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LIBRARY ‘ Michigan State _ University THESlS This is to certifg that the thesis entitled MEASUREMENT OF THE VELOCITY OF SOUND IN WATER BY OPTICAL METHODS presented bg Arthur Jared Crandall has been accepted towards fulfillment of the requirements for Doctor of Philosophy degree in Physics' Major professor Date [7 Wu/ 76 7 0-169 MEASUREMENT OF THE VELOCITY OF SOUND IN WATER BY OPTICAL METHODS Arthur Jared Crandall AN ABSTRACT Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 ABSTRACT Several new optical methods have been developed for the measurement of sound velocities in transparent fluids. The visibility pattern from stationary, pro- gressive and pulsed progressive waves are detected by a fast response photomultiplier tube rather than visually, which allows the use of lower sound intensities. A theo- retical expression is developed for the errors caused by diffraction in the near field of a circular transducer. The velocity of sound in distilled water was measured to be 1517.70 : .20 m/sec. at a temperature of 3H.OOOC. MEASUREMENT OF THE VELOCITY OF SOUND IN WATER BY OPTICAL METHODS by Arthur Jared Crandall A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 ACKNOWLEDGEMENTS The author wishes to thank Professor E. A. Hiedemann for his guidance and patience. Throughout this work Dr. B. D. Cook gave valuable suggestions and provided many helpful discussions. Without the inspiration and help of Dr. F. Ingenito the theory section of this report would have been more complex and less exact. The finan- cial support of this research by the Office of Naval Research and the National Science Foundation is gratefully acknowledged. ii II. III. IV. VI. TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . DESCRIPTION OF EXPERIMENTS . . . . THEORY . . . . . . . . . . . . . . THEORETICAL RESULTS . . . . . . . DESCRIPTION OF EQUIPMENT . . . . . EXPERIMENTAL RESULTS . . . . . . A. MEASUREMENT OF THE RETARDATION B. SOUND VELOCITY MEASUREMENTS . . BIBLIOGRAPHY . . . . . . . . . . . Page 22 33 1+2 1+2 1+6 55 Figure l. 10. ll. 12. 13. 1h. 15. 16. 17. LIST OF FIGURES Standing Wave and Progressive Wave Block Diagram . Pulsed Progressive Wave Block Diagram -.. Pulsed Time Coincidence Block Diagram . . Zeroth Order Schlieren Photograph of a Sound Coordinate System . . . . . . . . . . . Axial Pressure . . . . . . . . . . . . . . Pressure Distribution . . . . . . . . . Wavefronts (Phase Distribution) . . . Relative Retardation Integrated along a Path the Center of the Sound Beam . . . . . Off-Axis Retardation . . . . . . Relative Axial Optical Phase . . . . . . . Optical Bench and Transducer Mounting . . Tank and Temperature Control . . . . . . Retardation (ZmHz) . . . . . . . . . . . . Retardation (lmHz) . . . . . . . . . . . Relative Phase (lmHz) . . . . . . . . . . . through 0 0 Relative Phase - Comparison of Optical and DelGrosso Result 0 O O C O O 0 O O O O O O O O 0 I 0 iv 0 Page 10 11 13 15 2h 25 26 28 29 30 3h 39 us an 1+7 50 LIST OF TABLES Various Sound Velocity Values Progressive Wave Measurements Standing Wave Measurements LIST OF TABLES Various Sound Velocity Values Progressive Wave Measurements Standing Wave Measurements I. INTRODUCTION In Spite of many measurements of the velocity of sound in water, made by various techniques, there is still no universally accepted value. Before the development of the piezoelectric trans- ducer the measurements were generally confined to audible frequencies, 1,2 which entailed prOpagation in a lake or ocean for free field results , or propagation in a closed pipe or Kundt's tube3. These early measure— ments were not satisfactory for the absolute determination of sound velocity because, in the first case, the physical parameters could not be accurately specified and, in the second case, the effect of confining the sound field was not accurately known. With the development of the piezoelectric transducer and the associated electronics, continuous high frequency sound beams could be generated. With the possibility of many sound wavelengths in a small volume, Hubbard and LoomislL developed an interferometric technique for measuring sound velocities in liquids. This inter- ferometer (based upon an interferometer designed by Pierce for measurements of sound velocities in air) consisted of a fixed quartz plate transducer and a movable, plane reflector. When the reflector is translated, standing wave resonances every half wavelength are indicated by variations in the transducer impedance. The interferometer, with many changes and improvements, achieved a great prominence for its high precision and small sample size. .muanmmu m.cmamamouo wean: 0000.0m cu pmumsmwm mHoB mmSHm> mmoga * mo..H HH.m0mH umumaoummumucH mmma ommouwamn oH._H mo.mOmH cowaumasoo manna mmma aaaaxmoz ma. H.m0.m0mH “mumaoummumucH :mmH mauswfiH om..H oo.m0mH eflmammmnm - madam seas munmuowsun tam Hwnmnzmz :m. H.oo.00mH mmaam owmfl axooum mo..H 4:.00ma uswwam mo mafia Nmma cmmmcmmuo mo._H mw.00ma HmuoaoummuwucH :mmH .Hm .um owmouoamn N. H m.mOmH Hmofluao mmma Hmsmunom H H.m.mOmH umumaoummumucH wmma mwaooq use cumnnnm mme on oaofl mane neasm mswa smegma can neasm Aomm\sv waHooqm> nomamz mean mz QZDOm .H mgm> A) was not 9 15 completed until 1966 by DelGrosso . Grabau observed the equivalent behavior at long wavelengths (diameter re A), as early as 1933, which 16,17 led Grossmann to a theoretical investigation of this behavior in l93h. The second world war provided the impetus for sonar and radar development which consequently brought both increased need for accurate sound velocity measurements and SOphisticated pulse generating and receiving equipment. There are a great variety of pulse-type methods, but only three which may be considered milestones, will be described. Of the three pulse-type measurements, Greenspan's National Bureau of Standards time coincidence pulse measurements are perhaps the most useful. The apparatus consisted of a 200 mm long steel tube terminated by transducers wrung on to the ends of the tube. His measurements are the first in which the sound velocity was measured over the entire 00 to 100°C range with a method of very high precision. The resulting velocity versus temperature data for pure 19,20 water are highly regarded by researchers in the field and are used to compare results from measurements done at different temper- atures. At a temperature of 300C he obtained a velocity l509.hh i .05 m/sec which is much lower than DelGrosso's earlier results but higher than his current results, by about .33m/sec. This difference is many times the specified probable error of either measurement. In a 10 x 5 x 5 ft. cypress tank, Neubauer and Dragonette21 performed an approximately free field measurement. A nearly spherical transducer radiated spherical wave pulses to a small probe-type trans- ducer which could be positioned at accurately Specified positions in the tank. The difference in time of arrival was measured for two probe positions colinear with the source. Unfortunately, tap water was used in the tank rather than distilled water and thermometry is somewhat more difficult in a big tank. However, if a correction for water impurityl8 is subtracted from their values, a pure water sound velocity of approximately 1509.0 i.