THE USE OF LIGHT REFRACTION FOR THE STUDY OF PROGRESSIVE ULTRASONIC WAVES by MACK ALFRED BREAZEALE AN ABSTRACT Submitted to the School for Advanced Graduate Studies 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 1957 Approved: 2 MACK ALFRED BREAZEALE ABSTRACT Progressive ultrasonic waves at frequencies lower than one megacycle have been investigated in liquids by observing their refraction of light. Both the wave form and the sound pressure amplitudes were studied. For determining the wave foritij a stroboscopic method and an oscillographic method were used. With either method, the deviation from a sinusoidal wave must be greater than 5 per cent to be detected. For the waves produced in this study the distortion was less than this limit. For measuring the sound pressure amplitudes, two methods were used. The first is an extension to progressive waves of the method developed by Loeber and Hiedemann[1]. With this method one observes the decrease in intensity of the center of a light beam which has passed through an ultrasonic wave. This method is shown to give accurate values of sound pressure for light beam widths of a quarter sound wave length or less. The second method is based on a technique described by Huter and Pohlman[2] who observed the broadening of a light beam by ultrasonic waves. This method can be used not only for narrow light beams but for ones a full sound wave length or even larger. Since both the optical methods give absolute amplitudes, they promise to be valuable for calibration of devices giving relative amplitudes. 1. A. Loeber and E. Hiedemann, J. Acoust. Soc.Am.,28,27(1956) 2. T. Huter and R. Pohlman, Z. Angew. Physik., _1,405 (1949). THE USE OP LIGHT REFRACTION FOR THE STUDY OF PROGRESSIVE ULTRASONIC WAVES by Mack Alfred Breazeale A THESIS Submitted to the School for Advanced Graduate £ tudies 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 1957 ProQuest Number: 10008611 All rights reserved INFORM ATION TO ALL USERS The quality o f this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete m anuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQ uest 10008611 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. ProQ uest 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 sincere appreciation to Dr. E. A. Hiedemann whose guidance has made possible the results which have been achieved. Acknowledgment is also due W. W. Lester, who helped in construction of part of the equipments and due other members of the Physics Department who have assisted by their discussions and suggestions. Especially the author acknowledges the financial assistance provided by a U. S. Army Ordnance Contract without which the work could not have been performed. M .A .B. TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION.......... Sound waves of finite amplitude. 1 . . 2 Measurement of sound pressure amplitudes. II. EXPERIMENTAL APPARATUS The oscillator . . . 8 . The transducers . 8 . Optical arrangement. 8 . 9 Recording instruments . . . . III. 6 EXPERIMENTAL PROCEDURE AND RESULTS 11 . 14 Waveform determination: Stroboscopic method 14 Waveform determination: Direct observation. 21 Sound pressure amplitude measurement . Method I: BIBLIOGRAPHY SUMMARY AND CONCLUSIONS . . 34 Broadening of image . Comparison of the two methods. IV. 34 Intensity of the undeflected b e a m ................. Method II: . . 4l . 46 49 51 LIST OF FIGURES Figure Page 1. Basic optical arrangement for light refraction experiments..................................... 10 2. Diffraction patterns obtained, for increasing sound intensities showing alternative maximum and minimum of central order characteristic of progressive waves............................ 12 3- Optical arrangement used In stroboscopic method of waveformdetermination ..................... 4. 1 5 Recorder traces of sound wave form In glycerin by stroboscopic method. (a) Transducer moving toward optic axis. (b) Transducer moving away from optic a x i s ...................20 5. Optical arrangement used in direct observation of sound wave form. The output of photo­ multiplier P Is observed on an oscilloscope . 22 6. Single slit diffraction pattern caused by Sl^ • 23 7. Graphical determination of Lissajous figure expected if second harmonic is present in sound wave. (a) Resultant of fundamental and 20^ second harmonic. (b) Plot of slopes of (a). (c) Lissajous figure of (b) inter­ acting with a sine wave in phase. (d) Lissa­ jous figure for phase difference of 7t /2. . 26 8. Lissajous figures obtained from water at three sound intensities. (a) In phase. (b) Phase difference of TT/2 ............................29 9. Lissajous figures obtained from glycerin at three sound intensities.................. 30 10. Lissajous figures obtained from water at distances of 10, 17* and 24 centimeters from the face of the transducer...................... 3 1 11. Sound pressure amplitude in water as a function of the transducer voltage as measured by Method I. Frequency: lMc. Light beam width A / 3 ............................................... 38 gure Page 12. The variation of the ratio R with the width of the light beam through the sound field at three intensities. Curve A, 10 volts on transducer; B, 15 volts; C, 20 volts. Wave length of sound: 3*75mm......................... 40 13. Broadened image caused by passage of light through sound beam showing changes in structure as light beam width is increased. Light beam widths are: (a) X/3, (b) 2 X/3^ (c) X , and (d) 4 X / 3 .......................... 44 14. Recorder traces of broadened images for light beam widths of: (d) 4 X / 3 , . 