Absorption of Sound by Small Spherical Obstacles Citation Garvin, Walter William (1948) Absorption of Sound by Small Spherical Obstacles. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/E92F-YV72. https://resolver.caltech.edu/CaltechTHESIS:04062011-083923524 Abstract No abstract Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics and Mathematics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Epstein, Paul Sophus Thesis Committee: Unknown, Unknown Defense Date: 1 January 1948 Record Number: CaltechTHESIS:04062011-083923524 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04062011-083923524 DOI: 10.7907/E92F-YV72 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6279 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 06 Apr 2011 18:13 Last Modified: 13 Oct 2021 23:56 Thesis Files Preview PDF (Garvin_ww_1948.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

ABSORPTION OF SOUND BY SMALL SPHERICAL OBSTACLES Thesis by Walter William Garvin In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1948 ACKNOV~DGEMENT I am greatly indebted to Dr. Paul S. Epstein by whom the subjeot and method of this work was suggested and under whose guidance it was done, for his supervision and oonstant advice. SUMY~RY The following pages represent an investigation of sound absorption by small spherical obstacles to which we attribute, in different sections, the properties of rigid bodies, viscous fluids, ~and elastic solids. Both viscosity and thermal con- ductivity are taken into account. The general equations are derived, subject to the assumption that the size of the obstacles is small compared to the wave length, and are applied to the speCial cases of water drops in air, air bubbles in water, and to suspensions of elastic solid particles. The theoreti- cal results are compared with the experimental measurements in the case of fogs and found to agree very well within the accuracy to which these measurements can be made. TABLE OF CONTENTS PAR,LL I- Introduction • . • • II. The Basic Equations 1 ... 4 III. Calculation of kl and k2 • • • IV. Plane Wave Solution · . . • 15 · . . . . 17 .... • • 21 VI. Viscous Fluid Sphere • • . . . . . . . . . . • • 27 V. Rigid Sphere • . • • • VII. Elastic Solid Sphere • • • • • • 33 . . . . . . · . . . • • 38 IX. Calculation of the Extinction . . . · 46 VIII.Energy Relations • X. Applications • • • XI. Comparison with Experiment. ...... • 52 • • 61 PART II Appendix I • • • • • • • • • • • • • • • • . • • • • 66 Appendix II • • • • •• ••• 68 Appendix III • • • • • • • • • • • 70 Appendix IV • • • • • . • . 72 •••••• • .• 76 Appendix V • • • • • . • • • • • • • • • 84 Appendix VI • • • • • • • • • 86 Appendix VII • . • • • • •• Appendix VIII • • • • • ••• • • • •• • 91 •• • • • • • • • • • • 97 Appendix IX • • •• Appendix X . . . . . . . . . . . . . . . . . . . . . 98 Appendix XI •• ••••••••••••••••• Referenoes • •• •••••• 101 103 Table I . . . . . . . . . . . . . . . . . . . 104 Table II • • . . •. •• • •• •• 105 Graph I . . . . . . . . . .. . . . . . • 106 Graph I I . • . . . . . . . . . . . . . . . . • • . . 107 Graph III • • • • • • •• •••• ••• •• 108 LIer OF SYMBOLS The following symbols are used to indicate elementary quantities: A. = vector potential c< ~ extinotion per uni t length C ~ velocity of sound ely '" coeff. of volume expansion elf specific heat t= ratio of spec. heats. K. ,. isothermal bulk modulus ~I 10/2.:: soalar potentials h ~ number p'" hydrostatic pressure AIIt.t. '" elastic constants 1= ooeff. of viscosity fij '" stress tensor !AT '" angular '< ~ ~ of obstacles/unit vol. radial coordinate f ~ densi ty R ~ radius of obstacle CJ = thermal ..? ;. displaoement ~ .. :5 - surface 1;'. ... viscous T " absolute temperature frequency conducti vi ty absorption cross-sectien IJ tensor or elastic stress t " time V " volume (}". polar angle J!. '" velooi ty ( )' '" quanti ties pertaining to If '" azimuth angle obstacle ( ) '" unperturbed quanti ties () :¥ ( ) '" complex conjugate The following symbols are used to indioate compound quantities: a ='R~l l /:(1. .. "R~2. 1r ... '"RK /tCl!" ~'= (A+ ~)oty c1f = 6' b" f/ / V'" '1./f 1(= o/cr l € '" '1.hl -1- I. INTRODUCTION The first investigation of the effect of small spherical obstacles on the propagation of sound was performed by Lord Rayleigh (1). He calculated the scattering effect of the par- ticles in a non-viscous atmosphere and showed that the effect depends upon the ratio of the diameter of the particles to the wave length of sound. Some time later, Sewell (2) took viscosity into account and calculated the scattering and additional absorption due to a number of randomly distributed cylindrical and spherical obstacles for the case that the wave length was sufficiently long, or the obstacles sufficiently small, so that the phase Variation over the obstacle could be neglected. He assumed the suspended particles to be perfectly rough at the surface, perfectly rigid, and fixed in space, i.e. they did not partake of the motion of the surrounding medium. The resulting equa- tions were ~pplied to the particular case of the absorption of sound in fogs. The