m l ( ll IH‘IMHIIM “WIN“ 4 133 359 THS £31 STU”?! 93"“ {BE SD:S“‘C-‘U"Q3hz RA$OH “ wwzs N " I u? E‘" mi T1: Ami-é: was L“?! h Tbssis $32 the 9091'” ‘7" M 3‘ wag-4- GAN $143.1": UNEVE RSE‘E‘Y 1435:2234: Ad’ém‘ 395:3? Imumnmfifit'fim M 41114441411114 mu 3 1293 017 L I B R A R Y Michigan State ‘1 University A STUDY or THE msvrom'xon or rum: mm unmrc I wms In Lmums Lune Adlor A Thule Sub-1th“ to the College of Science and Arts nation 3““ university of Agriculture and Applied Science in pea-tie]. fulfill-eat of the reqtnremt for the degree of “ASTER 0? mm mm of Milan 1961 s» \‘1‘1 NW The euthor Viehee to express his thanks to Dr- E. A. Hiedennn for his guidance in thin study. Thanks ere else due Dr. K. A. Male, Dr. H. G. layer and B. D. Cook for their eid end suggestions. Espeeieny, the euthor eeo bounds" the timid. eeeietence given under the U. 8. Ammanecmtmtvithmtvhichthemkeoudnothfle been ecec-pnehed. L. Adler ABMOFTEEDIS'I'ORIION WMWUHPRABONIC VAVESINLIQUIDS hula Adler An Abetrect Submitted to the College of Science end Arte Miehim Stete “amenity of Agriculture end Applied Science in pertiel fulfills-at of the requirement for the «me of mascm, Wofmeiee 1961 W ‘IhereteetvhichthehernoniceeredcvelOpedduringthe propegetion of en initiem sinusodiel ultreeonic were depends on the noelineerity of the liquid. A steinlees steel plete us need ee e filter to peee on]: the desired Mimics. The fete of growth of eech of these henonics, es e hmction or distenu end pressure, vee neesured by eeene or the diffrection pettern produced by the were. Fro- the result or these neeem‘e-ents, the Mr B/A (describing nonlineerity) vee celculeted to be 6.2 i .6 end 9.6 i' l for veter end n—qlene, respectively. Il‘hereteorgrcnlthofthehex-Ilonicseeensured, deviete fluent free the velues predicted from the theory for e dis- eepetimlees .diu. An epproainte theory wee suggested to consider both generetim end ebsorption or the hermice. Peir easement vee obteined between this epproxinte theory end the unsecured velues of the Mel. Chepter I. II. III- TABLE OF CONTENTS MROMIOIVO e e e e 0 e e e e e e e e e e e mom. 0 O O O O O O O O O O D O O O O 0 O 0 Theory out londieeipetive Liquids . . . . theory or niseipetive Liquids. . . . . . WW. . . . . . . EmerlmtelArrengmt. . Preesmlheeurenent. . . . beam-tittiuocfflemice. celibntimofl’lete . . . . lleeeument of Seccnd Remonic Meow of Second Bunnie. heeurenent of Third mimicjeter. . . lvelmtimoi’B/Avelues. . . . . . . . Weeeeeeeeeeeeeeeeeee BIHJMRAPHI................. L“: 15 18 19 23 23 26 3O ’1‘- fig. 1. 2. It. 7. 9. FIGURES Exponent-e1 Arrangement. . . . . Denonstntion of Second Bennie. Denonstretion of mm Hex-Ionic . Second Ber-Ionics vs. I: muss, Vets: Wt“. . . . . . . Second Mics vs. 1: veluse, Heteg'meoreticel....... W of Second Harmonic vs. n-xylene W. . . . . mm Ber-mic vs. 1: velue, Veter, prerimtel end Woe]. . mum of B/A, Heter. . . . . lvelution od’ B/A, n—qlene . . . -12 .15 .17 .21. .25 -27 .28 CHAPTER I WON In generel, e sound been chenges its wave shepe es it propegetes in e fluid whose pressure density nhtimship does not follow Roche's leaf. m. .ens thet hernonics ere genereted es the weve progresses. In the ones of infinitely lull enplitude, this cheese in ueve shepe iseosnllthetonecenneclectit; however, intheeeseefefinite enplitude send weve, this effect hes to be considered. the distortion ofefiniteelplitudeveve iednetothenonlineerpropertyofthe nsdiu in which the ueve propesetes. The positive and negetive incre- nentsinpmsureereinpressedonthenssofnedim. IIlaschsngein thsvelmocfthenssnotbeingeqxnl,thevol-schengeforthsposi- five pressure will be less then the Mt. chenae for negetive pressure. nus-u, Jenkins end out-111 studied finite mums. effects in gesee. nephevepointedwtthettheeqmtienofetete foredie- betic processes shove thet the lime: relationship between pressure endvolansdoesnothold, endemsequsnth, leedstothsdistortim of the ultresonic revs. nae: gene on epproxinte solution for the generetion of hereonics es s function of distenoe, findenentel pres- sure end frequency. Pox end Wellecez, follwin; the seas ergusent for liquids, as- sued thet the following power series expresses the reletion betveen pressure end density ‘P:E+A.