A STUDY OF THE INTENSITY DISTRIBUTION OF THE LIGHT DIFFRACTED BY ULTRASONIC WAVES t>y Robert: Bruce Miller AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University of Argiculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1956 Approved 1A Robert Bruce Miller ABSTRACT The diffraction of light by an ultrasonic wave, predicted by L. Brillouin (1) and discovered independ­ ently by Debye and Sears (2) and by Lucas and Biquard (5), is an interesting phenomenon. The mathematical difficulties arising in any attempt to formulate an adequate theoretical explanation of the intensity dis­ tribution of the diffracted light has led to derivation of several theories. The simple theory of Raman and Rath (4, 5 & 6) is outlined and the predicted region of useful application given. The somewhat more involved and mathematically rigorous theory of Mertens (7) is also outlined, and a procedure siiggested whereby it may be experimentally checked. The rather detailed computations needed in the application of the Mertens' correction terms are carried out. The results of these are included in the appendix. The usual optical method for the detection of the ultrasonic diffraction pattern is described, and methods for using a microphotometer for actual intensity measure­ ments are outlined. Results are presented for a frequency range of 2 - 7 Me, and for sound field depths of ^ and 1 inch. Distribution curves, relating the intensity of the diffracted light to the sound field intensity, are given. In the more interesting cases the first five diffraction orders are shown. These curves are com­ pared to the theories of Raman and Nath and of Mertens. Suggestions are made as to the regions of usefulness of each. Robert Bruce Miller Literature Cited 1. L. Brillouin, Ann, Physij^i, 17; 103* (1921) 2. P. Debye and F, Sears, Proc . Rat. Acad. Sci. , 18; 409-414, (1932) 3. R. Lucas and P. Biquard, Comptes Rendus, (Paris) 194; 2132-2134, (1932) 4. C. V. Raman and R. S. Rath, Proc. Ind. Acad. Sci., 2; 406-412, (1935) 5. C. V. Raman and R. S. Rath, Proc. Ind. Acad. Sci. , 2; 413-420, (1935) 6* C. V. Raman and R. S. Rath, Proc. Ind. Acad. Sci. , 2; 75-84, (1936) 7. R. Mertens, Mededeling; ter Zetting, 12; 3-33 , (1950) A STUDY OF THE INTENSITY DISTRIBUTION OF THE LIGHT DIFFRACTED BY ULTRASONIC WAVES "by Robert Bruce Miller A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Argiculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1956 ProQ uest Number: 10008528 All rights reserved INFORM ATION TO ALL USERS The quality o f this reproduction is dependent upon the quality of the copy subm itted. 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 10008528 Published by ProQ uest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This w ork is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQ uest LLC. ProQ uest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 ACKNOWLEDGEMENTS The author wishes to take this opportunity to express his sincere thanks to Dr. E. A. Hiedemann, who first suggested the problem* and whose interest and guidance have made possible the results achieved, A debt of graditude is also due Dr. C. D. Hause for his valuable assistance in connection with the optics of the system. Other members of the department have also aided with valuable suggestions and discussions. The author is also indebted to the National Science Foundation, whose financial cooperation made possible the procurement of much of the necessary equipment for carrying out the investigation. TABLE OF C O N T E M N INTRO DUCT 10 M i THEORY 7 Raman and Nath. Theory Mertens Theory 7 11 EXPERIMENTAL APPARATUS Experiment?.! Proceedure PRESENTATION OF DATA EXPERIMENTAL CURVES 13 22 25 27-4-0 ANALYSIS OF DATA 41 Raman—Nath Theory 41 Mertens* Correction (zero order) 41 Mertens* Correction (higher ordei's) 42 OTHER RECENT THEORETICAL WORK Graph No, 15 44 Standing and Progressive waves 46 SUMMARY AND CONCLUSIONS 48 APPENDIX 50 Squared Be3sel Functions 50 V'nl values 52 V/n2 values 53 Even powers of v 54- Odd powers oY v 56 Correction Multipliers 58 BIBLIOGRAPHY 59 LIST OF FIGURES Figure I - Showing Raman—Nath.Theory 8 Figure II — Showing MertensTheory 12 Figure III - Optical Set-up 19 Figure IV - Ultrasonic Tank 21 1. INTRODUCTION In 1921 it was predicted by L. Brillouin (1) that a light beam, upon passage through a transjjarent medium in which a sound beam of sufficiently short wavelength was present, would be diffracted* That this was the case was demonstrated experimentally in 1932 by Debye and Sears (2) in the United States and by Lucas and Biquard (3) in France. Of great interest, however, was the observation, not of a single diffraction line as predicted, but of multiple diffraction orders which obeyed the simple grating formula, n X = d sin e where d becomes the wavelength of the ultrasonic wave. This observed multiple diffraction was not in agreement with the original theory of Brillouin (1). He had by making use of the method of retarded poten­ tials predicted only zero and plus and minus first orders. The intensities of which took on maximum values for angles of incidence satisfying a relation analogous with the formula established by Bragg for the diffraction of X-rays by crystals. A similiar result was obtained by P. Debye (4-). After the experimental investigation a more rig­ orous treatment of the problem was undertaken by L. Brillouin (5)* hut mathematical difficulties restrict 2 the application of this work to low ultrasonic energies. Debye (6) suggested that the multiple orders might arise from non-linear relationships between density and die­ lectric constant, or that the presence of harmonics might produce the observed effect. Lucas and Biguard (7, 8) pointed out the unlikelyhood of both proposals, the first, due to the relatively small pressure ampli­ tudes invo3.ved, and the second due to the fact that a piezoelectric crystal will resonate only on odd har­ monics. These latter men in the same work develop a theory based on a mirage effect. This theory predicts multiple orders the number and intensity of which in­ crease with the path length of the beam in the medium, and the ultrasonic intensity. However, their work in­ dicates that the relative intensity of the orders would decrease monotonically with increasing order number. That this was not always the case was shown experi­ mentally by R. Bar (9). If we now define a parameter, where = the ratio of the maximum density change to the average density of the medium. X ~ wavelength of the light. = wavelength of the sound, it is possible to divide the theoretical treatments into two rather broad divisions. First, the case where 3 & << 1, this cooresponds to high ultrasonic frequencies and in general the theories here have been patterned after the original work of Brillioun. We shall discuss only briefly any theoretical treatment in this region, For 1 no satisfactory theory exists, except perhaps that the work of Exterman and Wamier as extended by Nath (10) is applicable for intensities where only zero and plus and minus first orders appear. Our chief interest lies in the region 1. The first theory having any real success in des­ cribing the observed phenomena forl was first pub­ lished in 1933-30 in a series of three papers by Raman and Nath (11, 12, 13)* Their work follows closely the method of Lord Rayleigh in his treatment of the diffraction of a plane wave incident normally on a periodically corrugated surface. The three papers are quite complete treating both progressive and standing waves for cases both of normal and oblique incidence. They not only give relative intensities of the dif­ fraction orders, but also describe the observed angular dependance and the frequency variation effects in the several orders as observed by Ear (9). In two later papers (14-, 15) Raman and Nath give a somewhat more general treatment. These start from a differential equation, but the final results are the same as for the simplier theory. 4Experimental confirmation of the above theory was reported in 1936 hy Sanders (16), His results show good agreement between theory and experiment, and this work is widely quoted and reproduced in many publica­ tions dealing with ultrasonic diffraction. While no claim is made to the contrary, it should be pointed out that the region chosen for this experimental work was in the range best adapted to fit the theory, and that the theory is not in general well suited to explain diffrac­ tion effects over the entire frequency range, the entire range of sound intensities or of sound beam widths. An exact theory must allow in some manner for the relation­ ship between these three variables. Various authors have attempted to do this, most notable among them have been Extermann and Wamier (1936), David (1937), Nath (1936, 1938), Van Gittert (1937), and Mertens (1931)* It is the efforts of the last of these men that shall be the chief concern of this invest­ igation^. In general all of the above mentioned theoretical treatments have been directed at improving the approx­ imations of Raman and Nath made by assuming that terms of the type n / £ could be neglected, where n is the diffraction order and £ is the parameter previously defined. It is very difficult to treat this last term theoretically because of the number of variables involved. Thus the success of a given theory and the region in which it is applicable can best be determined by direct experiment. In this connection we note the validity limits of the elementary Raman-Nath. theory* out by Nath (10). These are pointed It is shown that these conditions are either, (>v2 < 1 if we accept the assumptions of Lucas and Biquard, or; % Q v2 <. X according to the work of Extermann and Wamier. n _ Where A and ^ /\J _ - £ TTyU. L ----- a where the following notation is used, ^ * wavelength of the light. = wavelength of the sound. jj = index of refraction of the medium. ^ - ma3d-mu“ variation i n ^ . L = thickness of the sound field. It is now possible to calculate validity limits for this theory if we assume a given maximum value for v. We notice that the two conditions differ by a factor of two and will produce this difference in the calcu­ lated limits. If one takes the most rigorous of the 6 restrictions, those of Lucas and Biquard, and assumes for the wavelength of light; 5 -X 10~^cm the following results are obtained; for maximum v =3 upper limit = 1.8 Me for maximum v=4- upper limit = 3*6 Me for maximum v -2 upper limit = 7*2 Me It is the purpose of this investigation to recheck the actual intensity of the diffraction pattern for progressive waves over a wider range of frequency and field depth than that reported by Sanders (16). Special consideration will be given to the recent work of Mertens (1 7 ), both as to the region in which it applies and to the actual improvement it may offer to the work of Raman and Nath. 7 THEORY In this section we shall outline the theories which shall "be of interest in this discussion. Our pur­ pose in doing