sound fieldcan be shown to bep(x, y, z) = 2ri wQ {cos (kxx- x ) cos (kxx - x )pxy) Qkxky  k2-k 21/2xy xycos (kyy- y) cos (kyYo_ y)} exp(ikz(z-zo)-it), (2)where p is the sound pressure at the field point, Qo is the strength of the source, ois the frequency, kx and ky are solutions to tan (knL/2) = --ikp /k n for the even modes,and cot (knL/2) = ikpn/k n for the odd modes. Here, n represents x or y, and L rep-resents a or b. A and A are normalization constants, and k = (k2-k-2 1/2. Also,x y z xy/QPR No. 74(VIII. PHYSICAL ACOUSTICS)n = -tan- 1 (ikpn/k ).For hard walls in a duct ( n = 0), and the power output of the source can be foundby integration of the acoustic intensity in the propagating modes. For example, theratio W/W 0 is 2rr/k2ab for the plane-wave mode, and can be found easily for the highermodes. Thus, W/W 0 can be found as a function of k for a constant-strength source.If the source is of the constant-pressure type, it is necessary to evaluate Eq. 2 nearthe source, and remove the asymptotic eikr/r dependence in order to find both the realand imaginary parts of the reflected pressure at the source. Both are needed becausethe mass flow produced by the source must adjust itself to keep the total pressure atthe surface of the source constant.If the duct walls are hard, or if a plane-wave reflection coefficient can be used todescribe the effect of the boundary, an image description can be used to find the reflectedpressure at the source. It can be shown that the ratio W/W for a constant-pressuresource of radius a is1W 1 + k Im (pr)Sr(3)W o 1+ + 2a Re (pr ) + a pr l 2Note that p is assumed to be of the form eikr/r, and therefore has dimensions 1/length.Image solutions have been found for a source in the center of a square duct with rigid3.53 k = 129IMAGINARY PART2 m:66n=66Sm 7011 0 REAL PART IMAGINARY PART n= 70CD_ m=63I n =40ARRAY SIZE--REAL PART m 70REAL PART n z7,-2Fig. VIII-2. Real and imaginary parts of the reflected pressureat the source for k = 12. 9. The abscissa is thearray size, the points being those that satisfy thecriteria that have been discussed.QPR No. 74(VIII. PHYSICAL ACOUSTICS)walls (side length = L) by programming an IBM 7094 computer to sum up the images.The program was written so that the pressure field from groups of images (at least fourimages to a group) were printed out when the total image strength was less than one.This procedure insured that the convergence would be reasonably rapid. The cumulativetotal was printed out for these groups. A typical calculation is shown in Fig. VIII-2.The calculations are cut off when the array of images reaches 70 X 70, and it can beseen that a 66 X 66 array (for the imaginary part of the pressure) and a 63 X 40 array(for the real part of the pressure) are much more consistent with earlier groups thatmeet the criterion discussed above than the 70 X 70 array calculation which does not.Thus, it is important to choose carefully the order in which the terms are taken toinsure -convergence of the partial sums of the series.The ratio W/Wo can be found for a constant-strength source either by use of theGreen's function method described above, or by the summation of images. The lattermethod was used in the calculation of the curves shown in Fig. VIII-3. This plot empha-sizes the difference between the power output of the constant-strength source in freespace and the power output in a duct. An alternative method of treating the data is toplot the power output relative to its value at k = 0. This has been done, and the resultsare shown in Fig. VIII-4. It can be seen that coupling to the higher order modes is con-siderably stronger than the coupling to the plane-wave mode (k < 6. 28 for the plane wavemode, the first cross mode is not excited).For a constant-pressure source the image technique is most convenient, and,109876WWo 5CONSTANT STRENGTH MONOPOLE SOURCE43-10 20 21 22 30 31 23 40 41 33 42 34 51 52 44 53 602 -iA i I I I0 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19kLFig. VIII-3. W/Wo vs kL for a source in the center of a hard duct.0QPR No. 74I I I I II0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19kLFig. VIII-4. W/Wk=0 vs kL for a source in the center of a hard duct.I a= OIL2 a= OO5L3 a=OOIL2' 23 3232 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19kLFig. VIII-5. W Q/Wp as a function of kL.QPR No. 74(VIII. PHYSICAL ACOUSTICS)according to Eq. 3, the ratio W/W is the same for either type of source if a = 0. Fora * 0, the image method is only approximate, but should be valid, provided that thesource size (a) is small compared with the x (or y) dimension (L) of the square duct.The ratio WQ/Wp has been plotted in Fig. VIII-5 for three values of a, a = 0. 01 L,a = 0. 05 L, and a = 0. 1 L. Here, Wp is the output of a constant pressure source, andWQ is the output of a constant strength (volume velocity) source.It can be seen from Fig. VIII-5 that for small sources the power output of a constant-pressure source is nearly the same as for the constant-strength source. For largersources, the difference between the two types becomes greater. Since the imagedescription is strictly valid only for very small sources, no calculations were made forvalues of a/L greater than 0. 