°2 m/sec is obtained. This value is somewhat low compared to recent interferometer measurements but is very close to a similar measurement by Brookszz which, when a small 7 free field diffraction correction is applied also gives a pure water velocity of approximately 1509.0 :,.3 m/sec. The last pulse type measurement to be described was 20’23 at Bell Telephone Laboratories. His physical done by McSkimin arrangement consisted of two precisely ground and polished fused silica buffer rods separated by a fused silica spacer ring which formed the sample cavity. A pulsed continuous wave is generated by a 20 mHz fundamental frequency transducer mounted on the end of the fused silica buffer rod. The wave is then interrupted and the echoes are observed on an oscilloscope. Interference effects from overlapping echoes allow the adjustment of the wave frequency so that an integral number of wavelengths occur in the cavity formed by the Spacer ring. The adjustment of the wave frequency,which yields results of high pre- cision, is very critical. However, the length of the Spacer ring was, at most, 12.7 mm, which when measured even with the best comparator, placed a definite upper limit upon the overall accuracy possible (rvl:105). Also, mercury-in-glass thermometers, which are generally less satis— factory for absolute temperature measurement, were used. These measurements were carried out over a temperature range of 200C to Yfioc. Comparison whlch Greenspan's measurements Show an essentially constant difference of .36 m/sec over the entire temperature range. The sound velocity versus temperature curve is parabolic with a maximum around 7&00. Since the difference between the measurements of Greenspan and McSkimin is constant,and not temperature dependent, thermometry is eliminated as a potential cause of this difference. McSkimin obtained a sound velocity in pure water which compares very favorably with DelGrosso's later results and is only about .1 m/sec higher than the corrected free field results of Neubauer and Dragonette and those of Brooks. Other referencesSiZbr’Z5 describe a great many more techniques for the measurement of sound velocities; these will not be discussed in this thesis. With this large group of carefully planned and executed experiments, and the differences and agreements which exist among them, a desire was expressed at the seventy-first meeting of the Acoustical Society of America for a new, independent method of measur- ing the velocity of sound. With the extended experience of optical methods for the investigation of acoustical fields here at Michigan State University, it was an easy and natural step to undertake another investigation of sound velocity measurement by optical methods. In the thirty years Since Schreuer's investigation, new equipment such as the laser, pulse generating equipment, and a fast response photo- multiplier permits the development of new optical methods. In fact, three new methods were devised, continuous progressive wave measure- ment, pulsed continuous wave measurement and pulsed phase comparison measurement. Also, with this new equipment Schreuer's experiments could be repeated at considerably lower sound pressures, which means that the heating effects which were not negligible in Schreuer's work were greatly reduced in the present work. II. DESCRIPTION OF EXPERIMENTS Bachem, Hiedemann and Asbachz6 and Nomoto27 have shown that when a collimated beam of light passes through a stationary ultrasonic wave, the wavefronts of the sound wave may be made visible. The stationary sound beam consists of condensations and rarefractions which cause a periodic change in the index of refraction of the medium. AS the plane light wave passes through the sound beam those parts which pass through condensations become retarded in phase relative to the parts which pass through rarefractions. As the light wave moves further away from the sound beam, the phase modulation produces an amplitude modulation which may be seen with a ground glass or a microscope. This is called a ”visibility pattern” and results from "secondary inter- ference”. This visibility pattern, as viewed by the eye, does not move in the direction of the sound wave because the acoustic wave producing it is stationary. For the case of a progressive wave or pulse, the visibility pattern moves with the sound wave, like a shadow. This will be described in a more mathematically satisfying manner in the Theory section of this report. One could use a Kerr-cell stroboscope, as Bachem did, to "stop" the motion. However, a photomultiplier capable of responding to magacycle signals is a valuable alternative because it permits an oscillosc0pe display of these variations in light intensity. The continuous wave and standing wave measurements were made with the apparatus shown schematically in Fig. l. The photomultiplier converted the fluctuating light intensity to an electrical signal which was amplified by a series of three wide band amplifiers. From the output of the amplifiers, the electrical signal was connected to the vertical input of the oscilloscope. the transmitter, was distributed by the transducer and to the horizontal plates phase of these two signals was compared pattern. The experiment then consisted The rf power, generated by matching circuit to the of the oscilloscope. The using the resulting Lissajous of observing the Lissajous pat- tern on the oscilloscope while translating the transducer. The translation was measured for an integral number of wavelengths. Frequency Transmitter Matching Quartz Counter Circuit %8' l l l l l l i—- l CRO H P.M. VI Amplifier Figure 1. Standing Wave and Progressive Wave Block Diagram. Transmitter Matching Pulsed Circuit Amplifier s r—w ‘ -llHlll II A P.M. Trig OLE H Frequency Variable Counter Delay 2 axi Pulse CRO Generator V p 1 ier Figure 2. Pulsed Progressive Wave Block Diagram. The circuit for pulsed continuous wave measurement is shown in Fig. 2. The transmitting circuit now includes a pulsed amplifier which generates a ten microsecond pulse every l/60th of a second. The photomultiplier circuit is identical to that in the previous discription. Finally, in order to observe only the effect of the pulse, a suitably delayed square wave from a Dumont pulse generator modulates the oscilloscope intensity (2 axis). Thus the oscilloscope trace appeared only when the sound pulse traversed the light beam. The measurements were carried out in the same way as the continuous wave measurements except that as the transducer was translated, the delay in the oscilloscope mod- ulating pulse was also changed. 