this is several fold; first, to review the actual theoretical treatments and point out the as­ sumptions that have been made; second, to put the theo­ retical results In a form which may be related to the experimentally measurable variables; and third, it is necessary that we use a uniform notation for the theoret ical treatment* In this latter connection we shall follow rather closely the original notation of Raman and Hath, adapting It to cover the work of Mertens, Theory of Raman and Nath, This simple restricted theory bears a close analogy to the theory of diffraction of a plane wave (optical or acoustical) normally incident on a periodically cor­ rugated surface, as given by Lord Rayleigh 0L9)* Pigure I may be used to illustrate the physical set-up. Here P represents a point on a distant screen where it is desired to find the intensity of the dif­ fracted light. The sound and the light are directed normal to each other along the x and y axis respectively. A B indicates the difTerence in path length between the two indicated paths to P. It is equal to x cos 0 . is the distance the light travels through the sound L 8 P/o Dirpption of sound Figure I field, p isthe length of the sound field. With no sound present a plane wave would pass directly through the medium ahd emerge as a plane wave. With the sound on it is assumed that the emergent wave will have a corrugated front as indicated in the figure, and that the phase change represented by this wavy front is merely the path length L multiplied by the index of refraction of the medium yJL (x) . yw.(x) in which; = jM. Where -j*. slyi x? (D 9^ refraction of the medium » maximum variation i n , ^ . =* wavelength of the sound wave. The following assumptions have been made, that there is, first, no deflection of the beam by the medium carrying the sound; second, no amplitude change in the light wave; and third, the assumption is made that the variation in^u.will he sirosiodal in. nature, this as­ sumption seems to he valid in many substances except for relatively high sound energies. The amplitude of the incident wave can he repre­ sented by the expression; Ae2^ H t (2) and that of the emergent wave by; Ae2 tn*{t where ^ - (3) - frequency of the light t = time c = velocity of sound in the medium Then the amplitude due to the corrugated wave at a point on a distant screen will he given by, f . xfrxl / jl A r * Jtx where (4 ) 1 = cos 9 A = wavelength of the light The time dependance is dropped since the velocity of light is much greater bhan the velocity of sound. The sound field is assumed to be of uniform thickness and intensity. Equation 4- is now broken into its real and imag­ inary parts and written in sine and cosine form. These 10 can be expanded in a series of Bessel functions which can be integrated# This reduces the imaginary part to zero and the solution can be written in the following series form; £7/1) - p T ,T m/1 f (*~ X (*.* -^ .e ) s.) x4^ok<\. P Z J~ » (~ where; n M f( ^ 7 T T - "■ f/)&) i z r ^ T l H j ' (5) u =* 2^9/^ b = 20/^* v = 27>>lL/^ and 5*(A) is the amplitude at a point on a distant screen, Examination of this series shows that for any value of n only one term in the series will give any signifi­ cant contribution to F(A). This is true when ul - nb (6) in which case the denominator reduces to zero, but for all other terms the denominator is large compared to the numerator and so we drop all terms but this single term. If we use equation 6 and Figure I we see that Ji r cos 9 = sin \y combining with 6 gives sin ^ = n A / (7) this is the grating equation, and gives the direction of the light incident on the screen. 11 To get the relative intensity of the nth to the rath order we note that by using our approximation the bracket in equation 5 becomes one for both n and m. Thus the ratio of the intensities of any two com­ ponents is simply the ratio of the square of the ampli tude functions (8) For experimental purposes the light for the zero order is taken as one, so a plot of the square of the nth Bessel function* for an arbitrary set of values for v gives the distribution curve for the nth order. These curves can then be fitted to the experimental curves without actually measuring the The Theory of Mertens. The development by Mertens is similiar to that in the previous section, but embodies a more rigorous math­ ematical formulation and solution of the problem. The sound and light again enter the medium at right angles to one another, see Figure II. The index of refraction is assumed to vary in the same manner as before and is given by; yUx.y.z,t) +yu.s±n [2TT^*t - (£•?)/ (9) wherey6/L (x,y, z ,t) the refractive index is a linear function of the density, and the following notation * see appendix Table I for these values 12 L (xyz) light so und Figure II applies; yotQ - refractive Index of undistrubed medium. = maximum variation of^ulq • -2) * - frequency of the ultrasonic wave. = propagation vector. X* = wavelength of sound in the medium. "r - position vector. L = thickness of sound field. The light waves entering the medium must satisfy c)H I. Curl E = -fc, dJt II. Curl H = -f* III. Div l! = 0 (10) IV. DxvyOL2^ = 0 If H is eliminated in the usual manner, we get 13 V =* p. div 7 grad (div E) p -* (11) E) = 0 a system of partial differential equations describing the light diffraction* We assume that since *$*<< t> we may c o n s i d e r i n d e p e n d e n t of t in the calculations and reestablish the time dependence in the final results. ‘ fhen, ^ V 2E = ^ H i-+ srad (div E) 2 -> div (^m , E) = 0 C12) now assume that the plane of the ultrasonic beam is parallel with the x~y plane. ^t(x,t) y U Q Equation 9 becomes, sin - y/^*) (13) and the second equationin 12 reduces to <) (m* £yj T *■ V « - 0 (14) since ^ a i s not a function of x or z. —i the div E, gives; div E Ev ^ which may Solving this for o' 05) y be substituted in the first equation of 12 giving the expression; v 2e _ ^ & *) i _ ? _ grad (^X£, Ey) (16) Brillouin (7) shows that the last term may be neglected leaving 2-S * - -£3. V “ ^ (1 7 ) but since there is no variation of E in the z direction this reduces to, A A JL A (DEx TJ*- +" 5 ^ ' C* <)* (18) How taking 3 = e277*1^ $(x,y,t) and substituting in the above equation we get, after p tex'ins containing a 1/c factor have been dropped, H - + Tp- r - $ VWN>*J (19) Because of the periodicity of the sound wave along the y axis we can write, y«*(y + p A % t) v"-(y,iO yU,(.y, t +t?*/L/x For f = >2/ ^ v m a *2 equal to zero the Bessel functions satisfy the equation and this is just the solution of Raman and Hath* That is, ^ n (v) fh Jn C O (22) Mertens, however, writes his solution in the fol­ lowing form; $ (v) = J (v) + ^ p P l7> np(v) [?=/ V ^ (2 3 ) where \p x np (v) is a function to be determined which must satisfy the boundary conditions Jo <0) = l J (O) = 0 n n t 0 V np(°) = 0 The desired function is found to consist of the sum of two series* The intensity may then be written as the square of equation 25, and has the following form, In (v) = J2 (v) + f 2£ y nl(v)j2 + 2 Jn ( v Y n2(v^ (2 A-) where for small values of (p the following expressions give adaquate values for the last terms in equation 24. 16 (25) Thus the intensity distribution in the nth order may be calculated from equations 24-, 25 and 26. The problem presented then reduces itself to checking the contribution of these series terms which is simply added to original results of Raman and Rath. The factor is the only term in the correction egr e s s i o n involving experimental variables. Thus it is possible to evaluate the series for arbitrary values of v chosen as before and to find the correction term for a given experimental situation by simply multiplyo ing by P • These series computations are tabultated in the appendix. We must bear in mind that for the correction to be of any value we must have p ^ 1. Recalling that (° is given by; and v by the expression, a.'iTS* L ?■ we see that the product P v will eliminate the trouble- 17 some factoryju . When this product is formed we have pv= V x1r Vy-M ;** (27) The terms on the right of this expression are all experimentally measurable, and since the v's have been arbitrarily chosen we can arrive at a cooresponding p for each v. This then allows us to compute the correct­ ion for each value of v and plot a theoretical curve which can be fitted as before. 18 EXPERIMENTAL APPARATUS The experimental arrangement consisted of the usual optical set-up for the observation of the diffraction of light "by an ultrasonic wave in a liquid. trated in Eigure III. It is illus­ The light source S was a 100 watt General Electric type AH—4 mercury vapor lamp. the condensor lens It and were housed in a light tight box so that excessive scattering of the light was prevented. The condensor lens focused the light on the slit . The latter was located at the focal point of lens L 2 * this gave parallel light through the tank. was placed between and Sl-^. A filter E This was a Central Scientific Wratten filter No. 87310E designed to pass the mercury 5&hl A line. Actually this filter was not necessary since a similiar filter was used in the photo­ cell housing, it did, however, aid greatly in the op­ tical alignment of the apparatus. The plane wave from L 2 was then passed through the tank T and was focused by means of slit Sip. upon the second This slit, ahead of the photocell, permitted one to pick out and measure the intensity of each of the several orders. The photocell was an RCA 931A and was used in conjunction with a Photomultiplier Microphoto­ meter Type 10-210 manufactured by the American Instrument Company. The readings from the microphotometer could either be observed visually and point by point obser- obtaining the light diffraction r-t pattern 03 -P Figure III. Optical arrangement for produced by an ultrasonic wave in liquids. t-H 20 vat ions made on it could "be used in connection with, a Brown Recorder. Both methods were used, but point by point readings were found somewhat more desirable, since readings could be made directly in percent of light transmitted, and a constant check could be made that the original light intensity or the sensitivity of the photo­ cell did not vary. was combination length. lens of approximately 8 cm focal Lg had a focal length of about 12 cm, and for a lens of 100 cm focal length was chosen so as to obtain greater seperation of the lines at the second slit • The tank T presented the most serious problem. The difficulty was to get the light in and out of the tank without excessive scattering by multiple reflections, and also to prevent the establishment of standing waves in the tank. The first difficulty was overcome by using V/z inchsquare plane parallel plates as windows tank. on the One side of each window contained an anti-reflec­ tion coating designed to transmit the 584-1 A mercury green line. By using the coated side at the air-glass surface reflections here were largely eliminated. At the inner surface no problem was presented since glass and xylene (xylene being the liquid used throughout the experiment) have practically identical values for the index of refraction. 