§.:L§E+%K3-§%9)z+nu (1) where P is the presume end 9 is the density. Po end 30 ere the presume and density of the undisturbed medium. B end A ere pere- nsters of the liquid which ere functions of tamperettme. For end Wallece, insteed of solVing thsqmtion of motion, sug- gested e grephioel enelysis of the distorted veve. As the: pointed out, the anthem steepening veve fruit would ultintely tons e discontinuity without sun stebilising nechenisn. This stebiliting snobnisn is the absorption of the ultreseund in the -diun. Pox end v.11... evehnted the nonlineerity pereneter BIA for 11mm. using compressibility dete. rubini-ahirm3 used e .mm treetmnt to thet of Fox end Uelleee, Minsteedofegrephioelnethod, housedenenslytioel treetnnt of the finite enplitude veve distortion. His theory essunu tint the nsdius is dissipetionless end the expressions for thsgrovthofhemics holduptothediscaltinuityoftheuve front. as. enelyticel method hes been redeveloped independently by Keck end Bsyerh, end Hex-groves. Hereafter, this Isthod will be referred to es the Pubini-Ghiron method, however, the notation of M will be used. mwulvorkves oerriedoutintheleetdsoedeto duonstrete end neesure the distortion of e finite emplitude veve in liquids. Zerenbo, Kresilnikov, end Shklovsksievxcrdi6 shoved the de- pendence of the ultresonic ebosrption coefficient on sound pressure. they used two different methods: first, they used e theml probe es a. receiver end enelyzed the hermit: stmctxme d the distorted unve; secondly, they observed directly the hernonics in e trevelinc veve. Musedenecoustic filtertoupmtetheflndenentelendits hermics. Wandmedennn7expleinedtlnttheeppeereneeofths esp-tryintherefreotionoflight byenultreeonic veveiseeused bythsdistortionoftheveve form. mi end niece-n3, in their study of 11m diffrectim by en ultrescnic been, explsined the es-ymetry of the diffrection pet- ternhythe distorticnoffinite enlitudeveves. neiruthodves en indirect vey of nseeuring hereunto structure. M essmsed e certein peroentego at the hereonic content end fitted it into the light intensity neesurenent of the diffrectiui pettern. monotherpeperg,theydeecnstretedthepseseneeofeseeond hemiebyuseofe-stelpletevhichservodesenecmticfilter. messccndher-sniovhichpessedthrcughthsfilterplete servedes ediffrectimgreting. The exPerimentel method of using a filter plate in combination with optical diffraction has nov been used to study quantitatively how the harmonic content of an ultrasonic wave varies with distance from a sinusoidally vibrating source. The results of this study, and the non-linearity parameter B/A for water and xxx-xylene derived from them will be given in this thesis. CHAPTER II THEORY mailmdissipetlwm The mat of distortion or 3 lug. amplitude sound wave is ex- pressed in turn of the aunt of bunnies pro-cut. M, Jenkins cndO'Noilobtcinedssolutionforth-mondnndthirdmiuby solving the caution of action for the aeolian-u ave. Their solutin is an ”predation for all pronouns not con- Iidoring my diuipstion of the ave. m expression for the mum of the second hon-mic .1 be given u 1’2“ 1M“ x133? <2) - 236Co mnxuthedimmthtmnittormdthemum. 