A THEORETICAL AND EXPERIMENTAL STUDY OF THE PROPAGATION OF PLANE FINITE AMPLITUDE WAVES IN REAL FLUIDS Thesis for the Degru of Ph. D. MICHIGAN STATE UNIVERSITY William Wrighi‘ Lester I963 ‘1:— 'MM“:Inlmllmn"I‘flmmnmmfln : 1293 01743 0392 This is to certify that the thesis entitled A THEORETICAL AND EXPERIMENTAL STUDY OF THE PROPAGATION OF PLANE FINITE AMPLITUDE WAVES IN REAL FLUIDS presented by William Wright Le ste r has been accepted towards fulfillment of the requirements for Ph D. degree in_Eh)L§i.C.5 Ximm Major professor Date November 13. 1963 0-169 LIBRARY Michigan State University ABSTRACT A THEORETICAL AND EXPERIMENTAL STUDY OF THE PROPAGATION OF PLANE FINITE AMPLITUDE WAVES IN REAL FLUIDS by William Wright Lester Longitudinal elastic waves of large amplitude prOpa- gated in real fluids exhibit a change in form and amplitude as they travel. An initially sinusoidal plane, finite amplitude wave is of special interest. A theory is pre- sented which predicts the harmonic structure of such a wave on the hypothesis that interactions between harmonic com— ponents of the wave are weak compared with the processes which generate and absorb harmonics. The result is given as an infinite sum of infinite series, and is a function of two parameters, one which specifies the initial conditions, and one which specifies the distance of travel in a reduced form. Numerical values of the predicted fundamental, second, and third harmonics are tabulated for general use; the numerical values of the second and third harmonics have been computed for a wide range of the governing parameters. The general behavior of the harmonic structure is as expected from plausible arguments and the results of other investigators. An experimental investigation making use of pulse techniques in water in conjunction with Optical methods William Wright Lester verifies essential aspects of the theory. The accurate transducer calibration required to specify both the initial value and reduced distance parameters of the theory is accom- plished by observation of light diffracted by the ultrasonic pulse at the transducer face. Temperature stability of the calibration of barium titanate transducers is demon- strated. The equivalence of the finite amplitude waveform for pulsed and continuous waves of identical frequency and initial pressure amplitude is also demonstrated, so that pulsed and continuous wave methods may be used inter— changeably. An investigation of the behavior of the fundamental frequency component of a finite amplitude wave is performed using a two transducer pulse technique at 5 MC. The behavior of the fundamental frequency component as a func- tion of pressure at fixed distance is obtained, and con- verted to plots of pressure versus distance at fixed initial pressure. The fixed distance method is found to work best for small values of the initial value parameter. The average and maximum values of the absorption coefficient of the fundamental frequency component of a finite amplitude wave are found to be linear functions of the initial pres— sure amplitude. A value for the nonlinearity parameter B/A of water is obtained from absorption measurements using weak shock theory. It is found that there is a maximum amount of fundamental sound pressure amplitude which can William Wright Lester be transmitted over a given distance. Distances as large as 50 cm and initial pressure amplitudes to 15 atm are used. The two transducer pulse technique used to investi- gate the fundamental frequency component is extended to the case of the second harmonic at 5.0 MC, and is found to perform well for larger values of the initial value para- meter in this case. The receiving transducer is calibrated by allowing a finite amplitude wave of previously measured second harmonic content to fall upon it while measuring the output with a tuned receiving system. The absolute measure— ment of second harmonic content is performed by light diffraction techniques using continuous waves. Light diffraction techniques are also used to verify the theory for the values of the fundamental, second and third harmonics simultaneously for a range of distances for one initial pressure amplitude. Initial pressures as large as 1.9 atm and distances as large as 80 cm are used. Compar- ison of the second harmonic with the theoretical predictions indicate satisfactory agreement. A THEORETICAL AND EXPERIMENTAL STUDY OF THE PROPAGATION OF PLANE FINITE AMPLITUDE WAVES IN REAL FLUIDS By William Wright Lester A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1963 ‘t; \ 6\\ S ‘c 1.)