10 Frequency Pulsed Counter Oscillator Audio Pulse CRO Oscillator Generator T . -fi rig. v Figure 3. Pulsed Time Coincidence Block Diagram. The pulsed time coincidence measurement is an attempt to do a measurement somewhat similar to that of Greenspan. The pulses of l microsecond pulse length and a center frequency of 6 mHz are generated by the Arenberg pulsed oscillator and the related equip— ment (Fig. 3). The pulse repetition rate is EDkHz and is controlled by the audio oscillator. The spacing of the acoustical pulses in the tank is, of course, proportional to the sound velocity and, for the repetition rate used, was about 30 mm. A different optical system was used; the laser beam was focused upon the acoustical axis, making a very narrow light beam at the position of the sound beam. A second lens focused the laser beam on the photomultiplier. The sound deflects the light beam approximately sinusoidally about its mean position so that a train of nearly sinusoidal waves was displayed on the oscilloscope. 12 A regular horizontal sweep was used on the oscilloscope which then displayed the oscillations in the optical signal when one of the pulses crossed the light beam. A small amount of the signal applied to the transducer was also fed to the vertical oscilloscope input through an attenuator. A measurement was made by first superimposing the transducer voltage pulse and the optical signal pulse, and then translating the transducer until another optical pulse was super- imposed. The distance translated, multiplied by the repetition frequency, gives the sound velocity. Diffraction has been mentioned as a possible source of errors in sound speed measurements. To illustrate one effect of diffraction, a zeroth order schlieren photograph of the sound field of a 2 mHz transducer of radius 11 mm is shown in Fig. h. The complicated pattern of light and dark areas is a direct result of the non-uniformities in the sound field which are caused by dif- fraction. For the case of a progressive wave, the effect of dif- fraction on optical sound velocity measurements has heretofore been treated neither experimentally nor theoretically. The next section will be devoted to this problem. 13 .HwosvmcmnH “new as mm «Mme N w aoum mcflumwwmm Emmm venom m mo sawquuOSm :oumwanom umvuo auouwN .: wusmwm THEORY In order to examine the effects of diffraction on optical sound velocity measurements, an expression for the sound field will be developed. Next, the effect of the sound field on the light beam will be calculated. Here we will assume that the light beam is not deviated as it passes through the sound field; therefore, only the relative retardation in phase of the light beam (retarda- tion v, for short) is calculated. This is equivalent to the Raman- Nath approximation. And finally, the light intensity measured at the photomultiplier is derived from the expression of the retardation of the light beam. A circular transducer of radius a, is mounted in an infinite rigid baffle on the x-y plane (Fig. 5). The transducer vibrates sinusoidally with angular frequency (D into a linear, dissipationless fluid. The resulting sound field may conveniently be described by the velocity potential m, in terms of which, the particle velocity u and pressure p may be calculated from the equations U=-V

=—kf f(r'),; r'rud (5) s 29 30 and also the expression of Bateman and King l6 ¢

= 21‘ 6“2 Jo g da, (6) u 2 2 where R = (z + r + r'2 - er'cose )l/2 J _ (a? _ k2)1/2, and a g =b/‘ f(r') Jo dr', <7) 0 where Jo is the zeroth order Bessel Function. If the transducer vibrates with uniform velocity uO over its surface, the function f(r') equals uo, so that Eq. (5) becomes Emu a . m(P) = ‘IEE k/m e'lkR r'dr'. (8) o R By substituting f(r') = uO in Eq. (7) and integrating, an expres- sion for g(a) may be obtained, which when substituted in Eq. (6) gives the velocity potential s = auoL/n e'“z Jo(ar) J1(Qa) 99 . (9) Later, some results obtained from these expressions for the vel- ocity potential will be discussed. Now that expressions for the sound field have been obtained, the retardation v of the incident plane wave of light may be calculated. Consider the light beam to be traveling parallel to the x axis in the x-z plane so that it passes through the diameter 17 of the sound beam. Then, if n is the index of refraction and s the entropy, the axial retardation is expressed as d 00 v : —nk'f 1) dx . (10) 00 a dp s _ Combining Eqs. (2), (9), (10) the expression for the retardation becomes dn Lmt ‘m -uz “m : _k'. _ e va dp unpouoae L/ H3 J1(a a) E/ Jo(a r)dx] dd s o -w (X) _. —uz J (a a) = [constant] e ubt Jf e 2 -l-——- da (11) u a a 0 Using the identity 2 J (a a)/(a a) = Jo(a a) + J2(a a) and the l . * integral U/w e'“z Jn(2 a) d1 = AB [H£}; (B) Jn/Z (A) ]’ (12) k "_""__"E k - where A = E [ A/zz + a2 - z], and B = 5 “£2 + a2 + z], the retardation may be expressed as Va =§ S—Ekrwpouoae'iwt new Ho”) (B) + am a (1) (3)]. (13) This expression involves only well-tabulated Bessel and Hankel functions. By taking the imaginary part of va (this is equivalent to assuming that the transducer vibrates with sin(-wt) time dependence) the * F. Ingenito, private communication. l8 retardation va may be expressed in the form Va = v0(z) sin (kz — wt + 9(2)), (1h) where 2 2 1/2 vo(z) = Iv) 2 [(Rev ) + (I Va) ] , and 9(2) = tan [(Im va)/(Re va)] - kz + wt With the optical retardation expressed in Eq. (1h), there remains the question: ”What does a photomultiplier light detector see when placed some distance x from the sound beam along the path of the laser beam?" The approach to the solution will be to first note from Eq. (1h) that the light beam has a nearly periodic variation in phase as it emerges from the sound beam; this variation in phase directs the light into diffraction orders. And finally from light travelling in these orders, a Fresnel diffraction pattern will be calculated for the experimental conditions at hand. Numerical analysis of Eq. (1%) indicates that the amplitude and relative phase of the retardation v oscillate rapidly near the transducer but their rate of oscillation decreases until at approxi- zx mately —E > .3 , they may be considered slowly varying functions a of 2. Therefore, the retardation may be expressed as V = vo sin(kz-wt + 60) (15) where v0 and 60 are slowly varying functions of z and may be considered constant over the width of the