1 5 . (a) . X/3, (b) 2 . . X/3, (c) X , 45 The variation of measured sound pressure with transducer voltage .......................... 47 CHAPTER I INTRODUCTION The optical methods which have received most attention in research in ultrasonics have been those which depend upon diffraction of light by an ultrasonic grating. Since dif­ fraction effects are increased by a decrease in the grating spacing, these methods have in general been limited to fre­ quencies above one megacycle where the corresponding wave length Is small. In the frequency range below one megacycle relatively few optical experiments have been performed. How­ ever, it was recognized already by Lucas and Biquard[l] that if one limits the cross section of the light passing through an ultrasonic beam to less than a half wave length, he finds, on imaging the light, that the image is broadened rather than diffracted. This condition Is most easily satisfied at fre­ quencies below one megacycle since the slit limiting the light cross section produces the usual slit diffraction effects if it is too narrow. Thus, this method is well suited for the lower ultrasonic frequencies. Although in a later theoretical paper Lucas[2] pointed out that this effect might be used to measure absorption and reflection coefficients, the first practical use of this effect seems to have been made much later by Hueter and 2 Pohlman[3] who used It to measure the absorption of ultrasonic waves in animal tissues. In 1952, Porreca[4] measured the distribution of light intensity across the broadened slit image and attempted to relate it to the ultrasonic wave shape. Loeber and Hiedemann[5] used the method to study standing ultrasonic waves in liquids and showed that the method could be used to determine the sound velocity, the sound pressure amplitude, and the wave form of the standing wave. The pre s ­ ent work is essentially an extension of the work of Loeber and Hiedemann to the case of progressive waves. This exten­ sion allows one to investigate a single wave, rather than the superposition of waves traveling in opposite directions. Sound Waves of Finite Amplitude In the usual derivation of the law of propagation of sound waves, one assumes that the particle velocity and the variation of the density are infinitely small. In this way one obtains linear equations which may be easily solved. The assumption of infinitely small amplitudes is equivalent to assuming that the particle displacements are small com­ pared to a sound wave length and that the particle velocity is small compared to the sound velocity. While these assumptions are sufficiently good in the audible range of frequencies at ordinary intensities, they might be questionable at ultrasonic frequencies, particularly at high ultrasonic intensities. Thus, there have been a number of investigations 3 in the ultrasonic region designed to establish the existence of a waveform distortion due to finite amplitudes. The mathematical theory of sound waves of finite amplitudes has long been a subject of investigation. Prob­ ably the first important step in the solution of this prob­ lem was made by Poisson[6] who showed that the particle velocity is described by a function of the form u = f [ x - ( a + u ) t] if one assumes Boyle's law p = a^p. If the particle velocity u is very small, then this reduces to the usual form u = f (x - at) in which the wave is propagated at a velocity a. Earnshaw[7] used this relation to discuss the propagation of a wave from its generation through its propagation and consequent change of type. Riemann[8] made a very basic study of the propagation of sound waves of finite amplitude. From his results It follows that the more dense portion of the wave travels faster than the less dense portion. This means that during propagation the wave is continually changing form. As the wave progresses the denser part continually gains on the less dense part until the front slope of the wave Is vertical, at which time a discontinuity in the wave sets in and the equations no longer hold. This raises the question whether 4 this wave is physically possible, continually changing. since the wave form is This would mean that any sound wave in air would become distorted and form a type of shock wave, which is physically not observed. Lord Rayleigh[9] showed that for real sound waves in air the viscosity and thermal conductivity of the medium must be taken into consideration. He showed that when one considers these, then it is possible to have a wave of a permanent type. More recently, P a y [10] considered the propagation of finite amplitude waves in air. The principal object of his analysis was to find the change in type of plane finite ampli­ tude sound waves propagated in free air. The solution of the exact equation of motion was obtained as a Fourier series. He found that, due to the non-linear relation between pres­ sure and specific volume, energy is gradually transferred from the lower frequency components to the higher ones. Since the higher frequency components are attenuated more than the lower ones by the viscosity of the medium, the increase in magnitude of any component due to non-linearity is balanced by losses due to absorption and transfer of energy to other components, so that a stable wave form can exist. ditions for stability vary with intensity, The con­ so that there is no stable wave form, only a ’’most stable wave form’1 for each intensity and wave length. Thusj for large distances it is found that the wave is attenuated and returns to the sinu­ soidal form found at infinitely small aplitudes. 