THE DIFFRACTION OF LIGHT BY ULTRASONIC WAVES PROGRESSING WITH FINITE BUT MODERATE AMPLITUDES IN LIQUIDS by KENNETH L. ZANKEL AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy College of Science and Arts 1958 Apnroved: 11 ABSTRACT KENNETH L. ZANKEL Two optical methods are used to establish the presence of finite amplitude distortion at moderate ultrasonic ampli­ tudes, One of these uses a transmission plate to Isolate the second harmonic component of the distorted wave. The other relates the distortion to the asymmetry of the diffrac­ tion pattern produced by light passing through a plane, progressive, ultrasonic wave. A method for quantitatively studying this asymmetry In order to determine the behavior of the distortion Is developed. An expression predicting the intensity of the diffraction orders for an arbritrary periodic, plane, progressive ultrasonic wave is derived. Expressions pre­ dicting the amount of second and third harmonic present under given conditions are corrected to Include absorption. Preliminary measurements of the light Intensities of the diffraction orders were made In water at 2 me./s. and me./s., at distances up to $0 cm., and at pressures between zero and 1.3 atmospheres. These approximate measurements showed the predicted dependences of the second harmonic on frequency, distance and pressure of the funda­ mental. The values so obtained agree, within experimental accuracy, with the values predicted on the basis of iso­ thermal compressibility measurements. More detailed measurements were made in carbon tetra­ chloride at frequencies of 2 and 3 me./s., at distances iii between 2.5 to 21 cm. and at pressures up to 0.6 atmospheres. The predicted dependences upon fundamental pressure, frequency and distance were, within experimental accuracy, verified. The values so obtained were consistent over a given range with those predicted on the basis of the iso­ thermal compressibility measurements. It is shown that, for given pressures at the point of measurement, and for large absorption, the amount of second harmonic Increases rapidly at first, and then more slowly, asymptotically approaching some limiting value. That is, at large distances from the transducer, the second harmonic Is, to an approxi­ mation, no longer a function of distance, but is determined by the frequency and fundamental pressure at those distances. It Is shown that the study of diffraction pattern affords a sensitive method for studying distortion at modest amplitudes. measured. Very small amounts of distortion can be The second harmonic can be measured when the intensity of the harmonic is less than 0*0\\.% of the inten­ sity of the fundamental. Distortion could be measured when the fundamental pressure was much less than 0.2 atmospheres. The agreement of the theory developed with the experimental results indicates that the type of theoretical approach used may be useful in approximately predicting the distor­ tion and its effects over a wide range. THE DIFFRACTION OF LIGHT BY ULTRASONIC WAVES PROGRESSING WITH FINITE BUT MODERATE AMPLITUDES IN LIQUIDS by KENNETH L. ZANKEL A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy College of Science and Arts 1958 ProQuest Number: 10008613 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10008613 Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346 ACKNOWLEDGMENT The author wishes to express his thanks to Dr. E. A# Hiedemann for his very helpful suggestions and guidance. Thanks are also due to Dr. M. A. Breazeale, Dr. L. A. Schmid, and other members of the department for their aid and suggestions. Especially, the author acknowledges the financial assistance given under a U. S. Army Ordnance Contract without which the work could not have been accomplished. K.L.Z. TABLE OP CONTENTS CHAPTER I. PACE INTRODUCTION ................................. Diffraction by a Sinusoidal UltrasonicWave . 1 3 Diffraction by Superposed Waves ............ 5> Pressure Measurements from Diffraction . • • 6 Distortion Due to Finite Amplitudes . • • • . 7 II. THEORETICAL INVESTIGATIONS..................11 Diffraction From Any Periodic Plane Progressive W a v e ...................... II Influence of the Absorption on Distortion . * 12 Adaptation to Diffraction Studies . ........ 13 III. DISCUSSION OF E X P E R I M E N T S ............ 15 Apparatus................................. 15 A Simple Method of Demonstrating the Presence of Finite Amplitude Distortion ............ 1? General Procedure for Observing the Diffraction Spe ctra .......................... 19 . Diffraction Spectra Obtained With Ultrasonic Waves In Water ......... 19 A) Distance Dependence......... 19 B) Frequency D e p e n d e n c e ................ 20 C) The Ratio B / A ....................... 28 Diffraction Spectra Obtained with Ultrasonic Waves in Carbon Tetrachloride............. 28 A) General Considerations • ............ B) Pressure Dependence, Second Harmonic . C) Distance Dependence, Second Harmonic . D) Frequency Dependence, Second Harmonic . E) Third H a r m o n i c ..................... 38 F) Absorption and Pressure Measurements . G) Determination of B/A and the Amount of Second H a rmonic................. 28 30 3I4. 38 \\.2 l\.6 H) Deviations at High Pressures and Distances........ ............ IV. SUMMARY OF RESULTS .................. APPENDIX A ................................ APPENDIX B ................................ BIBLIOGRAPHY LIST OF FIGURES AND GRAPHS Figure Page 1. Optical Arrangement . . . 2. Transmission coefficients of the steel plate at 3 and 6 me. /s. as a function of the angle between the olate and the sound . ............ 18 3. Diffraction from the fundamental and second harmonic components of an ultrasonic wave . . . Graph ..................... 16 18 Intensity vs. v. 1. First order, water; 2 mc./s. at 10 cm........... 21 2. First order, water; 2 mc./s. at 30cm. 3. First order, water; 2 mc./s. at $0 cm........... 23 .... 22 I|.. Second order, water; 2 mc./s. at 30 cm. • • • • 2l+ 5. Second order, water; 2 mc./s. at 50 cm. . . . . 25 6. First order, water; ^ mc./s. at 10 cm........... 26 7. First order, water; I4.mc./s. at 50 cm........... 27 8. First order, carbon tetrachloride; 3 mc./s. at 7 c m . .................... 31 9* Second order, carbon tetrachloride; 3 mc./s. at 7 c m . .................................... 32 10. Third order, carbon tetrachloride; 3 mc./s. at 7 cm. .................................. 33 11. First order, carbon tetrachloride; 3 mc./s. at 2.5 cm....................... 35 First order, carbon tetrachloride; 3 mc./s. at lij. cm........................... 36 12. 13. First order, carbon tetrachloride; 3 mc./s. at 21 cm..................................... 37 11+. Second order, carbon tetrachloride; 3 mc./s. at 2.5 c m . .................... 39 V| Graph Page 15. First order, carbon tetrachloride; 2 mc./s. at 9.5cm.................................... I4.O 16. First order, carbon tetrachloride; 2 mc./s. at 21 cm. ........................... I4.I 17. First order, carbon tetrachloride; 2 mc./s. at 7 cm. considering third harmonic . . . . I4.3 18. First order, carbon tetrachloride; 2 mc./s. at 9.5cm. considering third harmonic Ljlp CHAPTER I INTRODUCTION Sine© it was discovered that ultrasonic waves could produce diffraction of light, several authors have offered theoretical explanations of the details of this diffraction, Some of these theories *1*P 1 are rigorous but very complex and difficult to aoply to practical situations. In many cases it is preferable to use an approximate theory which is much sinpler to handle and which can be used in an easily obtainable experimental region* This theory, which was developed by Raman and Nath,^",^,^> gives satisfactory results in the region of moderately low amplitudes and frequencie s. All of the theories predict that there is symmetry about the zero order of diffraction when this diffraction Is produced by light incident at right angles to a plane, sinusoidal, ultrasonic wave. In the usual visual observa­ tions, one does not notice any asymmetry. However Sanders,^ using a photomultiplier, found that an asymmetry between the negative and positive orders appeared to be present. U Sette visually observed slight asymmetry at large distances from the sound source* Rao7 also observed asymmetry and tried to explain it by the generation of second harmonics in the ultrasonic source. Miller10 very carefully studied this problem and found that the asymmetry could not be eliminated by adjustments in the equipment. Breazeale and Hiedemann11 shoiked that a similar asymmetry observed with refraction could be explained by finite amplitude distor­ tion. Mikhailov and Shutilov1^ were able to show that for extremely high intensities (which at times approached shock waves) there was a marked visible asymmetry of the diffraction which they attributed to the finite amplitude distortion of the ultrasonic waves. Mikhailov and Shutilov mention that it should be possible to obtain some information about the shape of the distorted wave from the asymmetry, but at the enormous intensities that they used, this uroblem would be mathemat­ ically very complicated. However, since the observations of Sanders, Rao and Miller indicate that there is some asym­ metry in the region in which the Raman-Nath theory is useful an approach In this region appears to be the least difficult Zankel and Hiedemann 11J suggested such a study. Their preliminary investigations showed that this approach could be used to study smaller amounts of distortion at correspon­ dingly lower Intensities than that studied by other methods. ^ ^ A study at such low Intensities has the advantage of being not only In the region in which the Raman-Nath theory can be applied, but also in which theories concerning the behavior of ultrasonic distortion show promise. 3 Diffraction by a Sinusoidal Ultrasonic Wave The theoretical evaluation of the observed asymmetry for the purpose of determining the wave form is based upon the theories of Raman and Nath,^** ^ Rao^ and Murty.1^ The former predicted that for progressive periodic ultra­ sonic plane waves, and normal incidence of light, the light Is diffracted at angles -0- given by sin-©- = -nA/A* (1) where n Is any Integer including zero and negative Integers, A is the wavelength of the light, /T the fundamental wavelength of the sound, and 0 is the angle formed by the Incident and the transmitted light. If the wave is sinu- ,. 