laser beam. The diffraction integral expresses the amplitude of the light emerging in the direction 5: A(B) = C L/N exp i[klzsinB+ vosin(kz—mt + 60)] dz (16) L where k' is the wave number of the light and C is a complex constant. Inserting the identity exp [ivsin m] = E Jn(v) exp [in m] (17) nz-OO in Eq. (15), the amplitude may be expressed as: 00 A(B) = Ck/; exp [i k'zsinB] §=_w Jn(vo) exp [in(kz-wt + 60)] dz = C E Jn(vo) exp [in(-wt + 90)]L/; exp [iz (k'sinB + nk)] dz. (18) If the light beam has infinite width, the integral predicts discrete diffraction orders at angles am where sin Bn = nk/k'. TInr the measurements to be described later, the light beam was from two to ten wavelengths wide. Consequently this integral predicts diffraction orders which are somewhat smeared out. However, if one assumes that the orders are discrete, the Fresnel field may be easily calculatedsl. The diffraction orders may be considered to be plane waves traveling in directions am with amplitudes An = Jn(vo) exp [in(-mt + 60)] , (19) Which may be added at the position of the photomultiplier. 20 Thus the amplitude in the Fresnel field is given by A(z,x) = exp [-ikx] E Jn(vo) exp [in 8] exp [-in2 qx] (20) where k2 2k' 6 = kz - wt + 60 and q = In the measurement of sound velocity, local heating and finite amplitude effects although small, may cause errors. For this reason the transducer potential is adjusted to give a small but measur- able optical effect. This restriction held vo«\;O.l so that J1(.l) << 1 and J2(.l) << Jl(.l). Therefore, the approximate light intensity may be calculated very simply: A(x,z) = Jo(vo) + 21J1(v0) Sln 8 exp [-iqx]. (21) And thus, the intensity may be expressed as 2 2 . 2 . . I = AA* = Jo(v0) + H J1(vo) Sin 6 + MJO(VOJ1(VO) s1n631nqx or by taking the first term in the Taylor series expansion for Jn(vo), as I = l + 2vo sin 5 sin qx . (22) This simple approximation shows several important features of the Fresnel interference pattern. Remember that the relative retardation was vo sin 5; the fluctuating part of the intensity is exactly in phase with the relative retardation. In this approximation the intensity modulation is proportional to the transducer voltage and 21 the distance from the sound beam. A more complete analysis32 would show more complicated relationships at higher sound intensities;: however, the first prominent intensity peak is at 5 = n/2 for various values of retardation and at various positions from the trans- ducer as long as q x < 0.2. This means that the phase relationship between the transducer voltage (proportional to v) and the light intensity at the photomultiplier is constant for a given 2 position, independent of the magnitude of the transducer voltage. IV. THEORETICAL RESULTS The mathematical connections between the velocity distribu- tion at the transducer, the sound pressure at any point in the sound field, and the intensity of the light beam at the photomultiplier have been established. Now, from the theory, some calculations are presented which will point out some features of typical sound beams. Special attention will be given to those features which influence the accuracy of sound velocity measurements. The discus- sion will parallel the order followed in the preceeding section, that is, first the velocity potential is considered, which gives the spatial pressure distribution, then the Optical effects are described. From the theoretical work of Meixner33, Seki et al3u, 5 35 DelGrosso , Williams 136-38 and the early experimental work of Hiedemann and Osterhamme , a picture of the sound field has evolved. The sound field of a transducer whose diameter is many wavelengths, is very complicated, having some curvature of wavefronts and a very complex variation of pressure. In order to carry out a complete, although approximate calculation of a sound field and the resulting optical effect, a numerical evaluation of Eq. (6) was done with a CDC 3600 computer. First, to be discussed here, the program calculated the pressure and phase of the near field of a transducer of 5 wavelength radius. This transducer radius was chosen so that the computer time could be minimized, as well as approximating experimental conditions. 22 23 This calculation was simply performed by dividing the transducer area into small squares (1/3 x) and summing the effects at the observation points. To check the computer program, the axial pressure was plotted in Fig. 6 and compared to the well known exact integral. The agree- ment between curves could be improved by dividing the transducer into smaller squares. The nature of this curve, with its characteristic axial nulls, serves as a "precursor" of the optical results to be discussed later on. The calculated pressure distribution is shown in Fig. 7. Note that there is appreciable pressure amplitude for r greater than the transducer radius, which indicates a poorly collimated sound beam. For the transducers and frequencies used in this experiment, the collimation is somewhat better because r > 8K. The development of side lobes is also evident in this figure. The phase, or the wave fronts, are plotted in Figure 8. Note that the approximation to a plane wave is not very good — especially on the axis where there are the characteristic dimples. For r greater than the transducer radius, the wavefronts are strongly curved. When a light beam traverses a sound beam, it essentially integrates over the pressure distribution along its path; when passing through these strongly curved wavefronts, the rapidly varying phase tends to nullify the effect of the pressure amplitude outside the cylinder whose base is the transducer. :NI NM+NN vmcfim_ nowumaaoamo Housmaoo nu Fm u m munmmoum Hmfix< .w muswam 21+ 27 For light passing through the center of a sound beam, the retardation is given by the exact expression in Eq. (13). The amplitude of this retardation is plotted in Fig. 9 for the same transducer radius to wavelength ratio as the previous graphs. The effect of the strong oscillations of the axial pressure can be clearly seen. Although, because of the light beam's averaging effect, the oscillations are greatly suppressed. The maximum . zk retardation occurs at -E 1.1 and not at the transducer as one a might assume. The most important result suggested by this curve is that the exact effect of the near field on transducer pressure calibration by Optical methods is now specified. Since a light beam has finite dimensions, the resulting light diffraction is from light which passes through the sound beam somewhat off axis. In order to investigate the contribution of this off axis light, the same computer program, using the pressures already calculated, performed this optical integration numerically. With the solid curve being the exact solution, Fig. 10 shows the results of these computer calculations. The agreement between the exact solution and the computer solution for the integration through the diameter is better than might be eXpected. The various off-axis retardation values tend to oscillate with smaller amplitude. With a finite sized light beam of 2 sound wavelengths in diameter, for example, the optical effect would be similar to Fig. 9 but with less variation in amplitude. 