5 A direct experimental verification of the effects of finite amplitude waves in air was achieved in 1935 by Thuras, Jenkins, and O ’Neill.[11] They generated progressive sound waves in a tube and, by probing along the length of the tube with a microphone, they measured the amount of second harmonic for various distances along the tube. The existence of a second harmonic in the air when the sound is generated by a sinusoidally vibrating source is a direct verification of the existence of a distortion in the wave as it is pro­ pagated. This is a direct conclusion from the Fay theory. A distorted sound wave in air was observed by Hubbard, Fitz­ patrick, Kankovsky, and Thaler.[12] The existence of a distorted wave form before a high intensity source in air was shown also by the staff of the Acoustics Laboratory at Pennsylvania State University.[13] They not only showed that a second harmonic was present, but also demonstrated the wave form distortion in air on an oscilloscope. The oscilloscope traces showed the distortion of the wave during propagation and the most stable wave form for various in­ tensities at various distances from the source. They found also that there is apparently a. maximum pressure amplitude which can be propagated as a periodic disturbance in air. They found that after reaching a certain value, which d e ­ creased with increasing distance from the source, the sound pressure amplitude registered by the probe no longer increased linearly with increasing amplitude of vibration of the source, 6 but leveled, off and approached a maximum asymptotically. This implies that the particle amplitudes were so great near the source that the restoring forces on the particle were severely changed. The existence of a distorted wave form in liquids implies that there is a non-linear relation between the pre s ­ sure and the density. Fox and Wallace[14] used this fact to explain the deviation from the expected exponential decay of sound intensity with distance in water and in carbon tetra­ chloride. meter. Their measurements were made with a sound radio­ Finite amplitude effects have also been considered by Eckart[l5] in a theoretical discussion of the fluid flow near an intense source of sound (e.g. quartz wind). However, there has not been a direct experimental verification of the existence of finite amplitude effects in liquids. One aim of the present study is to establish whether wave form distortion due to finite amplitude effects exists to a m e a ­ surable degree. Measurement of Sound Pressure Amplitudes The accurate measurement of sound pressure amplitudes, particularly in liquids., is a difficult problem. Methods of absolute measurement are not accurate, although relative measurements can be very good. The various types of radio­ meters, including the cavity radiometer, which measure the sound radiation pressure are examples of absolute measurement 7 d evices. However, the inherent difficulties such as streaming of the liquid under investigation and instability of the torsional fiber with regard to mechanical vibrations have made this method one which can mislead even a very experienced expe rimente r . Probably the most trustworthy and least vexing method of measuring sound pressures is the probe hydrophone. This method requires an electro-mechanical transducer, usually a barium titanate ceramic, which is so small that it does not disturb the sound field under investigation to an appreciable extent. (This condition is increasingly difficult to fulfill as the frequency is increased.) The electrical output of such transducers can be calibrated to indicate the sound pressure amplitude directly. However, the accuracy of this method d e ­ pends upon a previous calibration of the instrument. The most successful method of calibration of such instruments has been static measurement. Thus, one might question the use of this instrument in extremely accurate measurements, since it is possible that the theory relating its dynamic behavior to its static behavior is not sufficiently accurate. Another method which has been used successfully in aqueous solutions is the thermocouble probe[l6]. calibration by some other means. This method, too, requires The optical method to be described in this thesis offers the possibility of absolute measurement of sound pressure amplitudes. An evaluation of this method,, a determination of its accuracy as well as its inherent difficulties, is a second purpose of this study. CHAPTER II EXPERIMENTAL APPARATUS The Oscillator The oscillator used in these investigations was a commercial instrument (Hypersonic Generator, Model BU-204 manufactured by the Brush Development Company). The maximum power output was rated at 250 watts, although the experi­ ments were usually performed at much lower power. The timing drawer for the oscillator used in these experiments covered the frequency range of 3 0 0 kc to 1000 kc . The Transducers The barium titanate transducers used were discs two inches in diameter. One had a nominal frequency of 400 kc, the other a nominal frequency of 1000 k c . One of the quartz transducers used was a disc 6 centimeters in diameter having a resonant frequency of 3 0 0 one inch on a side having kc; the others were square plates resonant mately 420, 600, and 800 k c . frequencies of approxi­ All transducers had either silver or gold plated electrodes. The impedance of the barium titanate transducers was low enough to match the out­ put Impedance of the oscillator; however, the impedance of the quartzes was so high that, as is