- :11: Q\c j‘ .49 3,44 ACKNOWLEDGMENT The author wishes to express his gratitude to Professor E. A. Hiedemann for his guidance in this work. Thanks are also due to Dr. K. L. Zankel, Dr. B. D. Cook, Dr. M. A. Breazeale, and Dr. W. G. Mayer for many interesting and helpful discussions. The financial support of the Office of Naval Research, U. S. Navy, and the U. S. Army Research Office (Durham) is gratefully acknowledged. TABLE OF CONTENTS CHAPTER I. INTRODUCTION II. THEORY OF THE PROPAGATION OF PLANE, FINITE AMPLITUDE WAVES . . . . . Fundamental Relations . The Phase Velocity of a Plane, Finite Amplitude Wave. . Solution for the Case of a Plane, Initially Sinusoidal Wave Discussion. . . . . III. AN EXPERIMENTAL STUDY OF THE PROPAGATION OF PLANE, FINITE AMPLITUDE WAVES . Pulse-Optical Methods Introduction Procedure Calibration of a transducer Waveform distortion in an ultrasonic pulse. The Fundamental Frequency Component General. Experimental apparatus. Experimental results The Second Harmonic Frequency Component General. Experimental arrangement Transducer calibration. Experimental results IV. SUMMARY. BIBLIOGRAPHY PAGE LIST OF TABLES Fundamental Frequency Component of a Plane, Finite Amplitude Wave P1(K)/Pl(O) Second Harmonic Component of a Plane, Finite Amplitude Wave P2(K)/P1(O) Third Harmonic Component of a Plane Finite Amplitude Wave P3(K)/Fl(05 Available c< must be multiplied in order to obtain the absorption coefficient of the nth harmonic in the medium of interest. Fundamental frequency. Discontinuity distance for dissipationless case. Reduced distance. lOX/L. Pressure amplitude of the nth harmonic wave com— ponent, measured at reduced distance K. Harmonic generation parameter (Fox and Wallace). Wavelength. CHAPTER I INTRODUCTION For a long time, it has been known that an exact solution to the equations of hydrodynamics predicts a change in form of a longitudinal elastic wave as it travels (l,2,3,4,5). These early theoretical investi- gations were performed for the simple case of a plane wave in a nondissipative medium, and showed that a simul- taneous solution to the nonlinear differential equations of hydrodynamics and the equation of state could be obtained. This solution was in a form such that the value of the prOp- agation velocity of phase points on the wave depended on the pressure-density relation at that point and the local particle velocity. Thus, the usual case is that the points of higher pressure (particle velocity) travel faster than the points of lower pressure, and the wave becomes dis- torted, or acquires various Fourier frequency components as it travels. These early investigators classified these waves as finite in amplitude, in contrast with the usual case of infinitesimal amplitude waves, because the solutions to the infinitesimal amplitude case were obtained from the equations of hydrodynamics and state by neglecting small quantities of second order, while it was necessary to consider second order quantities in order to show the change in wave shape one should actually have. Of course, it is never really correct to neglect the second order quantities, so that, strictly speaking, all waves are finite in amplitude. However, if one considers that the medium is dissipative, it can be shown that only waves of large amplitude exhibit the appreciable change in form predicted by the nondissipative theory; the absorptive processes in the medium Oppose the generation of higher Fourier compon- ents as the wave travels, cancelling the rather weak harmonic generation process in the case of small amplitude waves. Hence, one has the correct division of wave prOpagation processes in real media into two categories: large and small amplitude. Special interest is drawn to one case since it is easy to investigate and commonly encountered, and that is the case of an initially sinusoidal, large amplitude plane