28 N.H Jam HoumEmHn emom venom mcon woumummucH cowum©HMuwm o>wumfimm 04 m6 x5 mo do .m wuswwm m.O O.H N.H cowumummuaw mwxm o>onm Am nu cow umuwmucw mwxm m>onm Km 0 cowumuwmucw mwxm o>onm &H x cowumuwmucw HousmEoo menu so mv ma Am n m cowumpumumm mfixm mwo Amnuwcmam>m3 mo muwcsv N 2 m .oH mtsmam _ . a m _ 4 . _ _ a . _ _ _ A 29 N.H m ommnm Hmowumo Hmwx< m>wumamm .HH muswwm 2 I.“ u 30 fi. 31 The curve of the phase of the retardation, relative to the plane wave phase, is plotted in Fig. 11. FOrwa/A ratios greater than 5, the curves are found to be similar to the one shown except that in the region zk/a2 < .8 there are more oscillations. Since the light intensity at the photomultiplier has been shown to be proportional to the retardation for low sound intensities, the phase of the electrical signal from the photomultiplier equals the phase of the retardation plus some constant phase shift due to the electronic circuits. Using the signal applied to the transducer as a phase reference, a Lissajous pattern on an oscilloscope indicates the phase of the signal voltage relative to the phase of the photomultiplier output. Since phases are compared, the amplitude of the retardation is unimportant if the electronic circuits are linear. A sound velocity measurement might be carried out in this way: the transducer is translated until the value of z is such that a Lissajous pattern is closed; next the transducer is moved a whole number of wavelengths indicated by the Lissajous pattern again being similarly closed. By measuring the distance translated, the wavelength is measured, and finally the velocity computed. If the phase 90 [Eq. (1h) v = vo sin Ont - kz + 90)] was a constant over the range of measurement, then such a measurement would give the true plane wave phase velocity c sinde k =cD/c. However, measurements are usually made over a large number of wavelengths to increase the precision. Therefore, the fact that 90 is a slowly varying function of a 2 introduces a systematic error in the sound velocity measurement. 32 Because the value of 90 is know as a function of z, the distance from the transducer, including this correction will eliminate this systematic error from the absolute error. The correction is introduced in the fol- lowing way: At position 1 the spatial phase of the argument in Eq. 1h becomes kz + 9 1 = znml, when the Lissajous figure is closed; in the 01 same manner, at position 2 the phase is + kz2 + 602 = +2nno where the difference between n1 and n2 is the integral number of wavelengths measured. Then the velocity is calculated from these two equations giving c = (”AZ (21) where Az = z - 2 etc. Zmnmeo l 2 For instance, in Fig. 11 if the measurement was carried out for zlk/a2 = 1.2 and ZZX/a2 = 0.8 the corresponding phases would be 901 = 0.182 and 902 = + 0.005. Thus the measured velocityanz/ZmAn would be higher than the plane wave value c, by approximately 2A , . -—Q- = 0.3%. However, If a measurement was made between mfln zlA/a2 = 0.8 and zzk/az = 0, for instance, A6 = + .01 the measured velocity would be lower than the plane wave phase velocity "c” by an amount less than .01%. To sum up these results, when the optical phase curve 90(2) is known for a given transducer geometry, the effects of diffraction on the measurements of the sound velocity may be calculated and the measurements thus corrected to give the plane wave phase velocity "c”. Furthermore, this correction is valid for all distances from the transducer. V. DESCRIPTION OF EQUIPMENT This chapter contains a description of the equipment used in the experiments described in chapter II. A diagram of the optical bench is shown in Fig. 12. A stable optical system is required for the measurements described here. The optical bench was constructed of oak, bolted and glued in bulkhead type configuration. Supports (not shown) for the tank of water and the lathe bed were built in triangular form for rigidity. The light source was a He-Ne laser (Spectra Physics 131) which has a collimated beam of high intensity radiating from the front and a slightly diverging weaker beam from the back. The laser was firmly mounted on an optical rail fastened to the bench. All other optical parts, including the windows of the tank, were aligned to the primary beam of the laser. Light from the rear of the laser, directed by three right angle prisms, illuminated a Michelson inter- ferometer used for the length measurement. The photomultiplier was located behind the tank 20 to 50 cm from the sound beam, depending on the frequency. Although the photomultiplier tube 1P21 has an s-h surface which is not par- ticularly sensitive to the red line of the He-Ne laser, the sensitiv- ity was adequate. The output impedance of the photomultiplier was controlled to obtain maximum signal response for the bandwidth required. A further description of the photomultiplier can be 31 found elsewhere . 33 31+ :30 m uo: muuommnm mm Hw>o wowucsoz emu: mamHH cm onmm moflumo .NH wuswfim n p a w p . e H HwoswmcmuH umquonmmumucw cowamnoflz \\ can magma 35 Unlike Schreuer's arrangement, which moved the entire tank and transducer assembly, only the transducer was moved in this experi- ment. Thus, any curvature or imperfections in the tank windows (care— fully chosen and mounted optical flats) will not effect the measure- ments. In these sound velocity measurements, the translation of the transducer must be measured to the accuracy desired for the velocity measurement. In addition to accuracy, the problems of precision, convenience, and finance must be considered. The transducer mount was attached to the carriage of a miniature lathe bed. A screw, which could be turned either by motor or by hand, drove the carriage along the dovetail guide of the lathe bed. At one end of this guide, the beam splitter and fixed mirror of a Michelson interferometer were attached. The moveable mirror was attached to the carriage on the lathe bed. The illuminating light for the interferometer came from the laser as previously described. The fringe system was monitored by a photomultiplier tube which was also mounted on the dovetail guide. As the transducer assembly is moved, the interferometer photo- multiplier produces pulses in accordance with the detected fringe pattern. This output was amplified and shaped into uniform