1 1 soidal, the Intensity of the n order is given by Xn = Jn2 <2) where v - 2TTyHL/A (3) jm being the maximum change of index of refraction caused by the sound pressure, and L the path length of light through the sound. Jh = (-1) Jn is the order Bessel function. Since , the symmetry predicted is obvious. This development is an approximation which Is only valid when the sound beam is narrow enough and the sound intensity low enough so that the light rays are not curved very much while passing through the ultrasonic beam. This condition may be stated mathematically as 2TTLX v/,u0 A*1 < K (1*.) J^o being the index of refraction of the media when the sound beam is not present. Various values of K, on the order of one or two, have been suggested by theoretical 1 '19 and experimental 20 studies. It is clear that the value of K allowed would depend upon the desired and obtainable accuracies of the measurements made. For £14-61 A mercury light, this limitation is LvF2 < 87K (5) for water and LvF2 < for carbon tetrachloride. 37K (6) L is expressed in cm. and F is the ultrasonic frequency in mc./s. The parameters used In the above and following calculations are listed In Table I. 5 Table I Some Measured Characteristics of Water and Carbon Tetrachloride Water Carbon Tetrachloride Density, ^ , gm./cm. 0.997a Sound velocity, c, m./s. iU95b 938° Index of refraction,yU0 1.33d l.ltfd a. Handbook of Chemistry and Physics 21 b. E. Schreuer c. Lagemann, McMillan and Woolsey^ d. International Critical Tables^ l.S9Sa pp Diffraction by Superposed Waves There are two different ways in which Raman and Nath developed their expressions. The first, usually called the "simple theory", utilizes the diffraction integral. method already includes the approximations. This The second method, called the "generalized theory", consists of an approximate solution to the wave equation for light in a medium of varying index of refraction. The diffraction integral was used by Rao to discuss diffraction from the superposition of two separate ultrasonic waves, one of which is the second harmonic of the other. Murty extended this work to any two separate ultrasonic waves with a frequency ratio l/s, s being an integer. We will write their results In a much more compact form than was 6 published by these authors , since this form will become useful later* = j The intensity of the n ^ order is given as M (7) A.U where a is the ratio of the pressure of the s to the fundamental and A harmonic is the phase between them* The expressions of Rao predicted an asymmetry of the diffracted orders for two such superposed waves for even s. these expressions in qualitative experiments* He verified Zankel and Hiedemann pointed out that since a distorted wave can be resolved into its Fourier components, one can consider the light diffraction as being produced as simultaneous diffraction by the Fourier components* The expressions of Rao and Murty are therefore applicable to the study of finite amplitude distortion* Pressure Measurements From Diffraction In order to discuss the behavior of finite amolitude distortion, it will be necessary to discuss pressure ampli­ tudes. Fortunately, any detailed study of the diffraction yields these amplitudes quite easily since the voltage on the quartz is proportional to the pressure, which is in turn proportional to v* Therefore, by comparing the diffracted light intensity vs. the voltage to the theoretlcal values based on eauation 2, one can assign a value for # Murty*s expressions contained typographical errors which were pointed out in a private communication. 7 v for each point on the curve* Prom these values of v one can calculate the pressure* by Willard* This procedure is summarized The equation relating these quantities is (8) where (9) and P is the peak pressure. The parameters listed in Table I give for this P - 0.56 v/L atm./cm.^ (1 0 ) P =. 0.25 v/L atm./cm.^ (1 1 ) in water and in carbon tetrachloride. Distortion Due to Finite Amplitudes In general, the equation of motion for a sound wave propagated in a fluid predicts a change of shape of the sound waves as it progresses. Although it Is usually assumed that a sinusoidal wave retains its shape, this is theoretically true for only two cases. The sinusoidal wave will not be distorted if there is a Hooke*s law dependence between pressure and volume or if the sound pressure amplitude is Infinitesimal. The dependence between pressure and volume for gases is determined by the equation of state 8 for adiabatic processes, and thus the first condition Is certainly not satisfied. This led to the -prediction and observation of distortion of finite amplitude waves in gases. Although It was pointed out that the first condition is not completely obeyed for liquids, 27 it was usually assumed that the deviations were small enough so that the second condition was very nearly satisfied. However this Ignores the fact that the distortion should be greater at higher frequencies. Since the absorption is less In liquids than in gases it should be possible to observe this distortion in liquids in the ultrasonic region. An observation of one of the effects of distortion, the Increase in absorption with pressure, was made by Fox and pQ Wallace. They assumed, Instead of a Hooke1s law depen­ dence, the power series expansion for the pressure, and were able to favorably compare their experimental results to the value of B/A obtained from fitting the available static compressibility data. the pressure and density and PQ and at one atmosphere. Here P and ^ are are these quantities This type of expansion had previously been used by Biquard.