assumption of stationary obstacles proved to be quite a bothersome point particularly since Sewell's result yielded a finite attenuation at zero frequency, and in an important paper in 1941, Epstein (3) extended Sewell's theory by employing a method which automatically included the oscillations of the particles in the acoustic field dua to their finite denSity. This step was partly neoessitated by the experimental work of Hartman and Fecke (4) on aqueous suspensions, since in their -2- Case the density ratio was olose to unity and Sewell's theory was not at all valid. Epstein oonsidered spherioal obstaoles and assumed them to be: (a) rigid, (b) visoous fluids, (0) elastic solids, and showed that for the oase of fogs, and suffioiently high frequenoies so that the osoillations of the drops oould be negleoted, Sewell's equation was a very olose first approximation. Reoent measurements on the absorption of sound in fogs by Knudson (5), using the reverberation ohamber technique, indioated, however, that Sewell's formula gives an absorption whioh is too low but of the right order of magnitude. Sewell considered the effects of viscosity alone but did not take into aooount thermal conduction. The same must be said about Epstein's paper referred to above. In a later paper Epstein took heat oonduotion into aooount by an indireot and not very aoourate method. Kirohhoff was first to point out in 1866 that the influenoe of heat conduction is of the same order of magnitude as that of visoosity so that for an accurate solution (or shall we say a more accurate solution) both factors must be taken tnto acoount. This was done in the plane wave solution of Kirohhoff (6), and as expeoted gave a more strongly damped wave than that oalculated by Stokes using visoosity alone. With this in mind, Dr. Epstein suggested to me to try to - extend his method as used in (3) so as to inolude any temperature effeots. The faot that the available experimental data gave a greater absorption tha~ that oaloulated by Sewell's -3- equation had been for some time interpreted as due to heat conduction, and as the results of this essay will show, this interpretation was entirely justified. A great deal of mathematical detail had to be gone through in order to arrive at the final results. paper is divided into two parts: For this reason this The first part is mainly de- scriptive in character, containing only the final results and a minimum of mathematics, except for the derivation of fundamental relations (such as II). The second part in the form of appen- dices contains most of the algebra, and references will be made to them as the need arises. -4- II. TEE BASIC EQUATIONS (a) Viscous Fluid The equation of motion of a viscous fluid is given by: (2.01) Where ~~ is the viscosity stress tensor with components: 1:.. "" 'J tz (O"i + 017j) OXj dX i i+j (2.02) ali.- 2't.. '" 2 ,,~- ... 11 Vo_lT U L en'i oJ £. It can be shown (7) that (2.02) is actually only a ;first approximation which neglects additional terms depending on the square of the coefficient of viscosity. These extra terms contain, a- mong other quantities, the thermal stresses which may exist due to lnequalities of temperature. The kinetic theory of gases, as developed by S. Chapman and D. E.skog, shows, however, that if the number of gas molecules contained in a cube of dimensions intrinsic to the problem, suoh as the wave length of sound propagation)is large, then (2.02) is a sufficiently close approximation. Since in our oase we shall not be dealing with rarified gases, but gases under ordinary conditions of pressure and tarot> perature, and shall furthermore be concerned with frequencies - r up to only about one megacycle, we shall su~ress the additional . terms in the stress tensor and adopt (2.02) throughout our whole investigation. -5- In the aooustic oase with which we shall be oonoerned, the velocities will be rather small so that we oan keep only linear terms. This means that we can delete the seoond term on the left of (2.01) and substituting the expressions for the stress tensor from (2.02), we obtain the usual form of the Navier-Stokes Equation: which can be transformed into: (2.03) An additional equation is furnished by the equation of continuity: (2.04) where again we have discarded seoond order terms. The third equation whioh we need for the mathematical formulation of the problem will pertain to the oonduction of heat. Let us consider at first a unit mass of fluid. If the tempe~a-· ture is ohanged by dT and the volume by dV, then the internal energy will be changed by: dU.",. (dU) dT + (C)'U) dV ;}T v ()V r Making use of the two well known thermodynamio relations: .. C (2M) dT " 11 £Y.l ~ -p + T (~) (aV T ~T >f -6- we obtain fo~ the change in inte~nal ene~gy: By the Fi~st Law, the quantity of heat abso~bed will be: ])Q = dU T peN =: CIT dT + T(~)t cit = = C11 dT- rT (Ee) 01" OJ'{ } Hence, the amount of heat abso~bed pe~ unit time by a unit volume of fluid will be: This heat must come f~om two sou~ces: (a) heat will be oonducted into the volume elemen~ f~om outside, (b) heat will be produced inside the volume element