usual, an impedance matching transformer was used to match their impedance to 9 the output impedance of the oscillator. In all cases the transducers were air backed for radiating in one direction only. Optical Arrangment The basic optical arrangement is shown In Figure 1. Light from the source S is imaged by lens 1^ on the slit Sl^. The lens then forms an image of the slit Sl^ on the plane of the slit Sl^. Between the lens and the slit Sl^ are placed the tank T,containing the liquid under investigationt and the slit Sl^. The purpose of SI is to limit the cross section of light passing through the tank to less than a wave length of the sound in the liquid. Since the width of this slit enters into the calculations, it was a slit having a micrometer adjustment. was 0.01 mm.) (The least count of this micrometer The width of the slit Sl^ Is critical insofar as one must be careful to see that its width Is small com­ pared to a sound wave length and that it is not so narrow that diffraction effects broaden the image in the plane of Sl^ enough to affect the measurements. In one case, however, It was easier to make measurements of sound pressure ampli­ tudes with Sl^ wider than a half wave length of sound. The possible error introduced by this width of Sl^ is discussed later. The tank T was designed to reduce any reflection of the sound beam to a minimum. It was made of metal and had 10 plane parallel glass windows. mately 1.8 meters. The total length was approxi­ One end was tapered, as indicated in Figure 1, and lined with cork. This reduced the reflected wave to a negligible value as was shown by the following experiment Bar[IT] has shown that with standing waves in the usual diffraction arrangement one finds on increasing the intensity of the sound that the central image (zero order) intensity decreases monotonously with increase of sound intensity. If, however, the sound wave is progressive, one finds that the intensity of the central image passes through maxima and minima as illustrated in Figure 2. These pictures were made by opening Sl^ to approximately 1.3 sound wave lengths and placing the film In the plane of 81^. picture (a) was taken. With the sound off Then, increasing the sound Intensity, picture (b) was taken when the central image passed through its first minimum of Intensity. Picture (c) was taken at the first maximum of intensity of the central image, and so on. This behavior of the central Image Intensity was taken as indicating that the reflected waves in the tank were negligibly small. Recording Instruments When a progressive sound beam passes through the tank T, the light reaching the slit Sl^ sweeps back and forth at the frequency of sound. The light passing through Sl^, then, 12 Figure 2. Diffraction patterns obtained for increasing sound intensities showing alternate maximum and minimum of central order characteristic of progressive waves. 13 is modulated. This modulation may be detected by placing a photomultiplier tube in the position P, amplifying the out­ put signal and displaying it on an oscilloscope. In these experiments an RCA 1P21 photomultiplier tube was used. The signal was amplified by a high frequency amplifier made in this laboratory. model 1 5 0 The oscilloscope was a Hewlett-Packard A. Rather than detecting the modulated beam, it was im­ portant for some of the experiments to obtain a time average of the light intensity in various positions along the plane of SI . This was done by using a photomultiplier micro­ photometer made by the American Instrument Company which used a 931A phototube. The phototube was again placed in the position P. In some of the experiments the phototube and Sl^ were moved horizontally by a screw arrangement driven by a synchronous motor. On these occasions the readings of the microphotometer were continuously recorded by a recording potentiometer made by the Minneapolis-Honeywell Regulator Company, The paper in the recorder ran at a rate of three divisions (one Inch) per minute. The synchronous motor, going at three revolutions per minute, drove the slit and phototube at three millimeters per minute. Thus, one divi­ sion on the recorder paper corresponded to one millimeter of travel of the slit and phototube. It was found that the error incurred In measuring the distance traveled by the phototube by referring to the recorder chart was less than 0 .1^. CHAPTER III EXPERIMENTAL PROCEDURE AND RESULTS Waveform determination: Stroboscopic Method The broadening of an image after light has passed through a sound beam offers the possibility for determination of the wave form of the sound in the liquid, and therefore the possibility of determining the extent of distortion of the wave due to a non-linear relationship between the pres­ sure and the density of the liquid. One method used in an attempt to record the wave form of sound in liquids is shown schematically In Figure 3* Neglecting the apparatus to the left of Sl^ for the moment, let us consider what happens to the right of it. As has been indicated earlier, the light passing through the tank T^ is refracted because of the gradient of index of refraction in the liquid caused by the sound emitted by the transducer Q^. As the sound progresses, the light beam is refracted first in one direction, then the other as the sound wave fronts pass the slit Sl^. d Now, suppose the light, rather than being continuous, were Intermittent such that the burst of light passed through the sound beam during the same phase of each wave length. This would mean that the image at SI would be broadened by an amount characteristic of that portion of the sound wave through which the light traveled. 0 (X 15 to CO £ d o Cm CO °0 CD > cd % CO hco CO Cm O rd O -M CD £ o *i—I a o o CO O XI CO o d 4-^ CO d •1—1 fO Td CD CO d -M <*0 CO _o CO d CD £ CD bO d cd d d cd • d o •i-c 4-» cd d i—i •H cd d o *i— i CD 4M 4-> d a o o p CO P CO d o O Q-h d CO o *H •P P Pm cd > P P P CD •iH CO i— 1 rQ PH o ■rH p i— I O P CD £ P o •rH d o d d Ph ■H