^ Fox and Wallace showed'r that by using the above expan­ sion one could adapt the work In gases by Thuras, Jenkins An error in sign was later shown by Zarembo et a l . ^ 9 and O'Neil^ so that it could be applied to liquids. They pointed out that such a treatment would be good only for small amounts of distortion and negligible absorption. The ratio of second harmonic to fundamental resulting from this is Pa/Pi = (13) r o x p j and the ratio of third harmonic to fundamental is (lli) where (15) £0 As has been the case in previous expressions the P's are peak pressures and the subscript zero indicates "at atmospheric pressure". Pox and Wallace made no measurements of these ratios. The first direct measurements of finite amplitude distortion in liquids were made by Zarembo et a l . ^ ,^C’,^ » 1 7 Using separate quartzes to receive the fundamental, second harmonic and third harmonic components of the distorted wave, they were able to show an increase of distortion with pressure and distance. Absolute measurements were made at only two distances, for the same pressure. Zarembo et al. pointed out that equations 13 and lip can only be used for 10 small distances from the radiator where it is oossible to neglect losses. CHAPTER II THEORETICAL INVESTIGATIONS Diffraction From Any Periodic Plane ProgressIve Wave In order to study the finite amplitude distortion, it would be preferable to consider more than just the second harmonic. However, it would be difficult to adapt the procedure used by Rao to obtain the theoretical diffraction spectrum even If one were to consider only the fundamental and the first two harmonics. Consideration of more harmonics would be Increasingly difficult. By putting the results of Rao into a more compact form, it was possible to show that the expressions he derived were solutions of the wave equation obtained from the “generalized theory" of Raman and Nath.*’ This had two advantages. One of these was that it demonstrated that these expressions are valid in about the same region as the original Raman-Nath expressions. The other was the indication of a method of solving the diffraction "problem for an arbitrary number of harmonics. The results of this study show that the amplitude of the n th order is given by (16) ■Jt-Thls was obtained independently by R. Mertens and was described in a private communication. 12 where as is the ratio of the pressure of the s^1 harmonic to the fundamental. Appendix A. This relationship is derived In This may look formidable but in practice the higher harmonics are usually small, and the Bessel functions converge quite rapidly. Influence of the Absorption on Distortion Zarembo et al, because of losses due to absorption and because they had large Intensities with correspondingly large distortion, did not make a more detailed experimental study of equations 13 and lip. Since diffraction studies are well suited to much lower Intensities, they allow an experimental study of equations 13 and lip. To eliminate the other difficulty of absorption, it would be desirable to at least approximately correct for the effect of this absorption. Assuming that each component of the wave is absorbed independently, and that the Increase of absorption due to finite amplitude effects are small, It Is possible to correct for this absorption in a manner similar to that used by Thuras, Jenkins and 0*Neil for gases. The increase in absorption, although perhaps large In Zarembo*s work, is small at low intensities. It is found that (17) 13 and _ foot X -4- (o CX X — where 1^ P( (1 8 ) ig the absorption coefficient of the fundamental at low amolitudes and x the distance from the transducer. The derivation of these equations is found in Appendix B. Equation 17 predicts that the behavior in highly absorbing substances is different from that in media having little absorption. It is seen that for a given pressure the percent of second harmonic approaches a fixed value asympto­ tically. That is, after a certain distance, the second harmonic is no longer highly dependent on x, but depends mainly on the pressure of the fundamental at that distance. At reasonable distances in a medium with a small absorption coefficient, (such as water at moderate frequencies) equations 17 and 18 may be aoproximated by ' HFx(l -(Kx)P^ (19) 3H2F2x2 (1 - Srtxjp-L2 (2 0 ) and a3 = Adaptation to Diffraction Studies It has been stated previously that values for v are obtained more directly from diffraction studies than are values for pressures. To simplify calculations It is much 11+ simpler to work in terms of v. Of course, the pressures are proportional to these values, and may easily be calculated from them. It is therefore desirable to nut the equations of the nrevious section In the form ___ G(FjL) Z« / ^ ( I- J aT (2 1 ) and 5 f'rVpL')/ 6 * \ ^ 4.6<*x-l)er (22) Using the parameters in Table I along with equations 10, 11, 15>> 17 and 18, one obtains G(F,L) = 2.6 (10)-tj-(B/A +■ 2)F/L (23) G(F,L) = 3.0(10 )"hB/A -t-2)F/L (214-) for water and for carbon tetrachloride. CHAPTER III DISCUSSION OP EXPERIMENTS Apparatus The experimental arrangement used is similar to that used by Miller and is explained In detail in his thesis.