due to viscosity. Denote this ~ate of production of heat inside per unit volume by F. Then, fo~ ene~gy balance we must have: oc dT _...r. (~) l£ _ (Tv2.T + P ) IT cit P at y dt - We oan write this as: (2.05) But since: d c) -::. -+17'.V' cit at .. -7- we can bring (2.05) into the form: (2.06) We shall show later in VIII that: Ii' =0 It Zl'J Lij Lij so that it consists of terms quadratic in the velocities. Hence, if we consistently neglect seoond order te~ms, we oan 'write (2.06) as: But the fourth term vanishes, and from (2.04) the second and third terms cancel, so that: (2.07) where we have 1st: -ae=~ fOC lT Equ. (2.07) gives us the relation between the temperature and the acoustio field. The system of equations which we must now solve simultaneously is given by (2.03) (2.04) and (2.07). If we differentiate (2.03) with respect to the time and make use of the fact that the pressure is a function of both ~ and Tso that: -8- then we obtain: and eliminating ~ by (2.04): We now express the velocity in terms of a scalar potential~ and a vector potential ...... A by the relations: (2.09) ~ representing the longitudinal vibrations and ~ the transverse ones. Substituting (2.09) into (2.08) and (2.07) we obtain the three equations: Let us now assume a periodic state with time dependence e- iwt • Then the three equations above become to the space dependent part only): and T now refer -9- (2.10) pOLI/~~ '" ~ LW~Vl~- (~)T~o~i~~ + iw ($. )(T -LLO'T -; ~ ..ylT + To (2.ll) (~) Vl~ (2.12) -iWo~ - [t~ + ~ (~+J\74~ (2.13) e" po dT 'f From (2.ll) we obtain: T= (~)y and substituting this into (2.12): (2.14) But: so that: T(~)J.. =:. - (c - c ) (~) aT y l' 11 dY T and hence: -10- But since: we can write: Substituting this into (2.14), introducing the kinematic viscesity )l lby Y='~/fo and letting: (the longitudinal velocity of sound), we can bring (2.14) into the form: (2.14) We can write this as: (2.15) so that if Ji2.;I: .1:) a solution of (2.15) will be where: V2~1 + ~~~I: 0 v2q:l. -J. J:l.1.cj>:l. '" 0 1 (2.16) -11- also, from (2.10): 2 K==- ,W Y (2.17) We have thus reduced the problem to the solution of two scalar --and one vector wave equation, subject to certain bounday conditions. In this form the problem was presented by Professor Epstein in his lectures on IIMechanics of the Continuum ll in the fall term of 1946. (b) Elastic Solid. The equation of motion of an elastic solid is given by: (2.18) where L~ is the elastic stress tensor with components (2.19) These expressions for the stress tensor components neglect any additional thermal stresses which may exist due to inequalities in temperature. In the previous case, these could be neglected, since they were very small compared to the viscous stresses. In the case of a solid, however, this may not be the case and we shall not be able to negleot them. It can be shown (8) that -12if temperature effects are taken into account, the normal stresses beoome: (2.20) where and 0<" is the coefficient of volume expansion. The shearing stresses are not effeoted by temperature differences. Sub- stituting (2.19) and (2.20) into (2.18) and again neglecting seoond order terms in the dependent variable, we obtain for the equation of motion: or: (2.21) Since the derivation of (2.0?) applies to a solid as well as to a fluid, we oan write: Again, let us introduce a scalar--and ve~ potential (for the longitudinal and transverse vibrations respectively) by the relation: -13- Substituting this into (2.21) and (2.22) we obtain the three equations: oA.q, fo ~t2. == (A+~)V2~ + ~T dT = dfv.2T + To dt YoC (~)V2~ IT oT V ~t Assuming a periodic state with time dependence e- iwt , these equations become (.6, ~ and T now refer to the spacial1y de- pendent part only): -OUl~ :::: M.v 2 A J 0 "'- / .-v- (2.23) - fow2.~ ~ (tl+7'<)172~ + fir _ lWT '" ~V:z.T _ iwTo (~)~~ POClT aT y i (2.24) (2.25) Solving (2.24) for T and substituting into (2.25) we obtain: (2.26) where: -14- We can again factor this into: so that if It)2.+ i; a solution will be as before: where: 'i7Z~1 + ~I~~) == 0 V 2 "l, >: 0 'f2. + ~:z.th 2 j';l. (2.27) also, from (2.23): (2.28) Thus we see that the equations which we must solve in the case of a viscous fluid or an elastic solid are identical in form, differing only in the constant coefficients. We note from (2.14) and (2.26) that if we had not taken thermal conduction into account (whiCh corresponds to letting -at ---"" 0) then we would have come out with only a single scalar potential. The existence of the second scalar potential is something new and must be due to the effects of thermal conductivity. The deeper physical significance of the second scalar potential will become clearer later. -15III. CALCULATION OF kl AND k2 (a) Viscous Fluid Fxom (2.15) we get: and compaxing this with (2.14) we obtain the two equations: (3.01) Solving fox Jc~ and.