P h cd o LT\ P P P bO1 pH *1 — CO O CO O % d o 23 the tank T is small in cross section compared to a wave length of sound in the liquid. usually A/8.) (This cross section was As the sound wave passes, the light is swept back and forth at the frequency of the sound. Thus, since Sl^ is a semi-infinite plane, the photomultiplier P receives light whose intensity varies sinusoidally with time if the wave in the liquid is sinusoidal. After amplification, the output of the photomultiplier is put on the vertical gain of an oscilloscope and may be viewed directly. A signal from the oscillator could be used as a trigger to synchronize the oscilloscope trace with the signal from the photomultiplier. It was found, however, that at these frequencies the internal synchronization of the oscilloscope provided a stable trace so that an external trigger was unnecessary. That the final slit Sl^ must rather be a semi-infinite plane can be seen if one considers the intensity distribution in the image due to slit differation caused by Sl2. intensity distribution is shown in Figure 6. Figure 6. Single slit diffraction pattern caused by Sl^. This 24 Now, let us consider that SI width w indicated in Figure 6. is actually a slit of As the intensity distribution is swept back and forth across the slit, there will be a frequency doubling effect due to the central maximum’s sweeping through the equilibrium position twice during each cycle, but more Important, If the amplitude of motion of the image is large enough there will be contributions due to the diffracted Images I^,and possible I^. The repetition rate for this contribution will be twice that of the con­ tribution IQ since there are two images I . Thus, there will be a second harmonic component added to the photo­ multiplier output. This is the same type of distortion as is expected from the wave form distortion In the liquid. Therefore, it must be eliminated. This is done by placing one edge of the slit Sl^ in the exact center of IQ and removing the other. Now, when the Image is swept back and forth sinusoidally, the light passing one edge will be essentially sinusoidal if the amplitude is less than half the width of I . The output of the photomultiplier was displayed on the screen of an oscilloscope. By observing the trace on the oscilloscope, then, one could determine the wave form in the liquid under investigation. It was found, however, that this method of direct observation was not accurate enough since in most cases the wave form was very nearly sinusoidal. In order to Increase the ability to estimate 25 the deviation from a sine wave, a sinusoidal signal from the oscillator was put on the vertical deflection plates of the oscilloscope and the signal from the photomultiplier was put on the horizontal,, forming the familiar Lissajous figures. This method necessitated a graphical analysis in order to determine the change in the Lissajous figures introduced by any distortion of the sound wave in the liquid. This analy­ sis was made as follows. According to the Fay theory[10], one expects the wave form distortion to be such that the second harmonic term in the Fourier Expansion of the wave is the largest term other than the fundamental. Thus, assuming the second harmonic to be a certain fraction of the first and neglecting the higher harmonics, a drawing could be made of the composite wave which would approximate the expected distorted wave in the liquid. Figure 7 (a) shows the wave resulting from a fundamental and a second harmonic assumed to be 20$ of the fundamental. high. (This value of 20$ will be shown to be However, it will give an idea of the type of dis­ tortion one may expect.) Since the refraction of the light beam depends on the pressure gradient and not on the pressure itself, it was necessary next to take the slopes of the curve in Figure 7(a) and plot them, as is shown in Figure 7(b). Now, combining the curve 7(b) with a sinusoidal curve in the usual way, one obtains the Lissajous figure to be ex­ pected on the oscilloscope when the wave in the liquid is 26 (a) (b) (c) (d) Figure 7. Graphical determination of Lissajous figure expected if second harmonic is present in sound wave. (a) Resultant of fundamental and 20^ second harmonic, (b) Plot of slopes of (a). (c) Lissajous figure of (b) interacting with a sine wave in phase-. (d) Lissajous figure for phase difference of TT/2, 27 distorted. This is shown in Figures 7(c) and 7(d) for phase differences of 0 and Tt/ 2 respectively. If there were no distortion 7(c) would be a circle and 7(d) would be a straight line passing through the origin. The Figures 7(c) and 7(a) indicate the type of Lissajous figure to expect on the oscilloscope. With these figures in mind, investigations were made in the following manner. In order to get the edge of the semi-infinite plane Sl^ exactly in the center of the Image of SI , so that its image at Sl^ was very long. was ro-3 -de very long The edge of Sl^ was lined up parallel to the Image visually, then the image was again shortened. Next, the photomultiplier P was moved aside and a photomultiplier microphotometer was put in its place. With the plane Sl^ completely