wave in a real (nonlinear, dissipative) fluid. The more recent theoretical approaches to this problem have been of two general types. First of all, approximation methods making use of some assumption regarding the manner in which the harmonic components of a wave are created and absorbed in the medium have been used (6,7,8). Second, approximate solutions to the nonlinear differential equa- tions have been sought, either by perturbation technique or other means (9-14). In general, these theories predict that an initially sinusoidal wave of finite amplitude develops a Spectrum of higher harmonic Fourier components as it travels, the second harmonic being the most important, and that these higher harmonics first grow rapidly, rising to a maximum at some distance from the source, and then decline slowly. Since the higher Fourier components all come to a maximum in the same neighborhood, and then decrease slowly (as does the fundamental frequency component), one may Speak of a region in which the wave shape is compara— tively stable, or a "stabilization distance" of travel from the source. The absorption coefficient for a finite amplitude wave is also different from that of a sinusoidal wave. Generally Speaking, the absorption loss for a sin- usoidal wave travelling in a Simple fluid is due to two causes; one, viscosity, and two, heat conduction between the hotter, compressed parts of the wave and the cooler, rarefied parts. Analysis of each of these mechanisms predicts that the losses for such a wave of pure frequency should be prOportional to the square of the frequency. In the finite amplitude case, energy must be taken from the fundamental frequency component of the wave in order to support growth of the higher harmonics, and Since one ex- pects them to be absorbed much more strongly than the funda- mental frequency component, the overall energy loss in the wave per unit of distance may be much greater than in the case of a small amplitude wave, and must also depend on the distance. Likewise, one may speak of a partial absorption coefficient for a given harmonic component, and it will in general depend on the distance and pressure ampli- tude of the wave. Experimental investigations of the prOpagation of plane, finite amplitude waves have been performed in both gases (6, 15,16) and liquids (7,17-27). Generally Speaking, these investigators have made use of either spectral analysis of the waveform or observation of the actual pres— sure waveshape or intensity as a function of the distance, thereby obtaining the harmonic structure and absorption coefficient of the wave. However, the amount and range of useful experimental data available is still quite small for several reasons. First of all, it must be emphasized that the theoretical problem at hand is an "initial value" prob- lem, that is, at zero distance a sinusoidal wave of known amplitude and frequency is postulated in a medium whose prOpertieS are known, and it is the task of the theory to predict what becomes of the wave at other times and dis- tances. It thus becomes the first task of the experimenter to provide an accurate measurement of the waveform ampli- tude at zero distance; the measurement of the wave frequency and prOpertieS of the medium are comparatively easy. Confining our attention for the remainder of the dis- cussion to the case of liquids, measurements of the waveform amplitude have been performed by a number of methods, such as calorimetry (23,17,19), thermal probe receivers (23), radiometry (23,7), and theoretical transmitter reSponSe (27). However, there are many difficulties inherent in these methods. For example, two methods may be used simultaneously and yet give grossly different results (23). Also, it is difficult to make an estimate of absolute error, which is essential if one is to make accurate Judgments when com- paring results with theory. Again, Some methods do not work well at small distances of travel from the sending transducer, forcing extrapolation of pressure versus dis- tance curves to zero distance (7), a very dubious procedure in the finite amplitude case. These problems can all be resolved with the use of Optical techniques for absolute pressure measurement. One very