'p \o mdiltrubodnediu. tie the immortheultmund, coiotbo isthonmdmt-lpnumntx-o. 9° isthdmityottho sound velocity, and B/A is the nonlinea- punmter of liquid. Exporiuntnhowsthotthelimrrohtionbctmntheuplim ofthemondhmnicnnddiutmholdsonnfornvewmllngion or distance. At motor dist-non, the «and We levels off. Buchebohlvior'il umbythethooryotl‘ubini-Gbim3. non- ‘thcory, intudofsolvinatheoqmtimotwtim,mlmlytiul methodmcerriedthrcughbetveenx-Oendthepointofdiscon- tinuity; thet is, vhre the veve would reeoh discontinuity due to the distortion. With the essmtion thst the cxest propeates tester then the trash, mbinioGhiron3 obteined en expression for the up- nm or the nth hum-uh in e dissipetionlese medium which my be given es 1:“. 12.310. 3h(nk) (3) when pn is the mum at the nth uremic, p10 is the nude- utel wessure, J’n is the nth order Bessel reaction, 3 is the much-J. penneter of the discontinuity distenoe krlé , end where L a 3.c: {2 [(qumhfl" m 'nlistheerygivestheright shtpe forthehsreenie generetiou, esitlevelsorfthecurve,butitstilldevietesfmtheexperinentel mm. Inthefirst ennui-Moe tormlldisteneesenduellpm- sures the tee expressins («nation 2 end 3) ere identieel. i‘he theory offlhid-Ghironvillbemdinemdifiedfmtodesoxibetheprope- anion of sound in e liquid. ' m of Dissipetive Liguids me we: to eecomt for dissipetion of s finite mlitude ultre- sonio wave is the nethod used by Thures, Jenkins end O'Neil. my essxned thet eeoh hes-sonic would be ebsorbed independently of the other. Here, however, we essuse e slightly different nodelvhioh hes ledtoebetteregreenentbetveenthem-ysndexperi-ent. Hedivide the discontunity distenee into mu intervels end «an. thst et mdisteneetheaeneretedhenoniooenbeeeleulstedfrathetheou ofFubini-dhiru. 'l'hedissipetionoftheherunioisintrodneedby usuinzthetintheintervsldktneeverecevelnsofthegenereted hereoniovillmderaotheebsorption. nuntheeetmleeenntof mispresentinesehintervelisthemtgeneretedeinusthe mtebeerbed. Let us consider the expression for the nth hernonic es expressed by Pubini-ahiron3. Pn‘%%g In (nk) (5) The nth order Bessel function csn be expended into the following in- finite series . J..(x)=f 44F (1’9sz (6) 3‘0 3! (144“! Impending emtim (6) end neglecting terns higher then the seeond, we obtein for equetion (5) 3'1: nk n-l - nk NJ 7 ‘P,o 2"" n! l'mirH-Ii! ( ) Io find the smut of nth her-onio dissipeted, we nee. en wtisl ehsorption of the oversee hensnie present, thus 1: “15;. -«‘.. x J = - - I -—e (8) (pm)dass;pa&ed ‘Pw ( ) where OInistheebsou-ptionooeffieientofthenu‘henuieinthe -61“!- meevereaevelueoftherelstivemmtofnthhermieis 31me (9) 15.. .. 133:3: 3M?“ - the“ W 13“, ydk 2""n n! 2.““(h+2)! Substituting equetim 9 into eqmtion 8, we obtein K35) =“( k)“. -(HKY‘H 0-50%“) (10) pm dl$S|PG+Cd g";"‘ n n! 2n“ (n+2)! besetulenenntofnthhennniopresentistheenountofhsrnonie generetion (emetic: 7) nimas the e” of her-mic dissipeted (eqmtion 10). (1-1) in nk““ (nk)“”' _[nk ““ nk “" -cxnkL) 'P ‘LzT‘LT 2"”(nm‘ gfifi'WQ’e ‘0 wheres-kl .mtheseeondhernnnie, equetimllheeuses P k k3 -O< kL 1a., 4 s' *[4 2413 2 ”2’ end for the third hsmonic p 320 640 |O B : J'i-z. 8‘ 4+ [153... Lani—3:] Q.(x3kL' (13) 10 thistheoryis linitedtovelues of 0c , k,endL, suchthetthe generetion of the hemics is not influenced by their ebsorption. It is else lilited to disteneee less then the discontinuity distense. By sstisfyina these conditions experimentally, welues of the hermeios es ef‘netionofdistenee whiehvere neesured cwldhe emperedwith this theory. These results will be given efter e description of the experiment. CHAPTER III EXPERIMENTAL METHODS Experimental Arrangement The experimental arrangement used is shown in the schematic illus- tration in figure 1. Light from the mercury vapor lamp 8 was condensed by lens L, on the source slip B. Lens L2 was adjusted