away from the image, the microphotometer scale was adjusted to read 100; with Sl^ blocking out the image, the scale was zeroed. plane was moved until the microphotometer read 50* assumed to be the center of the image. Now, the This was The photomultiplier P was put back in place and the Lissajous figure was observed on the oscilloscope. No observations were made without first going through this procedure to position the plane Sl^. That this was necessary can be seen from the following consid­ erations . It has been shown [17] that one of the sources of error in optical methods of ultrasonics is the effect of local heating of the medium, particularly in the vicinity of 28 the transmitting quartz. This local heating causes local gradients of Index of refraction. Thus, any light passing through the medium will be refracted away from its original direction of travel. Let us consider that such gradients of Index of refraction exist In the experiments under dis­ cussion. Light entering the medium will in general be re­ fracted away from the optic axis. at SI SI This means that the image is no longer on the optic axis and that the edge of is thus no longer at its center. Thus, the variation with time of the light reaching the photocell resembles the curve shown In Figure 7(b), and the trace on the oscilloscope indicates an apparent wave form distortion when actually the wave may be sinusoidal. This effect was observed when the light beam passed very near the quartz face. Any misalign­ ment of Sl^ would cause the same type of effect. For these reasons extreme care was taken to see that the system was carefully aligned and that the effect of heat schlieren was negligible. Some of the results of these experiments are shown In Figures 8, 9, and 10. In Figures 8(a) and 8(b) are presented the Lissajous figures obtained for phase differ­ ences of 0 and 71/2 respectively at three sound intensity levels in water. The greatest intensity is of the order of two watts per square centimeter. By placing a straight edge along the figures one can observe the deviation of the figure from the straight line expected of perfectly sinusodial (a) Figure 8 (b) Lissajous figures obtained from water at three sound intensities. (a) In phase, (b) Phase difference of 71/2. 30 Figure 9- Lissajous figures obtained from glycerin at three sound intensities. 31 Figure 10. Lissajous figures obtained from water at distances of 10, 17, and 24 centi­ meters from the face of the transducer. 32 waves. The deviation shown in the last picture in 8(b) corresponds to a sound wave having a second harmonic com­ ponent of approximately 5$ of the fundamental. Figure 9 is a series of three Lissajous figures in glycerin for approximately the same intensities as in Fig­ ure 8 for water. In this case one can see that the sound is very nearly sinusodial. According to the theory, a sound wave originating at a sinusoidally vibrating source begins as a sinusoidal wave and travels a certain distance, which varies both with the liquid and the frequency, before reaching the "most stable wave form" characteristic of its intensity. It would be interesting to show the gradual change toward this "most stable wave form." This could be done by observing the wave shape at various distances from the source. Figure 10 represents an unsuccessful attempt to do this for water. The wave form was observed at distances of 10, 17, and 24 centimeters from the face of the transducer. The difference between the Lissajous figures is very slight, if It exists at all. This cannot, however, be taken as proof that the theory is wrong. For water at these frequencies the maximum distance required for stabilization, according to the Fox and Wallace theory, is approximately ten times the distance covered in these measurements. Therefore, the fact that these pictures do not show this change of wave form is not surprising. It might even be surprising that the distortion 33 is observable at all at these distances, except for the fact that one would expect the energy transfer from the funda­ mental to the second harmonic to be greatest near the trans­ ducer and to be smaller and smaller with increasing distance. The results presented In Figures 8, 9, and 10 are not presented as proof that distortion exists at these sound intensity levels. More experimentalIon is necessary before such a claim can be made. In particular, it will be neces­ sary to investigate other liquids which will exhibit the distortion to a greater extent than those Investigated. One liquid which suggests itself is carbon tetrachloride. According to the Fox and Wallace theory[l3], this liquid will be better in some respects. However, the wave length at the frequencies used will be shorter so that the ratio of the light cross section to the wave length will be less and the sensitivity of the method will be reduced accordingly. As the experiments were performed, the possible errors were of the same order of magnitude as the signal to be measured. This may also be true of carbon tetrachloride. Considering the possible error, one cannot conclude from the foregoing experiments whether a distorted wave actually exists in liquids at the sound intensities used. One can, however, conclude that the distortion Is probably less than 5% if it exists at all. Thus, it Is safe to say that the distortion in the liquids used by Loeber and HIedemann[5] was too small to account for their results. 