troublesome problem in the case of methods making use of continuous waves is the possibility of stray reflections in the apparatus, giving rise to interferences and standing waves which must be avoided in order to obtain accurate comparison with theory. It is, therefore, advan- tageous to use pulse methods, so that reflected wave effects can easily be separated in time Of arrival from direct wave effects. This introduces the experimental difficulty Of making absolute sound pressure measurements of an untrasonic pulse by Optical means, but it will presently be seen that this difficulty has been resolved (28). The choice of liquid which one wishes to perform ex- periments on will be dictated, apart from considerations of convenience and safety, by the necessity of knowing as accurately as possible those physical constants of the liquid which appear from the theory to govern finite ampli- tude wave propagation. An examination of the theory at hand (8) shows that such properties as the absorption coef- ficient for small amplitude waves, a parameter specifying the mechanical nonlinearity of the medium, and the sound velocity for infinitesimal amplitudes, are needed in an accurate form, as well as other parameters not usually troublesome. These prOperties are best known and most in- vestigated for water (29-31). The first attempt at verifi- cation of the theory was, therefore, made by means of measurements in water. It will, therefore, be the task of the present dis- sertation to formulate a theory describing the prOpagation of plane, finite amplitude waves in a dissipative fluid, and to investigate and tabulate the prOperties of that solution. In addition, an experimental investigation making use of pulse techniques in conjunction with Optical methods in water will be presented as verification of the theory. CHAPTER II THEORY OF THE PROPAGATION OF PLANE, FINITE AMPLITUDE WAVES Fundamental Relations In general, a plane elastic wave traveling in an infinite, nondissipative fluid exhibits a change in form “as it travels. This fact can be seen from the nonlinear form of the equations of motion and state. For example, in Eulerian coordinates one has the equations of motion dP/dx = -/o Eiu/dt) + u(du/d;§] (1) %g + 6% (flu) = o, (2) the first representing a force law, and the second, the conservation of mass. The adiabatic equation of state may be taken as the series expansion to terms of second was was or, alternatively, one may use the gas-like equation of order state mo -_- (f/fbf‘ (A) both of which are nonlinear, applicable to the fluid state, and equivalent under certain conditions. It Should be noted that in the case of a gas,‘3’is the ratio of the Specific heats Cp/Cv; in the case of a liquid, however, it iS an empirical constant. A solution to Eqs. 1 and 2 has been known in implicit form for some time. Simply stated, any initial wave func- tion F (X) can be prOpagated in the positive direction with a velocity C', where l c' = (dB/<30)? + u (5) is the velocity of phase points of the wave (4). C' is Simply interpreted as the sum of the sound velocity C = (dP/dffll and the velocity of the moving medium U at the phase point. The consequences of this are apparent: any function 1 u = F [X - ((dP/gflg + [1)] (6) represents a wave travelling the positive X direction1 Satisfying Eqs. 1 and 2, and Since in general (dP/dP)E + u :> CO for u:>»o, a discontinuity develOpS in the wave after a certain distance of travel, for the points of greater particle velocity overtake the points of lesser particle velocity. For an initially sinusoidal wave, this distance, the so—called discontinuity distance L, is 3 L _ B00 (7) ’ 7T(B/A + 2)Pl(o)7) At the distance L, the wave deveIOps an infinite SlOpe at its point of zero particle velocity. One notes that the group velocity of the wave is un- changed, as the points of zero particle velocity move at the velocity of sound CO, where i 00 = (dP/dp) . (8) u: o The wavelength is constant, as the points of zero particle velocity maintain their relative separation as they travel. The pressure and particle velocity are simply related in a finite amplitude wave, and one can introduce the acoustic impedance/F500 so that P - PC = prcou (9) The Phase Velocity