-k':-, we see that they will be the two solutions of the equation: (3.02) It is shown in Appendix I that fox fxequencies up to about one megacycle, we can wxite with accuxacy sufficient fox oux puxposes! (3.03) (3.04) We see that except fox the case of zero fxequency, kl is fax from being equal to k2 so that no degenexacy exists and our two solutions of (2.15) will be independent. -16(b) Elastic Solid In this case we obtain from (2.26) the two equations: (3.05) so that ~~ and i: will be the two solutions of: (3.06) It is shown in Appendix II that for frequencies up to about 100 megacycles, we have with sufficient'accuracy: (3.0'7) (3.08) Again we see that except for the mase of zero frequency, no degeneracy will exist so that our two solutions to (2.26) will be independent. The values of kl' k2 and K completely specify the nature of the potential funotions. The physioal significance of the results obtained in this paragraph will be discussed in the next section. -17IV PLANE WAVE SOLUTION (a) Visoous Fluid For the case of a plane wave moving in the positive x direotion the longitudinal waves are expressed by 0/-= Ae.'~\Ee'~.zX where A, B are constants, since we have the possibility of tw® different longitudinal waves and the general solution will be a linear combination of them. Since kl and k2 are complex, the amplitude of the two waves will decline exponentially. In order to get a quantitative measure of the rapidity of absorption, let us define the quantity 1 to be the distance in which the amplitude declines to lie of its initial value. Then, fDem (3.03) and (3.04), we obtain: (4.01) lJ. ~ rl~1 .t (4.02) wt where the subscripts L1 and L2 refer to the two different longitudina1 waves. We see that the absorption increases with in- creasing frequency, but not quite as rapidly for o/~ as for The quanti ties ILl fl . and ILl.. are tabulated in Table II as a func- tion of frequency for air and water, and it is evident that they are of entirely different orders of magnitude, ~ being damped only slightly but for f~ the damping is appreoiab1e. In the absence of conduction effects, we let 1t~0 and obtain: -18- so that in this case we get only one longitudinal wave instead of two, since o/~ suffers infinite absorption and hence does not exist. II.. agrees with equ. (9) of (3) in the limit as Cff.--,.O, J as i t should, because conduction effects were there neglected. Equ. (4.01) is identical with the result of Kirchhoff (6) who used a slightly different method of attack. For the case of air, V and if£ are about equal, so that, from (4.01), the additional absorption of ~I due to thermal conduction is of the same order of magnitude as that due to viscosity alone, a result which was already anticipated by KirchhofL But in either case, whether conduction is taken into account or not, the absorption for 0/1 is small, so small that the damping taking place in distances comparable to the sizes of the obstructing spheres we shall consider in the later sections is negligible. The absorption of 0/2 and, . ~n T face, the existence of ~2 is due entirely to t?e effects of thermal c'onduction, and it would therefore seem appropriate to call ~:l. the denote 0/1 by IJlongi tudinal wave". IIconduction wave". We shall We see from Table II that the conduction wave is very strongly absorbed in distances comparable to the sizes of the spheres we shall conSider, and we shall not be able to neglect this. We note from (2.17) that the inclusion of thermal conduction does not affect the transverse component. iT = r:z; We obtain: (4.03) -19- Thus the absorption of the transverse-- and the conduction wave are of the same order of magnitude. It therefore appears that the main energy loss occu~ng in the diffraction of acoustic waves by small obstacles is due not only to the partial conversion of the incident longitudinal wave into a transverse wave which is quickly absorbed, but also to the partial conversion into a longitudinal conduction wave which also suffers a strong absorption. (b) Elastic Solid As before we have , and since kl and k2 are complex, the two waves will again be exponentially damped. Defining 1 in the same manner as before, we obtain from (3.07) and (3.08): lL = I l.j £ (;'t)~ ) (4.04) wJ.(jf M (4.05) Again the absorption increases with increasing frequency but not quite as rapidly for see that 11.1 of ~I ' is very large) indicating a very slight absorption while the absorption of ~~ will not be negligible. absorption of duction. 'h as for ~,. From Appendix II we The 0/, is due entirely to the effects of thermal con- For zero thermal conductivity the longitudinal wave is not absorbed at all while the conduction wave undergoes infinite absorption and will hence not exist. The essential difference between the case of a viscous -20fluid and an elastic solid appears in the transverse component. Whereas in the former case the transverse wave was strongly absorbed, in the latter case it will not be absorbed at all, since from (2.28) K is real. Furthermore we see from (3.07) and (2.2~ that in an elastic solid the wave lengths of the longitudinal-and transverse waves are of the same order while for a viscous fluid the wave length of the transverse component is very much smaller than that of the longitudinal one. Summarizing we may say that for either an elastic solid or a viscous fluid, the absorption of the longitudinal wave is very small and in all our