34 The irregularities in the structures of the field of the stationary waves in some liquids revealed by their measure­ ments must be explained in another way. Sound Pressure Amplitude Measurement Light refraction can be used to measure sound pressure amplitudes in two ways. These methods differ In the way in which the broadening of the Image is determined. The first consists of a measurement of the decrease in the light inten­ sity passing through the final slit Sl^ (See Figure l) occurring when the image at Sl^ is broadened by a sound beam passing through the liquid under investigation; the second consists of a direct measurement of the angle of deflection of the light beam. With the first method one measures the ratio of the light Intensity passing through Sl^ when the image is broadened by sound to that when the sound Is off and arrives at the sound pressure by use of the relation derived by Loeber and Hiedemann[5]* With the second he measures the distance the image at Sl^ is displaced from the optic axis, determines the maximum angle of displacement, and arrives at the pressure by use of a very simple equation given by Hdter and Pohlman[3]. Method I: Intensity of the undeflected beam. In order to use this method to measure the sound pressure amplitude in progressive waves, It Is necessary to make certain modi­ fications of the theory for stationary waves given by Loeber 35 and. Hiedemann. They consider the superposition of two o p ­ positely directed sound waves and how light is refracted on passing through them. They find that the amplitude of the •pressure wave can be expressed in the form P sin (*° x/u) ^ A'1 (IT2 - ,232)1/2 (1) whe re A = ^ aco g f k \ru and k . ^ ( h 2 -1) (ng + 2 ) 6n The quantities used in these equations are defined as follows (See Figure l): P uj x u R a g 6 r n ^ Sound pressure amplitude Angular frequency Distance between light beam and face of transducer Sound velocity in medium Ratio of light passing through slit Sl^ when souhd is on to that passing through when soundis off Width of slit S l 2 Distance between sound beam and Slo Length of light path in sound field Light wave length Distance between S l 2 and Slo Index of refraction of medium Compressibility of medium The sound pressure is evaluated at the point in the station­ ary wave where sin (cux/u) is a minimum. = 1, that is, where the ratio Then, for the actual measurement one uses the formula P £= R A -1 (R-2 .232)1/ 2 x min which is more accurate for R < l / 3 . (4) 36 If one considers a single progressive wave and makes the same analysis made by Loeber and Hiedemann,, he will find that the sine term will average out and that the pres­ sure amplitude of a progressive wave may be expressed in the form Pprog. = *• O >3 ■a> £ O 1 g cd £ rd O v. hH ft 0 CD T} T3 CO P >5 ft £ ! rH T* P. 0 cB P ■n CQ 0 cd u 0 CQ £ on 3 0 0 > 0 cd rK P ft0 bO “ *ft cd *5 g -p ± ; B H ^ o o *5 CO > 15 0 u p bO (.uio/sduAp) sjnssdJd ft 39 taken at the center' of the image. This consideration rules out the use of this method for light cross sections greater than X/2. However, it is important to know whether the method can he used for light beams which are an appreciable part of X /2 since slit diffraction effects will set a lower limit on the light beam width. than If the light beam is wider X / 2 * errors will be introduced by the presence of ’’fine structure1'; if the beam is too narrow, errors will be introduced by the fact that the light is diffracted by the limiting slit and no longer passes through the liquid in a well-defined beam. It is thus necessary to determine whether there is a light beam width which will give the correct value of the sound pressure amplitude. A series of readings of the ratio R was made at a given sound intensity for various widths of the light beam. In Figure 12. Curves for three intensities are shown It can be seen that the ratio varies consid­ erably with the changing light beam width. This means that the value given for the sound pressure will be greatly affected by the light beam width. The sharp increase in the value of R for very narrow light beams can be explained as resulting from slit diffraction. However,, judging from observation, it is probably that these effects cease to be Important somewhere between X/8 and X/4 (between approximately 0.5 and 1.0 mm). There should be a point along these curves below X/2 gives the correct value of sound pressure amplitude. which This point was found in one case by a comparison of this method and Method II which will be presented later. It should be pointed Xo i— ! -U (D •H X X > X bfl ^ O CQ O $ X i— IO X 4-> • 50 to £ (L> Qj *H i —I X 4—1 O -H CO > x C cd 4-i Q) 0) ro X 4-> rd •H co 4-> X x 4—io OJ 4-> •i—I4—i 5 cd IX X E o a> 00 o o (Li 04 *i—I-H 4—| 4—i LPv •H bO C 4—’ d X bcd o X * co on X X X cd 4—1 > CD ^ X EH X oj CD — ■»— 00 CD + o O CC CJ o X bO •rH IX ^ B n o O CO 4i out that because of slit diffraction this method must be used with extreme care at wave lengths less than those used here (approximately 4 mm). Actually, it would be expected that if there were no slit diffraction, the narrowest light beam which would give a reading on the photomultiplier would give the most correct value of the sound pressure amplitude. Thus, it is expected that an increase in the wave length (lower ultrasonic frequen­ cies) would give an increase of accuracy. However, at lower frequencies it is increasingly difficult to get rid of re­ flected waves since the absorption varies as the square of the frequency. Method II: Broadening of image. If one considers that the light in the tank, if its width is smaller than a half wave length, experiences a bending due to the gradient of index of refraction in the sound beam, he