by autocolli- nation to render the light parallel. The light passing through the tank was focused by lens L on the entrance slip B2 of the photomulti- 3 plier microphotometer, a filter I which passes only the Sh6lA line of mercury was placed between slit 82 and the photomultiplier Ph. The ultrasonic transducer Q was an air backed 1 inch square X-cut quartz crystal. This crystal was excited at its fundamental frequency by an RF oscillator which had a.maximum output of 50 watts. The RF potential across the quartz was measured using a high-frequency vacuum tube voltmeter. A.specially designed tank T described elsewhere, 10 was used to eliminate reflection of the sound beam, P was a stainless steel plate approximately 1 mm in thickness, used as an acoustic filter to select the desired components of the sound wave. Pressure Measurement .As is known, the ultrasonic beam acts on the incoming light as an optical phase grating. The angle of the diffracted light in the 13 21th order is expressed as Sin 6 sin-a7t (1%) Where X is the wave length of the light, A * 15 the wave length Of the wound, and n can be any integer number. The intensity of the diffracted light in the nth order for small sound pressure was derived by Raman and Nath11 to be 2. In: 3.. (v) (15) where Jn is the nth order Bessel function. The term v is expressed as V: mg... A where a is the width of the sound field and/u is the variation of the refractive index of the liquid due to the density variation caused by the sound beams The variation of the index of refraction is caused by the condensations and rarefactions in the medium as a consequence of the sound pressure. This variation is assumed to be proportional to the sound pressure. The measurement of the intensity of the diffracted light leads to the value of the variation of the index of refraction and indirectly to the sound pressure. However, the difficulty is that the relation- 1h ship between pressure and the variation index of refraction is not very accurately known. A.theoretical relation was derived by Lorentz-Lorenz 12, 13 for the variation of the refractive index, with the assumption that the molecules are optically isotropic. It is known, however, this does not hold true for many liquids. Raman and Krishnan 1h tried to use some correction term con- sidering the anisotropic property of the molecules, but to apply their result in actual calculation is too complicated for the pre- sent purposes. Several empirical formulas were worked out for the variation of the refractive index with pressure, of which Eykman's 15 fermula seems to fit best with the experimental measurements for work on water and nnxylene, which are the liquids used in this work. From the Eykman formula the variation of refractive index is found to be #- W+i4flo+-4J (l6) .8,uo+| where lie is the refractive index of the undisturbed liquids. From this formula, the relationship between pressure amplitude and the Raman-Nath parameter v is P = 'r Oim- (17) 1. 61. 15 for water and P = .25 4‘:— o’rm. (18) for m-xylene . With these expressions, it is possible to determine the pressure amplitude from the light intensity in any diffraction order. The procedure used was as follows: The relative intensity of the diffracted light in the first order of the diffraction spectra was measured with the photanultiplier microphotoueter. From the measured intensity values, the Hanan-Hath parameter v was calculated using equation 15. From the evaluated v-values, the pressure amplitade, was determined by means of equations 17 and 18. genonstration 9f the Higher Hamonics As a demonstration that the higher harmonics of the fundamental are present as a result of the distortion of the wave, the following emeriment was made: the filter plate at the front of the sound beam was rotated through a 90° angle and the light intensity of the second order was recorded. Because the spacing between diffraction orders is proportional to the frequency (equation 11), any second harmonics present would contribute to the second order in the diffraction pattern. Ba on. H.572. @05qu .m> ZOHmmHZmEmB g 950w OHZOZEE 9200mm ho ZOHBEmzozmn HH arm ..