21 To prevent; the establishment of standing waves, the tank was constructed as shown in Pigure IV* The main body of the tank was 8 inches long, 2)4 inches wide and 3 inches deep* The wedge shaped tail used to absorb the sound beam by multiple reflections was also approximately 8 inches long and attached at about 30° to the main tank* Windows Pigure IV The wedge shaped tail was lined with cork as was the back and several other portions of the tank, from which waves might be reflected* Tests designed to show the presence of standing waves indicated that they had been eliminated by this construction. The sound was produced by quartz crystals of various frequencies and of several sizes, so as to observe both effects of variation in L and frequency, where L is the depth of the sound field. The R.F. source used to drive the quartz was an oscillator designed and constructed in the laboratory. It could be made to cover the frequency 22 range from 1—15 Me. The final amplifier consisted of two 807 tubes connected in parallel. Maximum power output was about 100 watts. The apparatus requirements were completed by the use of a surplus U.S. Army Singal Corps Frequency Meter Type BC-221-C manufactured by the Bendix Company, and of a General Radio Vacuum Tube Voltmeter Type 1800A for measurement of the R.F. voltage on the crystals. Experimental Procedure. In actually taking data the following method was found to giv-e the best results, and the following pre­ cautions were observed. The source slit was adjusted to ten microns. lens L 2 was adjusted by means of a telescope. The The latter was focused for parallel light, thus a sharp image in the field of the telescope indicated that we had parallel light coming through the sound field. Lens L- was then adjusted to focus the image on the slit Sl^. With the sound present the image was again viewed by means of the telescope, and a visual adjustment made to line the sound beam and the light beam normal to one another. This was done by observing when the number of orders on either sider of the zero order were equal in number and intensity. A final check on the intensity symmetry was made by means of the photocell. It should 23 be noted that an absolute intensity symmetry about the zero order can only be approximated. This same obser­ vation was made by Sanders (16), and is probably due to the decrease in amplitude of the sound wave both from absorption and dispersion as it leaves the transducer. After these adjustments had been made one was ready to make observations. It was often found necessary to allow both the microphotometer and the light source to "warm-up" for approximately an hour or one would observe a drift in intensity readings toward higher and higher values. Other precautions included the following; it was found that unless all equiptment was properly grounded the microphot7ometer was affected by the R.F. source. Another source of error in the earlier work resulted when light slipped by in the fringe of the sound field. This was corrected by blocking out a part of the exit window, so that the vertical depth of the sound field was greater than the window. It was also found necessary that a stirrer be in constant operation in the tank to prevent local heating effects, and resultant disturb­ ances in the light intensity. After these rather simple precautions had been taken, and after the coated windows had been mounted on the tank as discussed in the pre­ vious section^ it was found that the intensity distri­ bution curves could be readily duplicated. 24 In actually talcing data, the zero order was checked first. The-microphotometer was adjusted to read 100% transmission for no sound present, and then adjusted to read zero when the light was blocked out. Both of these adjustments were checked repeatly during the run and if appreciable drift was observed in either the run was started over. The sound field intensity was then varied over a sufficient range of voltage so as to coorespond to a maximum intensity at least as great as 6v. Thus the curves could be plotted out to values of 6v. Simultaneous readings were made for both voltage across the crystal and precent transmission. The crystal current was allowed to flow only long enough to make the necessary readings and adjustments and a short time lapse allowed between each reading. This together with the use of a blower on the tank and a stirrer in the tank prevented excessive heating. It was found that by this means temperature changes in the liquid could be kept to values of less than one degree centigrade. Similiar runs were made on the plus and minus orders out to the plus and minus 4th order, using the same voltage steps. Temperature and frequency readings were made during each run. 25 PRESENTATION OF DATA The information obtained in the manner described in the previous section was then plotted, percent in­ tensity against voltage, and this curve fitted to a theoretical curve which was plotted percent intensity against v. The necessity of treating the curves in this manner was due to the inability to measure the variable^yixas described in the section on theory. The curves were actually fitted by assuming that several of the minimum points were correct on both curves. Two such points were present on the zero order curve and several others on the higher order curves. In this manner a multiplying factor was obtained which made it possible to convert the voltage readings plot­ ted along the x axis into their cooresponding v values. The curves shown on the graphs included in this section were obtained in the manner described above. In each case the theoretical Raman-Nath curve is showen together with the experimental curve. Also, in cases where the Mertens* correction proves applicable, and of sufficient order of magnitude so as to distinguish it from the Raman-Nath curve, it is plotted to the same scale and for the same values of v. The curves are shown for several frequencies and for two different values for the thickness of the sound field. 