can calculate the maximum angle of deviation from geometrical optics consid­ erations. Since the index of refraction gradient is propor­ tional to the pressure gradient, one can arrive at the sound pressure amplitude expressed as a function of the maximum angle of deviation of the light beam and the parameters char­ acterizing the liquid. This expression is given by Huter and Pohlman[3] as P I X K (6) L whe re K 2 IT ^ (n2 - 1) 6n2 (n2 + 2) (7) U2 The quantities used in these equations are defined as follows: P Y A L & n Sound pressure amplitude Angle of maximum deviation of light beam Sound wave length Path length of light in sound beam Compressibility of medium Index of refraction of medium (Equation (7) was not given by Huter and Pohlman, but was derived from expressions given by Lucas and Biquard.) Then., one can find the sound pressure amplitude by observing the maximum angle of deviation of the light beam. This can be done by observing the maximum displacement of the image at the position of Sl^. There are in general two different ways to determine the amount of broadening of the image by the passage of sound (that is, the maximum angle of deviation). One may take a photograph of the image and make measurements directly on the negative, or one may place a photometer behind Sl-^, then,, with Sl^ very narrow, move both Sl-^ and the photometer perpendicular to the optic axis. With a method of making this motion very evenly and very accurately, one can record the output of the photomultiplier on a strip chart recorder and relate the distances on the strip chart directly to distances moved by Sl^ and the photocell. The latter method was used exclusively In these measurements. The distances measured on the strip chart record differed from the distances measured by a micrometer screw attached to Sl^ by less than 0.1$. It has been mentioned[3] that, although the theory is derived for a light beam whose width a satisfies the relation 43 a < A/ 2 , one may use widths greater than this and find that the broadening is the same and that the edges of the broadened beam are better defined--even though the intensity distri­ bution characteristic of a diffraction pattern is present. This is shown in Figure 1 3 . The pictures (a) through (a) were taken for widths of Sl2 equal to X/3, 4 X / 3 respectively. A/ 3 , X , and 2 Figure 13(a) is the picture usually obtained for a broadened image where the light cross section is less than a half wave length of sound. The succeeding pictures begin to show a diffraction pattern; however, it is to be noted that the bright bands near the edges are the same distance apart in all four pictures. Thus, it ap­ pears that the amount of broadening is independent of the cross section of the light passing through the medium. This can be much more accurately tested if one considers Figure 14, which is a series of recorder traces taken for the same situation. The heights of the peaks are relatively unim­ portant for these considerations. was caused by changing (The difference in heights the photomultiplier scale between recordings to keep the readings on scale.) What is more important is the fact that the two outer peaks in each pic­ ture are the same distance apart to within 4/. Thus, it is permissible to go to the larger light beam cross sections where the peaks are more sharply defined in measuring the sound pressure amplitude. The difference in these traces shows that the accuracy of this method is limited if one 44 (a) (b) ! Figure 13. ! ! ■ Broadened image caused by passage of light through sound beam showing changes in structure as light beam width is increased. Light beam widths are: (a) X / 3 , (b) 2 X / 3 , (c) X , and (d) k X/3- 45 iff “ I_ rt i ) . ^. *1 ; .. : 1 T ' — .. r r i i i i ♦\ t t * ? .J t I .1 P itm 1 t ■ .. ""j— i 5 I : [. ; ; t -1 1“ 1 4 i 1 ; t * : r ' •, i i.. ; . 1 i "i. " ; "— 1 o i , 1 w i \r 1 1 (a) ! t \ t I1 1i 1 ’ if _ ♦Y kj n u I ! t. .■t <. i % I:. . - >— I < 1 ~ Ti j Figure 14. Recorder-traces of ..broadened images for light beam^widths of: (a*) X/3, (b) 2 X/3, (cj X , 46 uses light beam cross sections greater than X/2. However * going to smaller cross sections means that the peaks are not as sharp. This broadening of the peak will introduce an error into the measurements which will probably be of the same order of magnitude as that introduced by going to larger cross sections. Comparison of the two methods. In order to compare the two methods the apparatus was set up and a recorder trace was made of the unbroadened images and the broadened images at a number of sound intensities and light beam cross sections. In this way the ratio of the height of the center of the broadened image to the height of the unbroadened image could be used in equation 4 to determine the pressure ampli­ tude by Method I, and the distance between the peaks of the broadened image could be used In equation 5 to determine the pressure amplitude by Method II. This procedure allows the direct comparison of the two methods from the same data. The results of this procedure are shown in Figure 15. It can be seen that the points obtained by Method II lie reasonably close to the straight line drawn through them. The maximum deviation is of the order of 15%> but the average deviation is much lower than this. (This deviation is in part due to the use of a barium transducer whose output changes noticeably with temperature.) On the other hand, the points obtained by Method I are consistent within any particular run, I.e., they are consistent for any particular 15. The variation CD in —