^ Figure 1 Is a schematic illustration of the optical arrange­ ment, The light source, S, is a 100 watt G. E. Type AH-lj. mercury vapor lamp. is a condenser lens, F^ is a filter which passes the 5I4.6I A line of mercury and B^_ is a source slit* The light is collimated by using auto-collimatlon. The tank used with water is approximately 2.5 meters long; that used with carbon tetrachloride Is approximately 1.5 meters long. The ends of the tanks are lined with cork to decrease the reflections. The tanks are made of sheet metal, with flat glass windows. The lens, L^, focuses the light on the slit, B2 • An auxiliary lens is sometimes used between and B2 to magnify the image on B2 * A filter similar to F-^ Is in the opening to the photomult in H e r . The photomultiplier-microphotometer is one manufactured by the American Instrument Co., catalog no. 10-210. The R. F. source used was built in this lab and has an output of about 100 watts. The voltages across the quartzes were measured using a General Radio Vacuum Tube Voltmeter (Type 1800A), allowing an accuracy of about Frequency Figure 1. Optical Arrangement 16 X CO 17 measurements were made with a General Radio Heterodyne Meter ( Type 620A). The transducers used were 1 M square x-eut quartzes. A Simple Method of Demonstrating the Presence of Finite Amplitude Distortion If the second harmonic component could be isolated and observed optically, one would expect (from equation 1) that the spacing between the resulting diffraction orders would be double that of the fundamental component. One method of approximating this condition is the use of ultrasonic transmission plates. Such plates were used by Zarembo, et al. in their measurements of distortion. Transmission plates will, when placed at a given angle to the sound beam, trans­ mit sound of certain frequencies much more readily than sound of other frequencies. In Figure 2 the experimentally determined intensity transmission coefficients of a stain­ less steel plate in carbon tetrachloride are plotted as a function of angle between the plate and the sound beam for two frequencies, 3 me./s. and 6 me./s. The coefficients were determined by optically measuring the pressure of the beam with and without the plate. It is seen that this plate has very sharp transmission peaks and therefore allows almost complete elimination of a 3 me./s. ultrasonic wave while transmitting a 6 me./s. wave. This plate was used with an ultrasonic wave having a fundamental frequency of 3 me./s. Figure 3& shows the 18 100 3 MC/S 6 MC/S z o to to « 50 90 85 80 75 a n g l e in d e g r e e s Figure 2, Transmission coefficients of the steel plate at 3 and 6 me./s. as a function of the angle between the plate and the sound, Figure 3. Diffraction from the fundamental and second harmonic components of an ultrasonic wave. 19 diffraction pattern which results when the plate was turned to one of the transmission peaks for this frequency. Figure 3b shows the pattern when the plate was turned to a 6 me,/s. transmission peak. doubles as was predicted. It is seen that the spacing Extremely small amounts of distortion can be detected by using, along with transmission plates, a photomultiplier to detect the light. In using this method for quantitative measurements, one must be careful to avoid disturbances of the sound field which might be caused by reflected waves. General Procedure for Observing the Diffraction Spectra Much of the experimental procedure used in this study is based on the work of Miller. Measurements were made of the intensities of the various orders (which were focused on B2) as a function of quartz voltage. Since the pressures are proportional to the voltage and to the values of v, one can adjust the voltage scale (by a best fit) to correspond to the various values of v. All measurements were made at moderate ultrasonic Intensities. The highest pressure used in water was less than 1.5 atmosoheres, and the highest in carbon tetrachloride was less than 0.7 atmospheres. Diffraction Spectra Obtained With Ultrasonic Waves in Water A) Distance Dependence In water the absorption is relatively low. Therefore, one would expect (from equations 19 and 20) that the 20 distortion would increase with distance. This would not be the case if the distortion were somehow introduced in the generating apparatus or if the asymm etry were caused by absorption effects or spreading. Graph 1 is a plot of the first orders vs, v for a 2 me./s. ultrasonic wave at 10 cm, from the transducer. As will be shown later, the differences between the positive and negative orders indicate the amount of distortion present. It can be seen from the curves that the asymmetry is easily detected. Graph 2 shows similar curves for the first orders at 30 cm. asymmetry may be observed. A clear increase in This increase is again Illustrated In Graph 3, which shows the first orders at JO cm. Graphs I4. and 5, which show the second orders of diffraction at 30 cm. and ?0 cm., also illustrate the Increase of asymmetry with distance. In these and all other measurements made In water, the asymmetry did Increase significantly with distance. It is seen that this asymmetry is detectable at fairly low pressures. B) Frequency Dependence Equation 19 predicts that in water the amount of distor­ tion at a given distance should increase with frequency. Therefore, to test this, measurements were made at I4. me./s. Graphs 6 and 7 are for the first orders for 10 and 50 cm. at this frequency. Comnaring these with the graphs for 2 me./s., a definite increase of asymmetry at the higher 21 1 atm vO o C\i iCq.Tsueq.uj q.ueouefT oII > Fj=0 atm. 0.5 atm. o 22 1 atm •H •H » • to > 1 atm >» p *H © J >© *rt T3 P U 0.5 atm. G QJ OO O Q O 02 ^q.-fSUQq.ui q-uaoJScj O pH =0 atm. 02 2k < rD *U d> •H rcf -c^d> u o tiO e ctf *H T3 - C^\ Graph 4. Intensity vs. a +c 3 d IT\ O a +3 cd o C\Z o «H iCq.xsuaq.ux q.uso.iaj o Oil i-H 0. 