of a Plane, Finite Amplitude Wave Expansion of Eq. 5 in series, making use of Eq. 3 and 9, yields the series in u to terms of second order _B_s___3_ 2.2.2.2 C' = CO 1 + l + 2%) CO 8 A2 CO + . . . (10) where the identification from Eq. 8 1 Co = (A900)? (11) has been made. One may evidently neglect the second order term in u for cases where lO 2 3 B u 8 A2 CO 1 12 B + 1 << < > 2A." B In typical liquids, we have E'ole and CO n~v105 cm/sec, so that 3 B2 u 3 8 A2 Co 10’ u E + l "’ "TR§" EA' (13) Thus, the second order terms in u become comparable with the first order terms for pressures of the order P - Po":f%Co (16 x 103)nu16 x 108 dyneS/cm2 (1A) that is, for waves of the order of 1600 atmospheres of peak pressure. For waves of considerably less than 1600 atmos- pheres pressure, one may take as the velocity of phase points of the wave B 0' = C0 + (l +'§K ) u. (15) If terms in u2 must be included in Eq. 15, the equation of state (3) would probably be inadequate, and a third order term could be included. A similar calculation, beginning with the alternate equation of state Eq. 4, making use aS before of Eqs. 5 and 9, yields 2 c'=co+(I—HJ+X(X';)M'3) (a) +. . .,(16) and one may again neglect second order terms in u if K(X—1>(i—3)u A ct I<< l (l7) Supposing for the moment that one is dealing with a gas, takeXN 1.5, and CO~3 x 105 cm/sec,JDO~ .0012 g/cc (corresponding roughly to the case of air at atmOSpheric pressure). Then xir- 12(X- p, =I1(X- nur- 3) (Pi—Poi]: 1, A (3+ l)CO I A 05+ lfico3 I (18) if P - POrNu’l.7 x 1014 dyneS/cm2. (19) That is, one may use the expansion for the phase velocity C' = C0 + (}:%—l—)u (20) in gases at atmOSpheric pressure if the peak wave pressure is considerably less than 1.7 x 108 at m. The equations of state, Eqs. 3 and A, are both applicable to either gases or liquids, and they are equiva- lent as far as the propagation of elastic waves is concerned in the approximation that the square of the particle velocity may be neglected. Comparison of Eqs. 15 and 20 shows that one may take l2 +l=X. (21) >un Also, since Eq. 7 follows from the phase velocity Eq. 5 as expanded in Eq. 15, one must have the alternate form for Eq. 7 as follows: 3 9%. L = 7T<‘o’ + mm); (22) Solution for the Case of a Plane, Initially Sinusoidal Wave Fubini—Ghiron (32), Keck and Beyer (13), and Hargrove (33) have given a series solution of the finite amplitude problem for the dissipationless case which describes the harmonic wave structure as a function of the distance. This solution, for the case of an initially Sinusoidal wave, is given by Hargrove in the form P(K) = 2F1(o) i (.i)n+1 Jn(nK) Sin 27mph— 3;) n = l -—jfi?—— (g3) and is valid only for distances X5; L, i.e., for1{:£lu It Should be noted that Eq. 23 is obtained from Eqs. 6, 9, and 15 or 20, and is valid only where these hold. In particular, as has been stated, Eq. 23 should hold in the case of simple liquids for waves of finite amplitude provided Pl(0)<< 1600 at. This result (Eq. 23) allows one to write a Fourier series for the pressure components of an initially sinusoidal wave in a dissipative medium. 13 In a nondissipative medium, examination of Eq. 23 (see Figs. 1, 2, 3 for c£.L = 0) shows that the wave begins with a pure frequency fl , and develops higher harmonics of fre- quency nfl at the expense of the component of frequency?) . Let this mechanism of the shift of pressure from one harmonic to another be called the "transfer" mechanism. Inasmuch as it is a function of the distance, one may write the Spatial derivative (dPnIK)/dK)transfer = 2Pl(o>(“1)n+1 g-KEQE'é—Hfl SianVt-fi) (24) In the absence of finite amplitude effects, the Space rate of change of the pressure due to absorption is simply prOportional to the total pressure of a given harmonic. If the absorption mechanism is heat conduction between the hotter, compressed parts of the wave, and the cooler, rare- fied parts, or if it is viscosity, the absorption is also prOportional to n2. For the sake of generality, let the prOportionality for the nth harmonic component be given by f(n). Then (dPn(K)/dK)absorp = -f(n) ocLPn(K)total (25) The total Space rate of change of the amplitude of a harmonic component is now assumed to be the sum of the rates of