(QEQNK ...Usmeflk ..1)..h/\..\Ml.s..\.\ 4‘Q\G.W\u\<\mp Qtfllklfigk “(Q 4(Q\QW\V\<\M¢ how. 13 Q 1Q Qm. Qm. M.\ D K.\ QM. QM. 1Q _ O .o.\ Gm. ..ToQ _ m l m a . _ _ _ _ _ _ c l _ 41 9%: 5 . . .. 17 a; 09 592 MDZWDHUZH .m> ZOHmmflfig Egg BSOm .959 u e\ 4TQ\D W\~\<\m. 3.. .2 Q .3 on. . _ _ _ 4 UHZCESH QmHmB mo ZOHEBWZQEQ HHH 85mg ..UNAKMJ h «K QHL Q95 QQ Wee e o a. _ . _ _ _ d .8? x . QVQBQQSM. 0m. 1Q Q .m.\ Qm. sum £52337: _ e _ _ . _ : :. 18 At first, a 3 mo quartz was used and placed close to the plate and a very low sound pressure was applied. under that condition the finite amplitude effect is negligible. The transmission curve is shown in figure 2a. Since the light intensity is a function of the sound pressure, the peaks show the transmission of the fundamental pressure. Then the transducer was moved back to 50 cm and the ultrasonic pressure was increased to l atmm under this condition, the experiment was repeated. .As the result shows, in figure 2b, the extra peaks are indicating the presence of second harmonic since they coincide with the transmission peaks of the 6'mc sound wave shown in figure 2c. A.similar measurement was made to demon- strate the presence of the third harmonics as shown in figure 3. This method gives the possibility to select the angle of maximum transmission for the different harmonics. It is also demonstrated that the harmonics present are not generated in the transmission pJUate, but in the liquid. Calibration of the Plate One cannot expect that the energy of the incoming soundeeam to the plate will be transmitted into the outgoing'beam.one hundred percent, for due to the multiple reflection inside the plate and the generation of shear waves inside the solid.medium, there will be a certain amount of loss. To detect the sound energy loss, or more correctly, the amount 19 of sound transmitted, measurements of the sound amplitude were made before and after the plate using the optical method. To obtain a calibration for the second harmonic of a 3 mo wave, a 6 me source was used. The ratio of the outgoing sound amplitude to the incoming sound amplitude gave the transmission coefficient of the plate. The trans- mission coefficients Obtained were .92 for water and .7h for'm-xylene. For the measurement of the second harmonic these coefficients were taken into consideration. Measurement of the Second Harmonic in water In the actual measurement of the second harmonic, a 3 me quarts was used to generate the fundamental frequency. The plate was placed near to the light beam with a selected angle to the sound wave, at which angle only the second harmonic was transmitted (assuming no fourth harmonic is present). The fundamental and the third harmonica were reflected and the second harmonic was measured, using the method described, at various distances and at various fundamental pressures. The fundamental pressure was measured by measuring the quartz voltage. The quartz voltage was calibrated to the acoustical pressure by an optical method using equation 12. The distance was varied from zero to the discontinuity distance and.the fundamental pressure was varied from .1 atm to 1.1 sum. 20 In figure h the relative percentage of the second harmonics was plotted vs. the fractional parameter of the discontinuity distance 5 fbr 3 different