later applications we shall be able to neglect it. • In either case the absorption of the conduction wave is appreciable and must be taken into account. For the trans- verse component we have strong absorption in a viscous fluid and no absorption at all in an elastic solid. To our degree of approximation, the velocity of propagation of an acoustic disturbance is not effected by the inclusion of viscosity or thermal conduction. -21- V. RIGID, ROUGH SPHERE WITH ZERO THERMAL CONDUCTIVITY Before tackling the more complicated situations, it will be instructive to consider at first the simplest of all possible cases. Consider a plane wave, propagating in a viscous fluid and impinging on a spherical obstacle. We shall assume the obstacle to be heavy and rigid, and rough at the surface and composed of heat insulating material. flui~however, The surrounding has a finite thermal conductivity, and our prob- lem will be to determine the acoustic field. Since the obstacle is assumed to be rigid, we can say immediately that inside the obstacle, all the potentials will vanish. Outside the obstruct- ing sphere we have the incident wave, represented by a potential ~t and the scattered wave represented by the scalor potentials ~) and ~4 and by the vector potential coordinate system (r,eJ~ A. Adopt a spherical ) and let the incident plane wave travel in the direction of the positive polar axis. We see from sym- metry that we shall have an Af component only and that there will be no dependence on Lf . Since ~I and tf2. will be solutions of (2.16) and A'f will be a solution of (2.17), we can write down the following expansions for the potentials (9): "" Z i.)\ (211+ I) 1\"h.~ (~)A.)~ (~ e) ... (J) 0/:1. ~ Z LI\(2~+I)CI\ ~~ (~2A)~ (CiY>e) ~I ::: o IJ o (5.01) -22where, from (3.04) (3.03) (2.17): ~=~} I C- ~.2.: (1+ i.) r!'?{ ) K::: (1+ L)f Wi 2v 2Qf jn.(X) and .l~ (x) are the spherical Bessel and Hankel functions defined by: (fx H....~ ,.lex) == ff;( I J (x)+ L Y (x)) .2x h"i ""i fl (,) (x)=- -1 /h. 1I il ) Where ~+!-{)() and ~..! Ill) are the two ordinary Bessel functions. 2. Some J- of the important properties of J~ and 1.~) are listed in Appendix III. 0 (2) 0(1) • t We use An and not~h since our time dependence was e-~w. and we desire an expanding wave. In all that is to follow, the superscript on l~). will be omitted. Bn , C)1 and Do are constants which have to be determined from the boundary conditions. Since the sphere is heavy, rigid, and rough at the surface, and has zero thermal conductivity, the velocity must vanish at the boundary of the sphere and so must the normal component of the temperature gradient. Let the radius of the obstacle be R. Then the boundary conditions become: ( 119)'1.",1<== 0 From (5.01» (5.02) I 1Tf is identically zero everywhere. Thus, we have three equations from which the three unknown coefficients Bn , Cn • and Dn can be determined. We must now express T in terms of the potentials. From -23- (2.11) we have: But since the scalar potentials are solutions of (2.16), this becomes: so that (5.03) where: __ -ip. lAi + C<I J l. - Now (5.04) so that: Hence: Since we are neglecting the absorption of the longitudinal wave, we are neglecting terms of order WJ'c'" and hence we must also neglect them here. So that we can write with sufficient -24- accuracy: (5.05) Also: (5.06) We see that: ):~ I== Wc~ (1- i)« I (5.07) for the frequency range we are interested in, and we shall make use of this fact later on. Using (2.09) and (5.03), the boundary conditions (5.02) now become: Making use of the fact that: indicating differentiation with respect to t he argument by a prime, omitting the superscript on .1~) and letting: -25- (5.08) we obtain, using (5.01): al!~(QI) B~ + gll~(Qz)c ... - h(V\TI)~Jg)J\, - ~}\{'11) Bit i- -a,j~(a,) ~h (aJ Cit - [.I~~(,g)+JJC)1])h = - j,,{QI) (5.09) o· ))11. This holds for n > o. If n = 0, only the first and third equa- tions are to be used. The second equation does not exist. Let us now examine the arguments of the Bessel Functions a little more closely. I!l I = C wR C ilL We have: a). '" R {Ii-i.} r~1 2-af Since -w ': -.:21T" , where At. is the wave length of the longitudinal wave, we see that al represents the ratio of the circumference of the obstacle to the wave length of the incident wave. In all that is to follow we shall consider this ratio to be small. (In the case of fog particles, R is about 10- 3 cm, so that even for w as high as 10 6 sec-1 , al for air would be of order 10- 2 ) • The assumption that wR« I C imposes an upper limit on the frequency beyond which our approximation will not hold. The quantities a2 and b may not be small so-we cannot make any approximations there. It is shown in Appendix IV that subject to the assumption -26- that al« 1, and alsq making use of the faot that I:' I« I ~ the coefficients turn out to be: B =_"!'i Q 3 o 3 (5.10) I C Co "" 0 H J cot. 2 h.h. Il :; -;;; (In)! (5.11) (5.12) Equs. (5.10) obtained in (3). and~.12) are identical with the solutions This is of oourse as it should be since to our approximation Bn and Dn are independent of the thermal conductivity, so that as far as