change due to harmonic transfer and harmonic absorption: l4 (dPn(K)/dK)total = (dPn(K)/dK)transfer 'fIHA9 ln [an + O.l)/Pn(KBtransfer = SAT), (29) but as can be verified numerically from Eq. 23 and the graph- ically determined values from Fox and Wallace, taking into account the change in Sign of 81(K) which they introduced. Equation 26 may, therefore, be taken as approximately equivalent to the Fox and Wallace equations, except for the l5 inclusion of the factor exp [81$] in their result for the second and third harmonics. Comparison shows, however, that <51 is small in comparison with 8; or 3 for all but the largest K values. Integration of Eq. 26, using Eq. 24 gives an integral equation of the second kind: K 2P1(O)Jn(nK) nK Pn(K) = — f(n) 04L Pn(K') dK' (30) O The solution of Eq. 30 by the method of successive substi- tutions (34) (assuming oéL approximately constant) is an infinite alternating series for the nth harmonic amplitude of the form Pn(K) = An(K) - Bn(K) + Cn(K) — Dn(K) + En(K) ..... + ..... (31) The first five terms are found to be An(K) = 2Pl(0) Jn(nK)/nK (32a) /'¢so Bn(K) = 2P1(0):(n) OCL E 2Jn + 2q (nK) _ Jn(nK) n ) q=0 2 2 2 2 {To (3 b) cn = ““03; (“KL 2 <2q-1) Jn + 2.1-1 (mo q=1 (32c) .0 g 3 3 3 Dn(K) = Pl(0): (n)0< L 2 qun + 2g (nK) 1’1 q=l (32d) l6 ob 4 4 4- End) = 16W“): (“M L E i2 S Jn+e(q+r)+i 1’1 r=0 q=1 (32e) The solution is thus expressible as the series 00 n+1 P(K) = E (-l) Pn(K) sin27Tn (7) t 7% ), (33) n=l where the Pn(K) are the harmonic amplitudes given by Eqs. 31 and 32. We must take note of the fact that, as Eq. 23 holds only for K fézl, Eq. 33 also holds only in this region. For constant «L, the series Eqs. 31 and 32 converge absolutely and uniformly in K é 1. Terms following those given in Eqs. 32 may be obtained by successively multiplying Eq. 32e by -2f(nk9(L/n, adding 2s+l to the order of the Bessel function, and summing over s from zero to infinity. As expected, the correction terms in Eqs. 32 are seen to cause the predicted pressure in any harmonic to be less for a given K value than that predicted by Eq. 23. Discussion For a given absorption law f(n) and reduced distance, the solution is evidently a function only of the product 0(L. Graphs of Eq. 31 for the fundamental, second and third harmonic amplitudes are given in Figures 1, 2, and 3, with the curves from Eq. 23 (dissipationless) for comparison. 17 The 9<3 mooeAAmza meAzAm .mzaAm A mo ezmzomzoo oAzozmA3 mooeAAmzA meAzAm .ozaqm A no ezmzomzoo HHH mqm<8 OHzozm.O5. The experimental work pre- sented below Spans the rangeio(l.= .l to 1.7, and compares results with theory for the second harmonic in that region over a range of distances. Experimental arrangement. The electronic apparatus is shown in Fig. 19. A pulser or a continuous wave source, 42 each set to 5.0 mc, may be chosen by means of a switch. The second harmonic component of the wave is received by a 10 m0 barium titanate transducer and passed through a filtered amplifying system so as to display the 10 me Second harmonic frequency component on an OSCillOSCOpe. The medium is distilled water; the transducers are in rotating mounts about vertical and horizontal axes to allow for alignment, and can be translated to a separation of 80 cm. The filtered amplifying system is prevented from overloading by means of the decade attenuator,which also provides a test 0f the linearity of the system while in use. Distance measurements are made by reading the time of traversal of the pulse between the two transducers on a radar range calibrator oscilloscope, which is accurate to about .1% of the elapsed time. The pulse velocity is assumed to be 1.5 x 105 cm/sec. The sending transducer is a l x 1 inch barium titanate ceramic type, and the receiving transducer is either a 3 x 3 mm or 1/2 x 1/2 inch barium titanate ceramic, depending on the sensitivity required. Transducer calibration. The receiver-tuned amplifier system for the second harmonic is calibrated in actual use by allowing a wave of known second harmonic component to fall on it, while measuring the actual voltage response of the entire system. Two methods of Obtaining a wave of known second harmonic content are used, and both give the same end result. 