fundamental pressures: 1.1 atm, .88 atm.and .hh atm, with the corresponding discontinuity distances: 39 cm, #8 cm and 97 cm. The solid line represents the theoretical curve for a dissi- pationless medium. .As one can see, the experimental points are in good agreement with the dissipationless theory for low k values. As it was expected, the medium.can be considered dissipationless if the sound wave propagates a small distance. However, with increasing k values, the experimental points deviate more and.more drastically from the theory, indicating that the medium cannot be considered dissipationless when the propagation distance is too great. It is significant to point out that the deviation from the theory is greater for smaller pressures than for higher ones. Since the discontinuity distance is inversely proportional to the fundamental pressure, the absorption becomes more dominating, as the sound wave has to travel farther to reach the discontinuity distance. Figure 5 shows the theoretical curves considering dissipation using equation 10 with the experimental points. .As we can see, there is a.mmch better agreement between theory and experiment, even if one considers only this simple model for dissipation. For small pressures the agreement is much better than for higher pressures. There is a . possibility that at higher pressures some fourth harmonics are gener- x .m> UHZOEEE Emooxm PH shaman ...... .. 3 e. m. a. e. u. e. m. m. l. I4 _ e _ _. _ _ _ _ n . 0 sen: .. sees... an e e e + 0 some. V ...Suemmwefl o o 0 Sum? .5. .aniuefi + + + deoiukooik amulet. .N +e ofiflhx e Elk) me. Q\.. me. Suede. "so. @ .33.? so. 3. anohlk ox PVQBQQI .II x edema «wedlock .3th .Sonenuwfixo 3..“er e e e «Sue m. “.8. .E\e\.\u9Q.+ + + as m ... u .. here; x .m> 320% 68on > an 2.x .3. ON. MN. on: 1...... Q‘. .m ated which also pass through the plate. But probably some more terms should be included into the dissipationless theory. The fact that absorption affects the generation of harmonics could cause some of these discrepancies. Measurement of Second Harmonics in.m-xylene The same method was carried out to obtain measurement in.mpxylene. The measurement was more difficult for higher k values than in water. When high ultrasonic pressure was used, the disturbance in the liquid was very high. This disturbance is attributable to the presence of a quartz wind. Plate vibration could also be observed in xylene more than in water. However, for small k values the experimental points are in good agreement with the theory (figure 6). fleasurement of ThircLHarmonics in Water To show that the method is applicable for measuring the value of harmonics higher than the second, the plate was calibrated for the third harmonic in the same way as was described for the second harmonic, and one set of measurements was carried out. .As was expected, figure 7 shows that the experimental points deviate more rapidly from the dissipationless theory than in the case of the second harmonics. Since the absorption is proportional to the square of the frequency, this can be explained. The approximate dissipationless theoretical curve shows good agreement with the experi- 2h x .m> 055985» 9200mm H> 0.5mm u n M be m. o. s. o. o. .v. m. m. A. 1 1 1 q 4 8 ~ . 4 q 1 a a. den 2wre\w.m1xm o oo . . usumeoMIL . . . L on. szmWV 0 e O . o o O O . I oo . he 00 0 d o o o o ..QN all: 0 W .. bu. I. one .. mm. OE m «k mzmux x12 a .m> neg Ema. H; enema he: 25 qqrnzmké QNQXM o o o \AQQNIF thudcumeo I II \AQONIK WWWVZnXK-TOQ Wmlx 0 ....III . Q2WQWM DEW u.