these two coeffioients are concerned, it should not make any difference whether thermal conductivity is included or not. What is essentially new is Cn pertaining to the conduction wave. Since o/~ is strongly damped we see that at points far away from the obstacle, the acoustio field is essentially the same as the one we would have obtained if thermal effects had been neglected. As 2 -..,.0, Cn does not go to zero but o/~ will and we are then left with the solution given in (3). Since al is assumed to be small, the series for the potentials will converge rapidly and it will be sufficient within the acouraoy at which we aspire to maintain only the terms wi th n = 0 and n = 1. Equs. (5.10)-(5.12) ooupled with (5.01) completely desoribe the acoustic field and represent the solution of the problem. -27- VI. VISCOUS FLUID SPHERE We shall now assume the obstacle to be composed of a viscous fluid of finite thermal conductivity. is the same as before. different waves: waves inside. The outer medium Here we shall have to deal with seven four waves outside the obstacle and three If primed letters pertain to the medium inside the sphere and unprimed ones to the medium outside, then the expansions for the potentials take the following form: I:: L\1(2MI)j~(~/I.)~ (Cf)')(;l) 00 (Pi: = ~) ,. <j>~ = o cO .~~>)1 (2n-+ I) Bn~J~IJ!)7; (c~e) o co L Lh(2n+l) Cn!1'> (i,/L)r:; (C&)(;l) () DO Af ".. L: ill (2V1T 1).Dh ~11 (KI?) ~ (c~ .9) I .. (6.01) ~/ == ~ In(2Y1+1)B~},(,~>L)~ (c~e) ~; = co L i (1Y1+ I) C~ j,,(~dlt)~ (C~ ~) 11 o A~ == co L l~(2tl+l) D~ jl1 (K\)~l(c~e) J Since the Hankel function diverges at the origin, we must use J~ inside the sphere (standing waves). We now must determine six unknown coefficients instead of only three, as in the previous paragraph, and this will make matters a little more complicated. The boundary conditions we must satisfy are that across the boundary of the obstacle the velocity, the temperature, the -28- heat flow, and the stresses Prr' Pre and Pr~ must be continuous. It can be shown that in our case, where there is no dependence on f and only an A~ component, these stresses are given by: (6.02) (6.03) The stress condition thus gives ~ two equations, so that we shall get a total of six equations to solve fbr the six unknown coefficients appearing in (6.0l). The term p. in (6.02) is the hydrostatic pressure and we must now express it in terms of the potentials. periodicity, we have! But: In view of the -29- and using (2.11), this becomes: so that from (6.04): then from (2.16): Now: so that we can write: (S.05) where: (6.06) (s.O'7) Using (S.02) (6.03) (6.07) and (5.03), the boundary conditions yield the following six equations (out;ide the obstacle we must again identify ~I wi th ~~"+ ~I • ) : -30- :BnG\I4'.~(al) -13~ Qijj(~:) + C.hQl.-l~ (t.l~) - C~ Q;jn.(an - ])ll'y\'(h+))~~(t)+ +1)~n(I'l-l-I)Jn[.gI) == -l4 j:(a l ) J (6.08) "B h ..t. I1 ('II)- ~~jh(tll)+ Ch-tIl(Qz)- C;}'I(~) - J)1l.[.E1~(-l)+){.h(.g)]+ +~ r.gJ·~(.gI)+ jh(~J)J::: -J~(().J) (6.09) (6.10) 'En 7("'A,K,~ [a,) - 15;«/ Gil· (~:) + CnXrli.l" --K,~ (a z) - Cn'",} Q~ F(o.~) ~ == - x()(\ a.1jn' raj) "EnE (6.11) I(l1!~(Q,)-ll1 (a,)] - -:B~ [Q/P (a:) - J~!a/)] + c~(; [ Q21~ (o,7-) - .lh {(.Il)]- -c~ [a1p (al)- jl'l(aD] - rJ)h € [-8:t~ (g) + (n'l..r h-l)lll (i)] + +fD~ [.D'J~J(~I)-j. (n1.+ Y1 _ 1 )jh(.KI)] = - ~ [a,j~(a,)- jn{a,)] (6.12) [.g2fJ!)t{~,)- .2aJ~~ (Q.)] - 13: [-KJ~~~ j» (Q:)-2atjll') (al ~+ L;,E [.g~),/al)-2tl.t2.1~J(alil­ 'BI1 G -C~ [.gJf1 jh(Qn~2atF{Ai)] + Zi1,6'~(11+1) [~1~ (~)-l"{,O](6.13) where: We shall solve these equations subject to the same approximations we made beiore, namely that: t:l, «I ) a~« I ) 01 )a~I«} 0/) J 1.,{;j«J. :;l -31- Since we have seen in the pxevious section that the expansions fox the potentials converge rathex rapidly, it will be sufficient to evaluate only the coefficients for n = 1. n = 0 and If n = 0 the second and fifth equation do not exist and in the xemaining equations the terms involving Dn and Db do not appear. The solution is indicated in Appendix V and the results are as follows: (6.14) ,,~gIG:~ . (-&t"'(~)tJs,(-8I)1.Wl{-&) (6.15) L Gl1W1 (-& tm(.gt~J~ (-G )lm (.t) b,m" 0, 1 1 (6.16) (6.17) I (S ~ - I) ai}I (al) co -= cXl.eI, ~ijda1)AD( X jo (an Jt,) Ql) - (Al ) x- 3 ~G1!I+l{1-A:)(I-~ )G1lJJ'I(Qi)+[~/ GBd- (1- S)GBJa~J·O(Q1) (6.19) 3 )IX,. J(x-I )j) (Q~)lJaz) + Gi jo(lli)lJLll J- x C!2.i (IA,,) jl (Il.D C :1.a ell I Q1 (6.18) [ 1-1' "'I I I _ ') I U 01 I c)= oi, fl i\(6a;-I)Q~-t\,(aJ (6.20) o ell Doi [. [o.l)lo(G;t) - )CIA], f~ (ai).,{Jt:ll ) ~ ",I 1 fl [(J+2x);i; Ga: - 3xJ~JQ,.) + X [1+ (1- o)G"B)- i, G-s/ Q.t4l.o(Q:t) ",I .2(x-l)jdQi)~,{a~)+ al J~{Qn.1JQz) - XQ2~O(Q2)J",{QD (6.21) -32- Bl whexe Gnm and Gnm axe abbxeviations fox the following quantities: G~I"I-E, G?'=3(6-1)-lJI.} ~:, ... 