25 € Id c> pu J (S D • r t rc -P Fn •rl O pa cd o * CO 1.0 atm > -P 5h •H r0 cd o 0.5 atm. bo o o cn, o ^^TSU0^UI q.U0OJ0J =0 atm. c\l 27 w > -p 03 © f O Ot Jh © -P (X, © •H (0 * I> S © 14 > © -P 14 •H rO O o U~N O o O CM C^N jCq.-jsuecmi ^uaoaed 28 frequency is seen* All other* measurements in water showed an increase in asymmetry with frequency. C) The Ratio B/A The measurements in water were made only to obtain qualitative information. The main concern was testing the dependence on the parameters listed above in order to establish this method as a basis for quantitative measure­ ments of distortion. However it was found that these measurements allowed an estimate of the value of B/A in water* Fox and Wallace, on the basis of isothermal compress­ ibility data, calculated that B/A should be approximately 7. The measurements made from the diffraction pattern, although aporoximate, lead to values of B/A which satisfactorily agree with those of Fox and Wallace. The method used in arriving at a value for B/A will be discussed in more detail later . Diffraction Spectra Obtained with Ultrasonic Waves in Carbon Tetrachloride A) General Considerations It was found from the preliminary studies in water that several experimental difficulties had to be considered. One of these was alignment of the quartz so that the ultra­ sonic beam was perpendicular to the light beam. It was found that although this adjustment was not very critical to the overall asymmetry of the diffraction pattern, it was 29 fairly critical to the details of this pattern. Approximate alignment of the quartz could be obtained by turning the quartz so that the maximum amount of light is diffracted from the zero order at low sound intensities. It may be seen from equation 16 that for very small sound intensities, and for small distortion, the first orders should be equal in intensity. Therefore, the quartz was aligned by making the first orders equal at short distances and low sound intensities. Accurate alignment of all parts of the optical system is very Important to this procedure. Another problem encountered was the fact that the sound field is not homogeneous. An average sound pressure could be obtained by allowing the light to pass through a large part of the sound field. However, this is not satis­ factory for comparison with theoretically calculated curves. Since different parts of the field are at different sound intensities, one would expect that the curves thus obtained would be flattened at the maxima and minima. Therefore a compromise was made and a light beam several mm. in width and height was used. The position of the light beam was adjusted so that it covered a portion of ultrasonic beam which was nearly homogeneous. This was controlled by observing the schlieren picture of the sound beam. An error Is introduced in this manner, since the Intensity of the beam depends on which portion of the beam is used, which did not remain the same as the distance was changed. 30 Other difficulties encountered in making accurate measurements were fluctuations caused by line voltage changes, by heat schlieren and by non-linearity of the photomultiplier. The effect of heat schlieren was lessened considerably stirring the liquid close to the light beam. Since curves are plotted, deviations due to line voltage fluctuations and non-linearity of the photomultiplier are readily observed. Therefore, these three difficulties presented errors smaller than the previously mentioned ones. B) Pressure Dependence, Second Harmonic If equation 21 is correct one should find that for a fixed frequency and distance a2v = K2v2 (25) By using equation 11, a^v can be directly related to the pressure of the second harmonic. calculated for various values of Theoretical curves were Graph 8 shows the theoretical curves of the first order based on equation 7 for K2 = 0.025. Experimental points are plotted on this same graph for 3 me./s. ultrasound at a distance of 7 cm. Graphs 9 and 10 are similar plots for the second and third orders under the same conditions. It may be seen that there is very good agreement between experimental and theoretical values over a large range. This pressure dependence can be 0.6 atm 31 cW •cP ©" •H * XJ •H -P ^ 0,4 atm. f H J3 •I CT M\ p4«,h o (A o £ h 0 3 II OOpti M W . 0.2 atm, .. * H T3 -P O o o O CM •ftt-Fsuaqixi q.uaoJ© u • fH > CD T3 * 14 CQ O atra. > ra >» a o o cn =0 atm. 0.2 p o •H o to •H •H P fH a o 0) I—1 o r~| f4 o a> csJ fX. fH • P CP to • p ■>v^ ir\ « . CVZ o p o 0 O o X 0 o • o £> >1 Co fH aj II fl O O Ph X tS* ^-psiz8q.ax q.uaoa©(3 33 •o S ■s -I?* o p € Rj P a> u > * h© P H o •H O -0r\ ■ !a ; P i 40 ) oi £ © •H •H P IH C O © rH O& rH ChX* 03 H c^vL" oII u 3 ii ii c a o fa x ^ s aJ o II CHrH p o O rH iCq.xsueq.ax Cj.ueo.ieti o 3k observed in all the following graphs. (Some small character­ istic discrepancies will be discussed later.) C) Distance Dependence, Second Harmonic that It has been shown previously that one would excect Ina highly absorbing medium such as carbon tetra­ chloride, K2 would asymototically accroach some limiting value as x increased. This limiting value would be M = G(F,L )/2c* (26) For the measurement at 3 me./s. and 7 cm, the calculated value of M, using equation 21 and K2 = 0.025, is 0 .05. (The used was measured to approximately I P 55(10)"*^/cm, me. ) Equation 21 predicts that at 2.5 cm., should be approximately 22% of M; at lij- cm., 75^ of M; and at 21 cm., 90% of its