pulses to insure con- sistent counting by a Beckman scalar. It is recalled that a Lissajous pattern on an oscilloscope indicates the relative phase between the optical effect and the driving signal. With a motor moving the trans- ducer at a nearly uniform rate, the scaler was switched on at one closing 36 of the Lissajous pattern and switched off at another, similar closing of the pattern. Thus the count of the fringes yields a length measured for an integral number of wavelengths of sound. The major advantage of this type of measurement is that it is an absolute length measure- ment because it depends only upon the wavelength of the He-Ne laser line (6328.17 A). The disadvantage is that the measurement takes a great deal of time and the precision of the starting and stopping is not high enough to obtain satisfactory statistics without many tedious repetitions. To crosscheck and to expedite the length measurement, an alternate system, using Johannasen gage blocks was used. Fastened to the other end of the dovetail guide, a Browne and Sharpe micrometer screw, calibrated to i .002 mm, was mounted in a sturdy aluminum block. Gage blocks could be inserted between the micrometer screw and the transducer mount. The micrometer screw was then used to interpolate between gage blocks.. With a Sheffield comparator,Fonda (i .2 microns) gage blocks, which had been recently calibrated, were used to calibrate the working set of gage blocks... The Michelson interferometer also provided a good check of the trueness of the dovetail guide. The circular fringe pattern is very sensitive to angular displacements of the movable mirror which follows the angular deviations of the motion of the transducer carriage. Since the fringe pattern remained centered over the measurement range used, the transducer motion was assumed to be only a translation with no angular deviation. 37 In the continuous progressive and standing wave measure- ments, the rf potential applied to the transducers was generated by a 100 watt transmitter. In the pulse measurements, two Arenberg model 650PG pulse units were used; One operated at a pulsed oscillator and the other as a gated amplifier of the signals from the rf trans- mitter. The gated amplifier was triggered by a Dumont HORR generator. The frequency of the transmitter was monitored by a Hewlett Packard 52h B frequency counter. Its standard frequency, accurate to l : 106, was checked against a more accurate laboratory standard (Hewlett Packard .S2h5L). The sound was generated by x-cut quartz transducers which were mounted with air-backing in holders of nylon. The transducer was aligned perpendicular to the light beam prior to each set of measure- ments. To permit angular adjustments in the transducer orientation, the holders were constructed with gimbal pivots for the horizontal rotation and knife edge pivots for the vertical rotation. A transducer of similar construction was used as the re- flector in the standing wave measurements. The air backing of the quartz gave the standing wave cavity a very high ”Q" so that the required driving potential was on the order of 5 volts. One important disadvantage of previous optical methods is that large acoustical pressures were required to produce an observable effect. The eventual dissipation of the acoustical energy within a resonant cavity resulted in the local heating of the medium. Schreuer, for example, found that his measurements of sound velocity in water 38 were raised as much as 0.6 m/sec at his highest acoustic power levels. In an attempt to account for this local heating, the sound velocity was measured at several acoustic power levels and the results extro- polated to a value for zero acoustic power. The sound velocity in water at room temperature changes by approximately 3 m/sec per degree centigrade; therefore, temperature changes of .01 degree centigrade are significant. Considerable thought was devoted to the design of the present arrangement to insure that: a) minimum acoustical energy was dissipated in region of measurement; b) heating from the energy dissipated in the transducer was not significant; c) absolute temperature was measured in a region in proximity with sound beam; d) temperature variations in the tank were minimized by adequate stirring and pumping of the medium; and yet (D v adequate flexibility was available to allow different configurations. A diagram of the resultant experimental arrangement is shown in Fig. 13. The tank (15 cm x 15 cm x 80 cm) is constructed of aluminium and is insulated on the sides, bottom and partially on the top with a h cm layer of styrofoam. The best location of the temperature control elements, determined by trial and error, is shown in Fig. 13. A pump circulates the water through a chamber containing an immersion 39 mammmm umummm o y _ I noumEoEHosH Honucoo HmLHOmn< venom Houucoo menumuomEmH can xcmH sevens emospmnme .ma seamen mME-i MO heater. Then the water flows past the control thermometer of a Bayley Instrument Company proportional controller which controls the heater power. Next, the water flows around the sound absorber and through the measurement area. In this area, the water is also stirred by one or two stirrers, depending on transducer configuration, to insure temperature uniformity. By working at approximately ten degrees above room temperature, no auxiliary cooling device was required. The temperature uniformity was investigated for a variety of conditions with a pair of thermocouples and a Kiethly micro- voltmeter Model lh9. The reference junction was placed in a water- filled flask which was placed in the measurement tank. The other junction, fastened to the end of a glass rod, was used to probe temperature differences. With the temperature control apparatus turned off, and the water not circulating, the temperature variation was about .0200. With temperature control apparatus operating and the water pumped and stirred, the maximum temperature variation over the entire tank was less than .OlOC. In the space where the measurements were made, the temperature variation was at most .00500. Next, the temperature variations produced by the sound beam and the heat dissipation of the transducer were investigated. In order to achieve a significant temperature change, a high input of rf power (70 watts at 300 v) was applied to the transducer. The temperature of . 