26 They axe grouped in the following manner* We first show three frequencies of approximately 3, 4 and 5 Me for the % inch square quartz* Then five frequencies of approximately 2, 3, 4-, 3, and 7 Me for the one inch square quartz* In all cases the zero order distribution curve is showen. Higher order curves are reproduced only for the 3 and 3 Me cases for the % inch quartz and for the 3 and 4- Me cases for the one inch square quartz* These are sufficient to show the agreement with the Raman-Nath theory, and the region in which the Mertens1 correction is of value. In all cases the exact frequency, quartz size, and the (Ov product is_ indicated on the curve. This latter product enables one, by using it in connection with the correction multipliers listed in Table VI of the appendix, to see how the correction behaves In cases where it has not been plotted. In calculating the p v values the following con­ stants were used; Velocity sound (Xylene 20° C) Velocity temperature correction 1340 m/sec 4 m/sec °C Index of refraction (Xylene) 1.305 Wavelight light 5.461 X 10 Field thickness quartz width —5 cm 27 © O CO in o • to in m N P G cd G O cd CO i—1 w • o G *H l G CD O 'G G CD o G cr* XJI CD ^ > O 1^ to -p G *(— i o p' CD > G G CD o F* i—I G cd ,G G P> -P o G cd 0) C O i=S s 1 G *i—1 G CD G cd -p CD G Ph Cd CD M « ^ W CO > p> *rH CO G CD P> G HH -P G <2) O P » cd /y >v /- £ o CO —h O <£> O o cvj 28 \§> \ / © V I © in in I M ®/! ® o S> U 2 o CO xi cv? <0 ft I *© \ -p V>{ \ <$>•- © Si / Fh *© o / 03 S Q—' CEj > \ © less 2nd order Frequency 3*053 Me 3/4 inch quartz i1 than * -p CQ / © © /* © to vs v. © '<0 / -© \ /© / Cv? — cv? cv* Percent /© intensity A i % at I © Mertens’ correction / \ t—i \© © •© \ © o o cv? Hf- o Tf< o cv? all . *—1 W- ^ 11 ei I: Graph No. 2 points. © 29 f5 t *<2> © I /® i<*> .© in Iin f Me 4th order Frequency 3.053 i \I ® © V \ e> \ CM CO d cd t )! f\ -I-cm I CM Oraph i-r No% 3 Percent \ Intensity vs CO o o CM o o CM 30 \ r, *\1 h i Oth order Frequency 4.056 e, & co s co /fe r' * i I? CO t\ : 1 Percent / ® /i / / ’1CV2 ' / a/ / / Intensity ! v vs i -" -L*H -fiT' O -+ co o * r-\ O O 43 tQ 'd fl o * t-4 «> G O 3 *H I O1 43{ «) > P| P p O CO ^ ! ° i • w 1 -o G €3 -H >* O G Ph 3 O rH «5 43 -P -P G 0? €> s d 1 -H G G 00 « a p. flu W 43 pq excessive. It 32 /© / /@ % \ fn 3 tin / ©*■. / S/ <0 3 cr"E ^Q I —I Si <£> o * •GH I \i0 \® \ ‘f o si A G +^> -SS§ / Ps t O G / f r-HCC{ ©•ii •i / / / *■ © -J O CO / \ \ \ i® \ ©i-*H 4-rH I \ \, o CM O Intensity 'to .® -p | ^ OT| CCS o CM No. 6 ^ > /® ” 40 — in ® Oraph 2nd order Frequency 5.137 Me T tSJ ;• * * 33 I, ®//.; in y 7 © CQ © p c v/' CO N > o P Fh f—1 Fh © O J >i —I » co p o a «> o Xi rH «j © CQ go ♦ u ® •grH » i G •pgH o g g Jh p © . ,a( ® ® > P § p| Jh OB © ^ O &4 g<^ p ; p 05 q a // •/<«) IU v, \'© v\ \l ©n Percent «i j &J‘ " //} / // 03 v* Intensity 1! vs CO No* 7 *?>• Graph J& /. ©«" // L o CO g- o <0 O o 03 34 ' ©\ , \ * ®l V <9 C- NI i —1 4-^ o ^ t-o 30 3w CO p o ox) w "O K) O • p © a I o 0 C7< f> 44,f-t c O'PVI O °''■ . ® >( i / o > o f-i Pi 0 c O > r“t cd -p ^ 2 +5 4-> o c CD © 02 i - i f * J i 10 c o j^ c o e (/ / 0 //® -1 ij® C *rH c u -p © (f© fe \\ ? v '© V / ',«r /®/ // CVI //(£>* Intensity Y\ vs v CO Percent \•N\ \' © l\ Cirtph No. Q *• s*S> S' , s® // f ps O /<3> _ /0 CM _cv? Percent © o\ \ \ <* \ \ —) rj—I No. tf' \ ii Graph \ \ \ 0 o o CM intensity \ * o o CM 9 3.017 2nd order Frequency in a M 36 $ ; / G (O O > g 3 a o i— i C3 Xj -P +3 G •3 C in / © /• (® G> Vs G o t3 G o 1 •H a G CC ID P: H / lO i» \ 'TS *Q G * 3 l to \ \0 \0 to o * \ <2> c\? \5»c\? < Oraph No, 10 I intensity vs V Percent CO o O CM O O CM 37 ~: I * ©: 0» *rH > n* c* o o *> 1—1 -p fH • o c r cr.. \ \& X p o p 'O c o U o a O cr « -P I U c O P4 o to m • .P p « 2 P o c « / I O *r-i « p id P e a U P< t i > Qp fS • ®( I , o w nv \ u too E> r—( © \ a o €> high • p value t for ©/ in Merten# *v cd C3 pq to 1 \ \© * I \ I •v / y y y / "© © '© ®y / / / / o CO o o sf1 o CvJ / h ; . 03 Grftph No. 11. Percent correction to be useful. © - intensity 1 vs v. Note 'c: 38 © shown. J A ©\ \ © ;j® 1 i > u 2 o e>, / o u CO €2 o C if) « o • U o G a t> ►> i —i \ / /© *\ i' PW »I }®» , / rH X / /® / -€> t* / CO CO ^// j. /V /'© «<* © A® 4" \ ',0 \ o\ 02 02 & \ <3 V \ v cl a 02 ^ 3 cm \1 v>**•\* J3 G p Jh -P O G V 0f C pU s1 cGa f^-t o c ® G P © ■Pi a G p. 001«J «> w / LO order y- 02 N P -1 Li 1 • «3 7S £>cr a> *p t' ! 