maximum value. To check this assumction, graohs for the first order, showing the compar­ ison between experimental and theoretical values at these distances and at these values of 11, 12, and 13. are plotted in Graphs Considering the approximations made in the theories used, the agreement here and in other such measure­ ments in carbon tetrachloride Is well within the experimental errors. The results indicate that for high values of 0 1^ i a -p - c<-\C\i ^ O & -P CCJ cv O S -p «s HoV o II iCq.xsu9q.ui quaouaj r- &4 0,6 atm 36 •o o m 0.4 atm, rH P •--< D P 02 rH xr o O O h MW OO C O O2 0 jCq.Tsn8q.ui queonej O Q Pt =0 atm, ♦H 0.2 atm, 0 )u >h < D * r a -p h P]i=0 atm. 0.2 atm. 0.4 atm 37 38 determined chiefly by the pressure of the fundamental. Similar agreement is shown in Graph lip for the second order (K2 = 0.011) at 2.5 cm. D) Frequency Dependence, Second Harmonic One would expect on the basis of equation 21 that the limiting value, M, would change with frequency. It is predicted that at lower frequencies the limiting value would increase, and would be reached at longer distances. At 2 me./s. the calculated limiting value, using the previous assumptions, is approximately M = 0.075. This gives as an approximation K2 — 0.025 at 9.5 cnu and K2 — 0.014-5 at 21 cm. Graphs 15 and 16 illustrate good agreement between theoretical and experimental values for the first orders. Further measurements at other frequencies are suggested. E) Third Harmonic Equation 22 predicts that a^v =• K^v^, (26) with the result that the third harmonic would be more pronounced for higher v values. Using the previously determined value for G(F,L), several theoretical curves, which included the third harmonic, were calculated on the basis of equations 16 and 22 for K2 =• 0.025 and for various values of K^. Two of the previouslymentioned experimental curves give this value of K2 * 3 me./s. at 7 cm. and 0.6 atm 39 0.4 atm. H 'O ©G > h •r< OD to u 'Tl O 01 o C D 01 < to " C o p XI to • rH -P \ G O s •—j • OH O & o a ua • PcX > • o cd p c<~\cv u o o"\ O CM iCq.TSuaq.ux q n a o - r a j =0 atm. h ,jm O^ OS3 hii X W 0,6 atm 0,4- atm, Oi —1 •O £{ < D ft* cd td OS u oIti ..•O U X 0,2 atm, O o h o "3- o CO o2 0 .u -P ^ atm, *H nd o o O ^C^xsusq,ux q.u0oa©x O rH O >4-2 2 me./s. at 9*5 cm. these cases Equation 22 predicts that in both of is approximately 0.002. (The fact that these are identical appears to be a coincidence.) The theoretical curve for the first orders thus calculated are shown (solid lines) in Graphs 17 and 18 along with the experimental points for these two situations. It is seen that in the region considered, the experimental points do not compare favorably with the theoretical curves. The comparison with curves drawn for Ky = 0.0005 (dashed lines) is more favorable. It may also be observed that the devia­ tions for the frequencies are similar. Here it should be stated that the possibility exists that equation IJ4. is not applicable. Further investigation Is needed to determine whether the successive approximations used are justified, and whether further terms in the expansions and further harmonics need be considered. F) Absorption and Pressure Measurements There seem to be large discrepancies in the literature concerning absorption and pressure measurements. Apparently some discrepancies are due to the fact that distortion was considered negligible. Values of absorption measured at high sound intensities should differ from those measured at low intensities. Also, considerations of the absorp­ tion and the pressure must distinguish between those measurements effected only by the fundamental and those k3 pS © vO O •H - p i —I P • © CO P '"v 8 o o * > IO I .. ~ CI I\*M II OOPn W u~\ O o s p o• o © - -sf rr\ - a p © CM CM i —I o CM O iCq.fSttQq.xxx o -I° ^ It a, 0.6 atm W- •H . The main part of the error is due to inhomogeneities in the sound field and the resulting errors In the pressure measurements. Because of this, It was found that some of the curves drawn above could be fitted more accurately with a slightly different estimate of K2 than the one chosen. The evalua­ tion of the amount of distortion, as measured by is accurate to about 10$ of the measured value. H) Deviations at high Pressures and Distances The experimental results are in general In very good agreement with the theoretical predictions. However, a few deviations are evident, particularly at large pressures. kl Some of those might be caused by experimental error; but it is more probable that they reveal the need for better approximations. The equations concerning the behavior of distortion contain many assumption. As yet, no detailed study of the validity of these equations in given regions has been mad©. The experimental Investigation presented here certainly indicates that these assumptions lead to satisfactory predictions over much of the range studied. According to equation 6, one expects that the diffrac­ tion theory used should not be accurate at these higher values of v. This effect seems to be small, since It is seen that the experimental deviations do not appear to be very frequency dependent. and 18.) (See, for example, Graphs 17 However, one could possibly study this problem further by using a better approximation perhaps similar to that used by Mertens^ for the diffraction of sinusoidal ultrasonic waves. CHAPTER IV SUMMARY OF RESULTS The presence of distortion at moderate ultrasonic amplitudes has been