1.G. C. Maling, Jr., U. IngardReferences1. U. Ingard and G. L. Lamb, Jr., J. Acoust. Soc. Am. 29, 743 (1957).2. R. V. Waterhouse, J. Acoust. Soc. Am. 30, 4 (1958).3. D. R. Grine, Quarterly Report, Acoustics Laboratory, M. I. T., July-September1957, p. 15.4. G. C. Maling, Jr., Quarterly Progress Report No. 51, Research Laboratoryof Electronics, M.I.T., October 15, 1958, pp. 107-110.5. G. C. Maling, Jr., J. Acoust. Soc. Am. 31, 115 (1959).6. G. C. Maling, Jr., J. Acoust. Soc. Am. 36, 781 (1964).7. R. V. Waterhouse, J. Acoust. Soc. Am. 35, 1144 (1963).D. STUDIES OF SHOCK STRUCTUREThe observation of hydraulic jumps to shed light on the structure of collisionlessshock waves in plasma was reported in the last quarterly report. I These previousexperiments showed that neither the "thinnest shock hypothesis" nor the "thickest shockhypothesis" has general validity. The experiments have continued. The new results,when carried over to the plasma and coupled with some new plasma theory, containsome novel features.Hydraulic-jump structure has been examined at various water depths as a functionof Mach number. At small Mach numbers, below M = 1. 25, the structure is charac-terized by waves whose length is fairly long, compared with the depth of water. Thesewaves are a permanent feature of the jump structure, in that they travel at exactly thesame speed as the jump, and so they stand still in jump coordinates. If gravity is thepredominant restoring force, these waves stand behind the jump; if, on the other hand,surface tension is the predominant restoring force (this can be obtained experimentallyQPR No. 74(VIII. PHYSICAL ACOUSTICS)by reducing the water depth to approximately 0. 5 cm), then they stand ahead.These results can be carried over to plasma. There are plasma waves, the mag-netoacoustic waves, which have all of the essential properties to make them a permanentfeature of the collisionless shock wave, like the long surface waves in shallow water.Thus, the existence of finite-amplitude magnetoacoustic waves that stand in shock coor-dinates is expected.Some simple theorizing about these waves and their role in the collisionless shockleads to the following predictions: If the magnetic field is perpendicular, or almostperpendicular, to the direction of the shock, the waves will stand behind the shock front,and will be of approximate length me/m i r., where m and m. are the masses of elec-e 1tron and ion, respectively, and r i is the gyroradius of an ion moving at the magneto-acoustic speed. If, on the other hand, the magnetic field lies away from the planeperpendicular to the shock direction, the waves will stand ahead of the shock front, andwill be of approximate length r . The transition between these regimes is smooth, andoccurs when the magnetic field is at an angle 7,me7mi away from the plane perpendicularto the shock direction.As the strength of hydraulic jumps is increased, the long wave becomes of such largeamplitude that it breaks, and the character of the hydraulic jump changes. Surface tur-bulence now appears, as does a different wave, of much shorter wavelength. Thissecond wave, like the long wave that appears for weaker jumps, stands still in jumpcoordinates. The excess energy carried by the supersonic flow into the jump front isno longer converted to waves that carry the energy downstream; instead, the energyis dissipated right at the jump front by viscous damping of the surface turbulence andstanding waves. There are two outstanding features in these stronger jumps. The factthat the long wave is no longer excited and its consequences in the case of plasmashocks was discussed in the previous report.1Another feature is the pronounced tendency for short-wavelength waves traveling atthe same speed as the hydraulic jump to be excited, as well as the fact that the scaleof turbulence is largest around this wavelength. The three ingredients necessary forthis behavior are: (i) the long wave breaks, if strong enough; (ii) short-wavelengthwaves traveling at the jump speed are possible; and (iii) standing waves are prefer-entially excited.These three ingredients are also present in the collisionless plasma. I can argueon quite general grounds that waves whose phase velocity is the same as the sourcespeed (the shock front being the source, in this case) are preferentially excited, nomatter what the medium. As the amplitude of long (that is, of order r. or /meimr.)1 e 1waves increases, because of increasing Mach number, such long waves become unstable,and so they "break" into smaller wavelength disturbances. The wavelengths that canstand are of order (me/mi)ri and of order (wc/ p)ri, where w /w is the ratio of plasmaQPR No. 74(VIII. PHYSICAL ACOUSTICS)frequency to ion-cyclotron frequency. The former mode is an electron-cyclotron wave;the latter is an electrostatic oscillation. Their presence is expected for strong col-lisionless shocks, and their decay by Landau damping and collisional damping shouldprovide the energy dissipation associated with the collisionless shock wave.J. M. WittingReferences1. J. M. Witting, Studies of shock structure, Quarterly Progress Report No. 73,Research Laboratory of Electronics, M. I. T., April 15, 1964, pp. 27-28.QPR No. 74

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