3(€_1)+~"g};t} G~I::9{I-f)+~-812(8-1) Goo:: (J-e)(l+S)} GOI = 9$(E-I)- ~1(b+2)} G10 = 3(S-I)(d-J.2)+ t($+2) Gil = .27S(i-€)+~e.j2(S-I) (6.22) whexe: It is shown in Appendix VI that these equations reduce to the results of V when E ::S", 0, x= 00 , because there we essentially considered an infinitely heavy and infinitely viscous sphexe of zero thermal conductivity. This serves as a useful check. Since we shall not need Dn and DA in oux later applicati@ns, these coefficients were not evaluated. It can be shmvn that (6.14)-(6.17) are identical with Epstein's results (3) which is as it should be because these coefficients are independent of the thexmal conductivity of either medium (to within our appxoximation) , and it may be safe to say that the same applies to the coefficients in the vector potential. As befoxe, the only essential diffexence between oux re- sults and Epstein's lies in the existence of the second scalax potential, ox the conduction wave. But, again, since the con- duction wave is stxongly damped, we see-that at points fax away fXom the obstacle, the acoustio field is essentially unohanged by the inclusion of thexmal effects. -33- VII. ELASTIC SOLID SPHERE Consider now the case where the diffracting obstacle is a sphere composed of an elastic solid, and let it be embedded in a viscous fluid. Again we shall have to deal with seven dif- ferent waves, and from II and III the expansions for the potentials assume the following form: (/() ~~ = 12 i~ (2n+J) jl1 (-&Ih.)~ (cp?&) o . 00 cpJ = L i.~(ln+l}BhJtJ/~J''l)r: (CIY>9) II ~2. = L C'(l~+J) C"~Y1 (~lA.)~ (cI.I')9) cD o co AI!(":. ~ ~h(.1~+J)DYlll1{K~)~ (cll?f7) J (7.01) L i"Un+J) c: jl1(.~;It)~(CD1&) co ~1 ~ - L~ o where: o _ u; ~ I - -C. -ll,:f.~'i{l+i.) ... 2fll . ) i/:wt . A+3-« £1 j I I ~~"" 1:1.\7 J '(1+ i) ldf J J K • = £Off' Y'" J The reason for introducing - iiV- into the primed potentials will become clear later. We must remember that ~ and Afare poten- -34- ,hJ tial functions for the velocity in the viscous fluid while ~ andA~ are potential functions for the displacement in the elastic solid. We must again satisfy the same bounday conditions as bef~re, ~T namely vr, va, T,6" dJt. ' Prr' and Pre must be continuous. As before T has to be expressed in terms of the potentials. This was done in V for a viscous fluid and we must now do the same for the case of an elastic solid. From (2.24) we obtain: and using (3.07) and (3.08): since interested. for the frequenoy range in which we are Thus, we can write: -) . (I,h) I,hl) J = - LW a(, j. + o(~ 'f). where: 12-* ~*I 0(;.. :: - } (7.02) -35Again, the reason for leaving -iw outside will become clear later. The stresses Prr and Pre for an elastic solid differ from the ones for a viscous fluxs, (6.02) (6.03), in the following respects: (l):!r is replaoed by i ' (2) -p in Pn is replaoed by AV'i ' (3)? is replaoed by /{,( , (4) There appears an extra term -~T in Prr due to the thermal stress. We therefore obtain: But: If we now define: aT and and set up the oontinui ty conditions for v r ' ve' T, 15'~ Pre (remember that VI = -iws) we see th~t we obtain equations "- - which are identical in form with (6.08)-(6.12). This was the reason for defining the primed potentials and cI./ J 0(/ in the way -36- The continuity condition for Prr becomes: that we have done. 1)1'& + [,glfoll l1 (tl,)- 2a.l.~J}(tll)] - 'Bill [) rR,'!~ (a,') - e e [-82.f1 {f iL)-.2o.;iJt! til)J + h tl CtJ} 14:1 (ai)] + 1111 [2a~fh {Q;D-r 2 {J;i~IJ (aD] + -r ])11 ~ ·lh(h-I- J) [.Bl~ (I) - "'-11ft)] - ~/·2n (VJ-I-J) [$ Ii f.&l) - j~ ti'}] = :: - € [~1~lf.. {fill)-2a/}I)(tl,)] (7.05) The definition of the primed arguments is in the present case: The expressions r~-tf I. and IL',. are simply the longitu- dinal- and transverse velocities of propagation of sound in an elastic medium. Since they are of the same order or even great- er that the longitudinal velocity of sound in a fluid, then if we assume, as before, that al is small, then this· implies that ai and b' will be small also. We also see that e as defined in this section is an extremely small number (for the frequency range we are concerned with), and, as before, J~I are «I and _d/ "'1 The evaluation of the six coefficients therefore proceeds along identical lines as in VI except now we have the additional simplification that not only al and a{ are small but also b' and E. The solution is indicated in Appendix VII -37- and the results are: (7.06) (7.07) (7.08) (7.09) (7.10) 01 and 01 are identical in form with (6.19) and (6.21). We see that Bo ' Bl' BJ ' Bi are identical with Epstein's result so that here again the only new thing is the presence of the conduction wave. -38- VIII. ENERGY RELATIONS Since the primary object of this investigation is to calculate the energy loss, or extinotion, due to small spherical abstaoles in the path of a sound wave, we must now turn our attention to energy oonsiderations. Ca) Viscous Fluid Assume at first that the obstacle is not present, so that we have a homogeneous fluid and consider a volume V of the fluid. Let us multiply (2.05) by dV and integrate over the volume of interest. Then: and using Gauss's Theorem and the Equation of continuity, this becomes: (S.Ol) where S is the surface enclosing V. order terms. In II we neglected second This accuracy was sufficient for the equations of motion but is evidently inadequate when energy relations are to be considered, and we therefore cannot make the same approximations here. If we now consider a periodic state and average (8.01) over a period, then the conduction term on the right will give no -39- contxibution. Wxiting of dt for the average of the left side, we obtain: (8.02) We must now tuxn oux attention to the quantity F which xepxesents the xate at which energy is dissipated pex unit volume due to viscosity. Multiply (2.01) by vldV and (2.04) by 2dV, add and integxate ovex V. Then: tv [[fE' ~; + p~. (£.v)£ + ±11~~+ :i:u2.17·(f~)+ v,vp- ~L.J lli~· L~]dV~ 0 " which can be txansfoxmed into: 11 (flT~V + 2. dt V [17' Vp dV - }~ 4;0 ~ 'i7:J 't'..J dV ... - 1. I t 1]: Y ~J 2 f ciS "l.1X~ 017 J 5 Let us now define the quantity f by: Then: so that: (8.04) -40For the viscous term we obtain: (8.05) where we have used (2.02). ~h7.