established by two optical methods. One of these, which isolates the second harmonic by use of a transmission plate, Is very sensitive for the detection of small amounts of this harmonic. The other clearly demonstrates that distortion of the ultrasonic waves causes asymmetry of the diffraction orders. A method for quantitatively studying the asymmetry in order to determine the behavior of the distortion is developed. An expression predicting the intensity of the diffraction orders for an arbritrary periodic, plane, progressive ultrasonic wave is derived. Expressions predicting the amount of second and third harmonic present under given cond itions were corrected to include absorption. Preliminary experimental measurements with water show that the distortion increases with distance, pressure and frequency, as was predicted. Agreement is obtained between the amount of second harmonic predicted on the basis of isothermal compressibility measurements and the measurements made by the study of the diffraction. This study indicated refinements which were later used In the study of carbon tetrachloride. i+9 The experimental study of carbon tetrachloride showed detailed agreement, over a large range, with the predictions for the amount of second harmonic* The second harmonic was found to be (within experimental accuracy) proportional to the nressure squared. Similarly, the distance dependence determined experimentally was as predicted. The percent of second harmonic at first increased rapidly with increasing distance (for a given pressure at the point of measurement), and then less rapidly, appearing to asymptotically approach some limiting value. The measurements at different frequen­ cies were consistent (within experimental accuracy) with the predictions. Quantitative measurements of the amount of second harmonic were in good agreement with those predicted using the isothermal comoressibility data. The agreement of the experimental results with the expressions developed suggests that the type of theoretical approach used may be useful in approximately predicting the distortion and its effects over a wide range. Methods for determining the absorption coefficient and for measuring pressures were suggested. considered the presence of distortion. These methods Approximate measure­ ments were made which were consis tent with other data. The main difficulty in these measurements was the inhomogeneity of the sound field. The study of the diffraction pattern affords a sensitive method for studying distortion at modest amplitudes. Very 5o small amounts of distortion could be measured* The second harmonic could be measured when the Intensity of the harmonic was less than 0.0\\.% of the Intensity of the fundamental. Distortion could be measured when the fundamental pressure was much less than 0.2 atmospheres. this distortion cannot be neglected. It is evident that APPENDIX A DERIVATION OP THE INTENSITIES FOR THE DIFFRACTED ORDERS OF AN ARBITRARILY SHAPED PERIODIC PROGRESSIVE WAVE The procedures and results of Raman and Nath^ for a sinusoidal progressive wave will be used here to derive the expressions for any periodic wave. Raman and Nath have shown that for parallel light incident normal to any periodic, progressive, plane ultrasonic wave, the diffracted orders appear at angles sin~^(-n A/X* ) (A-l) They give the differential equation for the light in the liquid as where x is the direction of propagation of the sound, z the direction of propagation of the incident light9^ A (x,t) is the index of refraction and ^ -2TT/.V’C = JL T t) being the frequency of the light. (A-3) They go on to show that for this physical situation, they can expand § in a double fourier series, which results in ©o * - r a.Tri'rx/A* -VTT-t.rtl t 4. * (A-I4.) 52 where A and ^ are the wavelength and frequency of sound and fr (z) is some function of z* It is possible to extend the work of Raman and Nath to include an arbitrarily shaped periodic wave* The assumption is made that since the pressure is arbitrarily periodic, and since the pressure is approximately proportional to the index of refraction t) = y M 0 + ~ X/A*) (a -5) Then from (A-2) and(A-5) it is possible to obtain where * 2. b = 2.TT/A’’' IX.- a-TT/A. and terms of A*1 are neglected. (A-7) Combining (A-l|.) with (A-6) one gets *.(r Ur-j'KbX-S')' 53 Equating like terms gives ^ 3 11 = Rt v * 71 ^ /t S .. f \ "♦j ' (A-9) Letting —A U-M-o Z. fv, = A (A-10) tfigz') and substituting Into (A-9) gives 2 a- ii lA §-^22 s>2 + -71 D ^ — - (A-ll) 1 Let Z A ar ^TTyA ' (A-12) then /“* ? £ - a* / r - £ = (A-13) t>o Omitting the M term gives „ o’^yi <■ a / y 2‘ “ fr, j L > , ' J - (j> ^ ^ 1 A (A-lU ** 5k For n not too large 2 -£<*->( - S^j ^ » O (A- For the given boundary conditions, the solution of this equation is equation 16 In the text. APPENDIX B DETAILS OF THE CONSIDERATION OF THE INFLUENCE OF ABSORPTION ON DISTORTION The derivations given here are based on the procedure used by Thuras, Jenkins and O fN e i l ^ in their study of* the distortion of gases. With no absorption present, it is assumed that P2 = HPxP^ . (B-l) This means that the rate of increase of p£ Is 37 = H F F ^ If each component is considered to be absorbed independently, It Is seen that pl * p0 «cp(-*x) where P0 is the pressure at x = 0, and coefficient of the fundamental. (B-3) p the absorption Also, the rate of change of P2 with distance which is due to the absorption of P2 is —14.