0 water at the surface of the transducer increased about 0.2 C above the M1 ambient temperature of the tank. At one centimeter from the transducer the increase was only .OSOC. For progressive wave and pulsed progressive wave measurements the rf potential applied to the transducer was at most one fifth the potential applied in this test; for standing waves the applied potential was less than one sixtieth of the test value. Con- sequently, the temperature variations should be reduced by the order of 10.2 to 10")-L of the temperature variation given above. The absolute temperature was obtained by placing a platinum resistance thermometer adjacent to the sound field. With adequate stirring, the temperature difference between the sound field and thermometer was negligible. The resistance of the platinum resistance thermometer (Radio Frequency Labs) was ascertained using a Mueller bridge (Leeds and Northrup type 01). This bridge was calibrated at room temperature using a ten ohm standard. Since the room temperature varied less than 30C from the calibration temperature, no attempt was made to correct the resistance readings for variation in bridge temper- ature. After all calibration procedures were considered, the absolute . 0 accuracy of the temperature measurement lS i.O°O3 C. VI. EXPERIMENTAL RESULTS A. MEASUREMENT OF THE RETARDATION In the introduction, the possibility of doing three new types of measurements, progressive, pulsed progressive and time co- incidence measurements was discussed. In the theory section, the retardation "v" was calculated as a function of the distance from the transducer and ratio of transducer radius to wavelength. Also, in the theory section, the effect of diffraction on sound velocity measure- ments was discussed. Before the discussion of sound velocity measure— ments, one check upon the correctness of the theory can be made by measuring the magnitude of the retardation ”v” as a function of 2. Sound beams from both the 1 mHz and 2 mHz transducers were investigated with a zeroth order schlieren optical system. From the photograph Fig. u (made up of a series of photographs) one can see that the near field radiation pattern is quite complicated. The relative retardation through the sound beam diameter (center of the picture) was measured by varying the transducer potential in such a manner that the retarda- tion was held at a constant value. This was judged by eye or by a photomultiplier microphotometer located at the center of the schlieren image. Then the inverse of the transducer potential so measured is proportional to the retardation. These experimental measurements are plotted on graphs along with the exact expression in Figs. 1h and 15 for 2 mHz and 1 mHz respectively. Agreement is good in the two figures except near the transducer; however, the variation in the measured 1.2 “3 . madmah max< monummmz :H . «. EB NN «NEE m How cowuwvuwumm H . mucflom sump Kma m mew . m. m. mm\xu :. m. N. w. J _ _ _ . a m a, I 10 Q Emu-How... R an O H.H ML .uoosvmcmuu ...me as JAN Kuse H m pom coaumwwmumm $3wa pwusmmwz .mH ounwfim m N m \K o.m wé 04 :4 NJ 04 w. w. x. N. 0 4| _ _ a . . _ _ _ _ % L15 retardation for the 2 mHz transducer is less then theoretical curve. Note that the scales on the two graphs are not the same. For the calibration of transducers this agreement between the mathematical expression for the retardation and that obtained experimentally is of vital concern. Also, those making attenuation measurements using optical methods may benefit from this ”diffraction correction". #6 B. SOUND VELOCITY MEASUREMENTS As indicated in the preceding section, the magnitude of the calculated retardation and the experimental retardation agree except very near the transducer. From this encouraging agreement one would hope that the relative phase of this retardation would also agree with experiment. This relative phase variation represents a small systematic variation in the measured wavelength intervals. In order to ascertain this variation, the position of every third wave- length (as indicated by the closing of a Lissajous pattern) was measured for the 1 mHz transducer. Transducer positions were recorded over a distance of approximately 1h cm. which means that the dimension- less parameter zk/a2 varied from 0 to 1.h. These points may be fitted to a theoretical relative phase curve by estimating the plane wave phase velocity, calculating the plane wave phase and then subtracting the plane wave phase from the measured phase. This attempt is shown in Fig. 16. There is quite a bit of scatter in the experimental points, however, the general trend is clearly evident. The sound velocity (extrapolated to 3h.0000) which was chosen to fit the curve was 1517.6 m/sec. The tail of the curve would fit better if this value was raised to 1517.7 m/sec. The scatter in the points is approximately .06 radians which would give an error in the velocity of i .2 m/sec over the entire range. 117 H\ .mH .vm souw pmumasoamo AmaHH pHHomv o>uso Hmowumuomns H.ee :.mm Houmsmww « NmaHH .ommnm m>eumamm Hmex< wouSmmoZ «Nb «we «ad .mH onswwm _ _ _ us The sound velocity measurements using pulsed and continuous progressive wave techniques are shown in Table 2. Perhaps one of the most important results is shown in section 8a) in which a comparison was made between the continuous wave and pulsed continuous wave measure- ments. The two sets of data were obtained essentially simultaneously as the transducer was alternately connected to the output of the pulsed oscillator and then the transmitter. The two sets of data are in agree- ment within the precision of the measurements. The first four measurements done with the Michelson inter- ferometer were attempts to find variations in the wavelength near the transducer by measuring a few wavelengths at a time. The resulting errors are quite large because of the start - stop errors in the inter- ferometer counting. With the exception of the 6 mHz data all the measurements in Section B were corrected for diffraction errors by using the technique described in the chapter titled Theoretical Results. The diffraction corrections were, in all cases, smaller than those which would be applied if the geometrically equivalent experiment were performed using two similar transducers. As can be seen in Fig. 17, for diffraction corrections near the transducer the results of DelGrosso show much larger errors. Also, because this curve is nearly monotonic it is much more difficult to detect systematic errors caused by diffraction in a two transducer arrangement, when the frequency is low and not varied. Table 2. Progressive Wave Measurements A. Michelson Interferometer Length Measurements Freq. (mHz) Technique C3h.000 (m/sec) 0.8 cw 1515.6 i 2 0.8 cw 1517.0 : 1 11.0 cw 1517.6 : 2 1.0 cw 1518.7 i'3 (individual A's) 1.76 cw 1517.6h i '25 5.28 cw 1517.76 .1 .20 5.28 cw 1517.81 i.°20 average of last 3 1517.73 :_.20 B. Gauge Block Length Measurement a) 2'0 CW 1517'56 : '20( comparison between 2.0 pcw 1517.63 i,°ZO/ pulsed and cw b) 1.0 cw 1517.70 :_.20 2.0 cw 1517.66 3?. .20 2.0 cw 1517.62 : .20 2.0 cw 1517.72 : .20 2.0 pcw 1517.66 i_.l5 2.0 pcw 1517.62 i .20 6.0 cw 1517.63 i_.20 6.0 pcw 1517.56 : .20 average 1517.65 : .20 1+9 .AKoH n «V muasmwu Hmowuao can 0mmouuaom mo comwummEoo u ownsm o>wumamm .NH wuswwm w; m4 :4 NA 04 w.o 0.0 2.0 N6 0 J I _ _ _ 14.. Jfi _ _ Am 0 0 Av 0 4m.. mu mu me 030.539 0. 