0 i i 1 Qtt i 1P G e •iH i 0/ First , Mertensf correction \ 59 * V Q <5> t0 ID 0 / V QQ O > P G ■3 * G P- £ tO ISJ O -«H > O ^ G * C! o ID ID ) ZS / 3 O CT*O ^ f— I -P CD G X0 ^ CO -P c €) G O • T3 aI G f> G O Zf I G 01 <3 .Gl « c > C + fl> » G IOFhO < ^ fxj ( /© <\ © V '0 \ CD \ © } <3>iH, / qy x* x© ; . -r CV7 Percent © \ intensity vs \ © No. 13 © Graph © /'(2) «v /r o CD —4" O to o -t" o C\3 H~ 40 Q * © ( in 1 09 4-3 Me oth order frequency 7.201 J 9 G © *rH > O +X> Pi © 3 Oi cr1rH .G O to • rH G 1■rH | O> © / ■V G P- G O rH © / ® XJ 4-3 sy G ■3 © 3 S 1 *r-4 GS f-t /0 © © 3 p©W 1© % 3 3 \ ©CO No. 14 © © Graph © X ©/ y x;ercent <£> intensity vs o o CO o o o w 41 ANALYSIS OF DATA Raman-Nath Theory. We can point out here that the simple Raman-Nath theory gives good results in the regions predicted* We note in the final curves that as the frequencies become higher and higher the fit becomes poorer and poorer for the high values of v. At 7 Me the agreement becomes very poor for values of v greater than two. This is - in accord with the prediction of Nath (10) as discussed in our introduction. Mertens* Correction (zeroth order), As indicated previously our chief interest is with the application of the work of Mertens rather than that of Raman and Nath. As noted in the theory this correc­ tion is limited to values of smaller than one* This is also the condition on the Raman-Nath work but in this latter case the restriction seems to be less severe than in the former. For the case of the zero order our results indicate the following; First, the correction is not useful for values of v less than two. This is to be expected, since v itself is a function ofy*. as is ^ , and since low values of ^ give low values of v but high values the condition that applicable in this region. be less than one is less 42 For values of v greater than two and for frequencies below 4 Me the Mertens* correction offers some improve­ ment to the original work of Raman and Nath* At or below 2 Me the order of magnitude of the correction is so small as to be of little use. Above 4 Me the correc­ tion tends to become an over correction for the lov/er values of v and pushes the region of usefulness toward higher and higher values for v. However, the usefulness of the correction for values of v above six is extremely limited, due first to mathematical difficulties encount­ ered in calculating the correction terms, and also due to the fact that the original Raman—Nath theory becomes less applicable in this region, Mertens* Correction (Higher Orders), For orders above the zero order the Mertens* correc­ tion terms become less useful. At low frequencies and low values of v the terms are mathematically to small to be of much significance. At higher frequencies they give some correction for the low v range, but for the higher v values the correction tends to take on the wrong sign. Concerning this sign change the following observation may be made on the curves in general. For high values of v there is a tendency for the intensity values of all orders to be lower than those predicted by the theories. It was noted during the 43 experimental work, that the higher orders appeared with sufficient intensity to he observed at a faster rate than one would expect from the Bessel function relation­ ship used in the theories* OTHER RECENT THEORETICAL WORK The other theoretical papers that came to our atten­ tion during the course of this work were directed at a frequency region above that in which our equipment was designed to operate. We should, however, mention several, among them is a paper by Mertens (18). This work is an extension of that by Nath (10), and points out that for values of (p much greater than one the first order intensities should be given by, 11 3 \ 2 sin2 ^ (v) for progressive waves, and by; Ix = 2/*2 sin 2 \ (v) for standing waves, where in both cases the sound inten­ sity mu3t be low enough so that we may assume that orders higher than the first are not present. It would seem that the 1 to 2 ratio predicted here might be easy to check experimentally, and we looked for this result at a frequency of 15 Me. true that While it is is certainly greater than one at 15 Me the requirement that be much greater than one may not yet be too well satisfied. Our efforts to check these results were carried 45 out in the following manner; a reflector was placed in the tank: and the quartz and reflector were adjusted to give the maximum effect for a standing wave pattern. The microphotometer was set to read the intensity of the first order diffraction line. Starting from a zero current reading the current was increased in very small steps until a previously determined value, at which the second order line was known to appear, was reached. JTnr each current reading a microphotometer reading of the light intensity was also made. The reflector was now carefully removed and a similiar set of readings made for progressive waves. The results obtained were plotted current vs light intensity and the results sire shown on the accompanying graphs. 46 progressive wave standing wave ■" #— > i In this series each successive higher order of m must 2 be multiplied by a different power of v. We reproduce here the multipliers for the zeroth and the first orders. m Oth (n = 0 ) 0 0 1th (n = 1) 2.41935 X 10"1 v2 1 -2.50000 X 10”1 v3 -5.04052 X 10-2 v4 2 5.12500 X 10-2 v 5 2.94019 X 10“3 v6 5 -1.50208 X 10-3 v7 -7.87550 X 10~5 v 8 4 2.71267 X 10-5 v 9 1.20520 X 10-6 v 10 5 -5.59084 X 10-7 V11 -1.18497 X 10“8 v12 6 2.82570 X 10-9 v 13 7*59597 X 10-11v14 7 -1.68197 X 10-11v15 8 7.50878 X 10-14v17 -5.84452 X 10_14v16 53 TABLE III Here we are concerned with, the second of the two series in the correction expression. - Lhl* Z_ Where A*™* (s k -I)! (sv*.+*.