\ WGKSS rule MN. 0m. mm 26 mental points, indicating the correctness of the assumption for dissipation. Evaluation of’g/A Values for Water and mexylene The equation of state for liquids expressed by Fox and wallace2 introduces the nonlinearity parameter B/A. The growth of the second harmonic of finite amplitude sound waves is related to the B/A values, thus measurements of the second harmonic can be used to determine the value of B/A. A.quick way to determine the nonlinearity parameter is to use the dissipationless theory for harmonics as developed by Pubini - Ghiron. Considering the expression for second harmonics and neglecting higher order terms, one obtains B = -. L. /A COHS‘L —% a (l ) Thus, measuring second harmonics at various distances and various pressures pa 2 leads to the value of B/A. Figures 8 and 9 show 40 X vs. T)“, fer water and mpxylene. The extrapolation of these curves give the B/A value. As one can see, the evaluation of B/A occurs at XPlo - 0, obviously satisfying the condition for the dissipationless case.‘ Although at higher XPlO values, the curves deviate from the theoretical curve, these points are not used for the determination of the Elk value because the extrapolation 27 damn u - nets .8 ohsmwm \Eu-k\\Q\ \om o‘ .3. 9.. MN 9.. .3 E m. D V. _ _ _ _ _ m ._IIIII.;. . I; V. M e e e +x 0 x x e e. eNSu-§o.\ + x l n. nu m? x‘mMR . X0 C O N\ . O . x O 1... X9. . 1% N.\ e Xe. ... Some e e I Te: 3 e VQ. .... meo O 0 g 2mm. HQ‘QX X X .Eku\.\u odm... + + 28 05 m H .w 0 vcmfiegsfi MO «\m ..6 zoflsamfimo an enema mmtuugxek X sq 3 2 2 Os a. . o ... N _ _ _ _ n _ _ _ IIJ oak. zwtfiuuxw e e e dautueouefi. O / . e e e e e Kin... neglects the effect of dissipation. Using this approach, the value of B/A for water was found to be 6.3:.6 and fur mpxylene 9.811. 30 CHAPTER IV SUMMARY OF RESULTS A method for quantitative measurements of harmonics as a result of the distortion of a finite amplitude sound wave was developed. This method used an acoustic filter and the optical diffraction method of'measuring ultrasonic pressure. Measurements were taken for the relative percentage of the second and third harmonic growth in water and for the second harmonic growth in mpxylene at various distances and various fundamental pressures. .As was expected, there was a drastic deviation between the dissipationless theory and the experi- ment, indicating absorption of harmonics. An approximate theory was suggested to consider the dissipation of the harmonics. A.reasonable agreement was obtained between this theory and the experiment for small pressures. It has been shown that one can evaluate B/A from distortion measurement even though the exact theory is not known. Iron these measurements the nonlinearity parameter B/A is evaluated as 6.31.6 for water and 9.6:}.0 for m-xylene. l. 7. 10. 31 REBELOGRAPHY A. L. Thuras, R. T. Jenkins and H. T. O'Neil, J. Acoust. Soc. Am. 6, 173-180, (1935). F. Fox and w. Wallace, J. Acoust. soc. Am. g_6_, 99u-1006, (1959) E. Pubini - Ghiron, Alta Frequenza, 3, 530, (1935). w. Keck and R. T. Beyer, Phys. Fluids ;, 3116-352, (1960). L. E. Hargrove, J. Acoust. Soc. Am. 2, 511-512, (1960). V. A. Krasilnikov, V. V. Bhklovskaia-Kordi and L. K. Zarembo, J. Acoust. Soc. Am. g2, 6h2-6h7, (1957). M. A. Breazeale and E. A. Hiedemann, J. Acoust. Soc. Am. 22, 751-756. (1958). K. L. Zankel and E. A. Hiedemsnn, J. Acoust. Soc. Am. 21, hit-Sh. (1959)- K. L. Zankel and E. A. Hiedemann, .32, 582-583, (1958). L. E. Hargrove, K. L. Zankel and E. A. Hiedemann, 11;, 1366-1371, (1959). c. v. Raman and N. s. Nath, Indian Acad. Sci. _A_2_, 1+06-h12, (1935). ‘53, 75-8h. (1936). ' 13. 1h. 15. 32 H. Lorentz, Am. Physics, 2, 6+1, (1880). L. Lorenz, Am. Physics, 1;, 70, C. V. Raman and K. 8. Krishnan, Ser. A0 589 (1927-28)’ J. Eykman, Rec. Trav. Chim. 13:, (1880). Proc. Roy. Soc. (London) 201, (1895) . L11, MICHIGAN STQTE UNIV. LIBRRRIES 31293017640180