0 1N; :0» O O accauco mu 0 1e- 0 >> 6 as? . -C < c / 1mo. 51 The pulsed time coincidence technique was used to make a velocity measurement at only one frequency because of the limitations imposed by the pulsing equipment. At high repetition rates, a pulse width of 1 usec was the maximum allowable, and in order to have several cycles in the pulse enve10pe a transducer operating frequency of 6 mHz was chosen. The sound velocity adjusted to 3M.OOOC was 1517.53 m/sec i .2h. At 6 mHz, the maximum difference between the velocities obtained by continuous wave measurements (phase velocity) and pulse measurements (group velocity) for our configuration, is -0.10 m/sec (caused by geometrical "dispersion"). The standing wave measurements (see Table 3) demonstrate the 13 same "dispersion” that was observed by Schreuer and has recently been noted in interferometer measurements by Ilgunas6 and others. The velocities made at 0.8 mHz and l mHz are considerably higher than the rest of the measurements made at higher frequencies. Neither the general diffraction correction suggested by Bass20 nor the interfer- 9 ometer calculations by DelGrosso predict such a large deviation at these frequencies and transducer configurations, Ilgunas has obtained similar large deviations in the acoustic interferometer. At higher frequencies, measured values fall about the value 1517.7 m/sec when the measurement cavity was long enough to permit a length measurement of adequate precision (about 70 to 90 mm.). One exception is the measurement at 7.2 mHz which is low. Table 3. Standing Wave Measurements A. Michelson Interferometer Length Measurement f(mHz) 52:;Eh (mm) C3h.000(m/sec) 0.8 119 1518.50 .1 1.5 0.8 87 1518.86 1: 1.5 0.8 87 1519.111 : .90 0.8 87 1518.28 :2 .50 2.11 87 1518.10 1 .50 2.11 87 1517.67 : .50 2.1+ 87 1517.82 i .90 11.0 87 1517.77 3: .20 5.6 87 1517.60 i .30 5.6 87 1517.68 i .25 7.2 87 1517.411 : .26 weighted average C3h.ooo = 1517.70 + .20 1 117 1519.211 : .20 1 87 1519.23 1 .110 1 23 1518.7 .1 .60 2 711 1517.68 i .20 2 21 1518.7 i 1.0 6 811 1517.73 1 .18 weighted average C o = 1517.70 i .20 311.00 52 53 The grand weighted average of all the standing wave measure- mentstis 1517.70 i .25 (at 3h.000C) where the error limits now include the uncertainty in the absolute temperature measurement. There was good agreement between the values obtained with the two different measuring techniques (Johannasen gage blocks and Michelson interfer- ometer) which would indicate that the systematic error inherent in the length measurements is quite small. DelGrosso's most recent measurements, when extrapolated to 3h.0000 with Greenspan's data give a velocity of 1517.79 : .02 m/sec. There is a difference of about .10 m/sec which is within the experimental error of the present experi- ments. The wavelength intervals for a given transducer, reflector and frequency were not constant over the length of the ultrasonic cavity. Near the reflector the intervals were longer which gave an apparent sound velocity higher than the average sound velocity by 0.5 to 6.0 m/sec. These differences were found to be approximately inversely proportional to the frequency. This strange behavior might eXplain some of the "random" fluctuations in Schreuer's measurements; he chose locations in the sound field where the visibility fringes were clear over 3-5 cm but didn't necessarily limit his measurements to those points far from the transducer. An explanation for this behavior might be sought in diffraction effects at the reflector. For all the measurements described here, the magnitude of the errors are these: a) temperature i.°08 m/sec; b) sample impurity + .02 m/sec; and c) length :_.15 m/sec. Thus, a grand average sound 5L1 velocity is 1517.70 i .25 m/sec. Extrapolated to 300C this value becomes 1509.03 : .25 m/sec. As can be seen, this value is closer to the work of Neubauer and Dragonette and that of Ilgunas, although, considering the magnitude of the uncertainty it is also near DelGrosso's current value. This work has demonstrated that optical methods may be used to give sound velocity values in transparent media which are in agreement with other methods. Also, the errors caused by dif- fraction in the near field of the transducer are small and easily calculated, for the case of progressive waves. For the standing wave measurements, the appropriate corrections have not been calculated. For low frequencies the errors are very large and the corrections of Bass and DelGrosso are too small. Also, the corrections of Bass and DelGrosso are too small for the case of an acoustic interferometer operating at low frequencies. Until better calculations are completed (although the author has nothing to add to the careful and extensive calculations of DelGrosso) it seems that, for frequencies below 1 mHz and for small transducers the progressive wave optical methods together with a diffraction calculation which is accurately known might be the best way to measure sound velocities. H [\3 H .1:- H H \n U) vvvvvv H O\ 19) 20) 21) BIBLIOGRAPHY A. B. Wood, E. H. Brown and C. Cochrane, Proc. Roy. Soc. n.3, 281+ (1923)- E. A. Eckhardt, Phys. Rev. 23, #52 (192%). A. Kundt and 0. Lehmann, Ann. de Pogg. 63, 1 (187M). Hubbard and Loomis, Nature cxx, 189 (1927); Phil. Mag. 5, 1177 (1928). V. A. DelGrosso et. al., N.R.L. Report hh39 (195k). V. Ilgunas, 0. Kubilyurere and A. Lapertas, Sov. Phys. Acoustics 1g, hh (196%). V. A. DelGrosso, N.R.L. Report 6026 (196M). V. A. DelGrosso, N.R.L. 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Pergamon Press (1965), useful for the Russian references. L. Bergmann, Der Ultraschall, Springer Verlag, (19#9). E. Hiedemann, Ch. Bachem and H. R. Asbach, Z. Physik 81, 73h (193A). 0. Nomoto, Phys. Math. Soc. Jap. Proc. 18, #02 (1936); 12. 337 (1937). Lord Rayleigh, The Theory 81 Sound, Dover Publications, N.Y. (19#S)- H. Bateman,Mathematica1 Analysis 91 Electrical and Optical Wave Motion, Cambridge Univ. Press (1915). V. King, Canad. J. Res. 11, 135 (193#). P. R. Pisharoty, Proc. Ind. Acad. Sci. 8, 27 (1936). A. J. Crandall, M. S. Thesis, Michigan State Univ. (196#). J. Meizner and U. Fritze, Z. angew. Physik 1g, 36 (19#9). H. Seki, A. Granato and R. Ture11,J. Acoust.Soc. Am. 88, 230 (1956). A. 0. Williams, Jr., J. Acoust. Soc. Am. 19, 150 (19u7). E. A. Hiedemann and S. K. Osterhammel, Z. Physik 107, 273 (1937). E. A. Hiedemann and S. K. Osterhammel, Proc. Ind..Acad. Sci. (A) 89 275 (1938)- s. K. Osterhammel, Akust. z. 8, 73 (19h1). 56 1(11(1))(1wmnt l 9 2 4| 3