43 First of all, at small reduced distances and small amounts of second harmonic, one would expect the Simple dis- sipationless theory to be valid. Then it should be the case that the smallest possible reading of second harmonic content at a given distance is correct, according to dissipationless theory, and this can be used as a calibration—point. Second of all, it is possible to verify a postulated finite amplitude waveform by observing the light diffracted by such a wave. The asymmetric ultrasonic waveforms in the finite amplitude case give rise to asymmetric light diffrac- tion patterns, and these patterns can be predicted from knowledge of the Fourier Spectrum of the wave. A number of light diffraction patterns making use of the theoretical harmonic structure of the wave for one value of the initial value parameter have been.computed. In this way, predicted light diffraction patterns have been compared with experi- mental ones every few cm. By inserting various values for the harmonic structure of the wave in the calculation, it is found that the (n -l)St orders of diffraction are very sensitive to the magnitude of the nth harmonic, at least for n = l, 2, 3. It is thus comparatively easy to separate the effects of various Fourier components of the wave from each other. Good agreement was obtained between theory and experi— ment if the fundamental, second and third harmonics of the wave were considered correct according to the theory for the 44 case 0(L = .370. This corresponds to an initial pressure amplitude of .5 atm in the case at hand. One thus has a wave of known absolute second harmonic content which can be used to calibrate the receiver. It is furthermore possible by this means to obtain a superior calibration for the initial transducer pressure. That is, agreement between theory and experiment can be forced by varying the transducer voltage at fixed distance and obtain- ing a fit between the predicted and observed light diffraction patterns. The voltage so obtained can then be used as a cal- ibration point. By averaging a number of such forced-fit calibrations, a calibration for the initial pressure ampli- tude can be obtained which has the advantage of compensating for near field pressure fluctuations. The value of the case at hand was found to be .054, i .0053 at/volt. This probable error was Obtained from an analysis of the deviations of the measurements from the mean. There is an additional possible error of about 10% inherent in any absolute Optical calibration caused by a lack of accurate knowledge of the piezo-Optic Coefficient of water. This calibration is in good agreement with calibrations carried out near the trans- ducer face. Experimental results. The system comprised of calibrated sending and receiving transducers, with the associated electronic apparatus, could now be used to obtain the second 45 harmonic component of the wave as a function of the distance by increasing the transducer separation while maintaining alignment. There are several difficulties inherent in this scheme. For example, it is difficult to maintain correct transducer alignment while in the process of moving them. A smoothed plot of the second harmonic of the wave can be obtained by means of the fixed distance method. This method and results obtained by it were described in the section on the fundamental frequency component. Fixed distance measurements in the present case may be made by aligning the transducer system previously cali- brated, and recording the received second harmonic as a function of the initial fundamental pressure amplitude. This can conveniently be correlated with theory by calculat- ing the 0(L and K value for each value of initial pressure at which readings are taken. One can then find the appro- priate second harmonic value for the theory by referring to Table II with both oo3 mooscApcOo CA poo Amommosov mooAso :A mOOSOAAoEo endomopmuoczom AoApAcA mo GOApocsm o no moApAmcopcA pcoAA 7:...» r... .. r Coo.” 24: A on ON 0. O (flog flatwo; noose u>.>¢¢' “Icons?" swan saunas .AA .oAm .Eo MA no AovoAHOV oo>o3 msoscflpcoo CH oco Amommohov momasg CA mOOSOAHQEo oASomoAQIocoom AoAOA:A mo OOApocso o mo mOApAmcopcA pcoAA popoogmmflo poosonpmgflo UONAAoEAOz .OA .oAm mo m u:o_2