-/) I This differs from the previous one in that the multiply­ ing factor involves only a single power of v for each n. We reproduce helow the multipliers for orders zero and one# Hote that for the zero order the summation starts for m = 2. m Oth (n ~ o ) 1th (n“ 1) 1 - -0.03654 2 0.31250 0.01713 3 -0.08593 4 0.00553 5 -0.00015 sum 0.23195 X v2 -0.00147 0.00005 -0.02083 X v3 V v2 TABLE IV Even Powers of v 4 V v6 v8 v 10 — - 0.25 0.0625 0.0059 0.0002 0.50 0.2500 0.0625 0.0156 0.0059 0.0012 0.75 0.5625 0.5164 0.1780 0.1001 0.0565 1.00 1.0000 1.0000 1.0000 1.0000 1.0000 1.25 1.5625 2.4414 5.8147 5.9604 9.3132 1.50 2.2500 5.0625 11.5906 25.6289 57.6649 1.75 5.0625 9.5789 28.7229 87.9658 269.676 2.00 4.0000 16.0000 64.0000 256.000 1024.00 2.25 5.0625 25.6289 129.746 656.840 3325.25 2.50 6.2500 59.0625 244.141. 1525.87 9536.74 2.75 7.5625 57.1914 452.510 5270.85 24735.8 5.00 9.0000 81.0000 729.000 6561.00 59049.0 5.25 10.5625 111.566 1178.42 12447.1 131,472 5.50 12.2500 150.062 1858.27 22518.8 275,855 5.75 14.0625 197.754 2780.91 59106.6 549,936 4-.00 16.0000 256.000 4096.00 65556.0 1048576 4.25 18.0625 526.254 5892.96 106,441 1922601 4.50 20.2500 410.062 8505.77 168,151 3405063 4.75 22.5625 509.066 11485.8 259,149 5847040 5.00 25.0000 625.000 15625.0 590,625 9765625 5.25 27.5625 759.691 20958.9 577,151 15907174 5.50 50.2500 915.062 27682.9 857,359 25331610 5.75 55.0625 1095.15 56141.6 1194931 39507399 6.00 56.0000 1296.00 46656.0 1679616 60466176 55 TABLE XV (cont.) V 0,25 v 12 v 14 v 16 — - — - — 0.50 0.0002 0.75 0.0317 0.0178 0.0100 1.00 1.0000 1.0000 1.0000 1.25 14.5519 22.7373 35.5270 1.50 129.746 291.928 656.838 1.75 825.005 2526.57 7737.62 2.00 4096.00 16384.0 65536.0 2.25 16834.1 85226.3 4-31,4-58 2.50 59604.6 372,529 2328306 2.75 187,065 1414679 10698510 3.00 531,441 4782970 4-304-6800 3.2-5 1388673 14667800 154-929000 3.50 3379220 41395500 507094-000 5.75 7753480 95250200 13394-56000 4.00 16777200 268435000 4-294-970000 4.25 34726900 627256000 11329800000 4.50 68952500 1396290000 28274-800000 4.75 131924000 2976530000 67158000000 5.00 244140 X 103 610352 X 10* 152588 X 106 5.25 438441 X 103 120834 X 103 333050 X 106 5.50 766345 X 103 231819 X 103 701253 X 106 5.75 130621 X 104 431867 X 103 14-2786 X 107 6.00 217678 X 104 783642 X 103 282111 X 107 56 TABLE V Odd Powers of v v? v9 v 11 - - - 0.0078 0.0019 0.0004 0.2575 0.1555 0.0751 0.0422 1.0000 1.0000 1.0000 1.0000 1.0000 1.25 1.9551 5.0517 4.7685 7.4505 11.6414 1.50 5.5750 7.5957 17.0858 58.4450 86.4967 1.75 5.5594 16.4152 50.2654 155.958 471.455 2.00 8.0000 52.0000 128.000 512.000 2048.00 2.25 11.5906 57.6649 291.929 1477.89 7481.81 2.50 15.6250 97.6562 610.551 5814.69 25841.8 2.75 20.7968 157.276 1189.59 8994.76 68022.8 5 .0 0 57.0000 245.000 2 1 8 7 .0 0 1 9 6 8 5 .0 177,147 5.25 54.5281 562.591 5829.87 40455.1 427,286 5.50 42.8750 525.219 6455.95 7 8 8 1 5 .6 965,491 5.75 52.7544 741.577 10428.4 146,649 2062250 4.00 64.0000 1024.00 16584.0 262,144 4194500 4.25 76.7656 1 5 8 6 .5 8 25045.1 452,577 8171060 4.50 9 1 .1 2 5 0 1845.28 57566.9 7 5 6 ,6 8 0 1 5 5 2 2 8 0 0 4.75 107.172 2418.07 54557.7 1 2 5 0 9 6 0 27775500 5 .o o 125.000 5 1 2 5 .0 0 7 8 1 2 5 .0 1 9 5 5 1 2 0 48828100 5.25 144.705 5988.57 109,929 5029920 8 5 5 1 2 2 0 0 5.50 166.575 5052.84 152,245 4605550 159512000 5.75 190.109 6285.49 207,814 6 8 7 0 8 5 0 6 .0 0 2 1 6 .0 0 0 7 7 7 6 .0 0 279,956 V v 3 0.25 0.0156 0.0009 0.50 0.1250 0.0512 0.75 0.4219 1.00 227167000 10077700 562797000 57 TABLE V (cont.) V 0,25 v15 v1? — — — — - 0,50 0.0001 0.75 0.0249 0.0140 0.0079 1.00 1 .0 0 0 0 1 .0 0 0 0 1 .0 0 0 0 1.25 18.1899 28.4217 44.4089 1.50 194.618 437.890 985.252 1.75 1443.77 4421.55 13541.0 2.00 8192.00 32768.0 131,072 2.25 37876.7 191,751 970,739 2.50 149,011 931,319 5870740 2.75 514,422 3890300 29423900 5.00 1594320 14348900 129140000 5.25 4513210 49670800 524648000 5.50 11827300 144884000 1774830000 5.75 29000400 407818000 5754940000 o o• v 1? 67108800 1073740000 17179800000 4.25 147590 X 105 266584 X 10* 481517 X 105 4.50 310287 X 105 628331 X 10* 1 2 7 2 3 7 X 106 4.75 626640 X 105 141386 X 105 319002 X 106 5.00 1 2 2 0 7 0 X 10* 305175 x 1 0 5 762937 X 106 5.25 230180 X 10* 634434 X 105 174866 X 107 5.50 421419 X 10* 1 2 7 4 7 9 X 106 335624 X 107 5.75 7 5 1 0 7 1 x 10 * 248323 X 106 821013 X 107 6.00 130607 X 105 470185 X 106 169267 X 108 58 TABLE VI Here we reproduce the actual correction multipliers, or the term inside the bracket in the equation below equation 25» This term when multiplied by ^ 2 gives the correction term that we want. ^ is a function of X > X * aiid^^and must be obtained for each individual case. V 0.25 0.50 0.75 1.00 1.25 1.50 1-75 2.00 2.25 2.50 2.75 5.00 5.25 5.50 5.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 Oth order 0.0285 0.1097 0.2552 0.4055 0.6271 0.9280 1.5152 1.74-60 2.1209 2.2724 2.0184 1.2421 -0.0090 -1.4516 -2. 5654-2.6676 -1.9745 2.4669 7.9875 14.74-58 21.5525 26.8667 27.1508 23.7801 1st order 0.0001 0.0020 0.0085 0.0195 0.0289 0.0245 -0.0098 -0.0851 -0.1892 -0.5006 -0.5691 -0.5559 -0.1508 0.2082 0.7220 1.2845 1.8941 2.54-81 2.5989 2.6525 2.6404 2.4205 2.6562 5.6764 59 BIBLIOGRAPHY 1. L. Brillouin, Ann. Physii^ 17; 103, (1921) 2. P. Debye and P. Sears, Proc. Hat. Acad. Sci., 18: 4-09-4-14, (1932) 3* R. Lucas and P. Biquard. Comptes Rendus, (Paris) 2132-2134-, U952) 4-. P. Debye, Ber. Sarks. ac. Wiss. , 84; 123, (1932) 3* L. Brillouin, La Diffraction De La Lumiere Par Lea Ultrasons « Hermann, and Cie, Paris, (1933) 6. P. Debye, Physik Z. 22; 84-9-356, (1932) 7. R. Lucas and P. Biquard, Jour. Phya. 3; 464-477, (1932) 8. 9. R. Lucas and P. 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