· ~ L).' "'rl i': -.! .....v. I O"L But, again using (2.02): "[2 (.;ltT;)2. ,,» v.y av; ] 2l (C)11:.)2.. ) --1. ::: n ~ h - . - 2. ( -' . L ;)>c. ~ , ;>"1'(;})<. I i ' - :l -Y] .3 ( ( VO,!! ):L. that: 80 '7s-'-' "t.. Cl17j _ :2tz..2.. " 7,. T. 1\ d)Ci - L.., II II and substituting this into (8.05) we obtain: ~ 11· \7,j y.'J dV:::. f7r..... d ct _..l. (~T.. T· dV )f.' .lJ, J~ 'J 'J IJ 5 {, Y I'J ( I "I 11 lh" )J (8.06) -41- Substituting (8.04) and (8.06) into (8.03), there results: Again considering a periodio state and averaging over a period, the first two terms on the left will give no contribution and we are left with: (8.07) This represents the rate at which energy is dissipated due to viscosity, so that we can identify: 2~ 4.'tij L: Lt.) j = F Substituting, therefore, (8.07) into (8.02) we obtain: (8.08) For the acoustic case, the velocities, condensations, and compressions are so small that their squares may be neglected. Hence: so that: elE' -dt == (8.09) -42- This equation represents the energy loss per unit time inside the volume V. If now, the system is inhomogeneous with surfaces of disoontinuity between its homogeneous parts, (such as we have for the case of an obstacle) then the surfaoe Swill be the outer boundary surface enclosing the whole system since from the boundary conditions, p Y-ll') and vn are continuous. The volume integral must be evaluated over both the inner and the outer medium. We shall see in the next section that if thermal effects had been negleoted, the volume integral would vanish, leaving a result identical with Epstein's (Equ. (59) of (3». The volume integral in (8.09) thus represents the additional absorption due to thermal conduction. (b) Elastic Solid From (2.18) and (2.20) the equation of motion is: d~' + 017· \7.17, = V: 't.. - 8v.. T P-' dt ) Jj r 'j I L I Multiply this by vidV and the equation of oontinuity by tv 2dV, add and integrate. f[f~' Then: i +f~·(t·v)! ~1)1.~{ + + i1>l.V·(f3[) - '( - .~=:Vi~7;j +~)!.vT]tlY= 0 J 1, which we can bring into the form: (8.10) -43NO\v, using Ga~s I s Theorem, we can write: Jl.>. ~ 'tij tAV = f IT,, Tn" d y I'j S- ff,1"ij ~ lli dV 5 " 'j Let us define the following quantities: as; (8,11) e·· =- a""rL II Then, since L~ is symmetric, we have: ~ 7,.. '!7.lJ .. w "Tj' -ae·· L.. ') i)' J l>t ,.'j. ~ J i , Consider n0W the function W, defined by: Then, we easily see from (8.11) and (2.18) that: so that: Hence: so that (8.10) becomes: i ~ [f V1c1Y + ~ {WdV y Y f~TV. !!.dV = frrf.~-fT- fpu7.)v;,dt 'I > -44- Considering, as before, a periodic state and averaging over a period, the first two terms on the left will not contribute. Since, for the acoustic case the velocities will be small we obtain: (8.12) This equation states that the average rate at which work is done on the boundary (right term) is equal to the average rate of productien of heat inside. If we had not taken tempera- ture effects into account, then the left side of (8.12) would have vanished and the term :~Tvh would not have appeared on the right, giving us the result that no energy is dissipated. We can identify (8.12) with r[FriY]D.'1. of (8.02) so that (8.02) becomes: (8.13) This equation represents the energy loss per unit time inside the volume V. Let the system now be inhomogeneous and con- sider the case of an elastic solid sphere embedded in a viscous fluid. The energy loss inside the obstacle will then be given by (8.13) where the surface integral is taken over the surface of the obstacle. The energy loss in th~ fluid will be given by (8.09) which will now, however, involve two surfaoe integrals: one integral over the surfaoe of the obstacle and a second in- -45- tegral over a surfaoe surrounding the whole system. The total energy loss will be given by the sum of (8.09) and (8.13). But sinoe, from the boundary conditions, ,..,...J Lhl' - ) ~T :: fllYl on the boundary, the two integrals over the surface of the obstacle will canoe1 and we are left with (8.14) where the surfaoe integral is taken over the boundary of the whole system and the volume integral is evaluated over both the inner and the outer medium. in form with (8.09). We see that (8.14) is identical The various ooeffioients involved, however, are of oourse quite different. Equs. (8.09) and (8.14) are the expressions we desired to obtain, sinoe they will enable us to calculate the extinotion of the incident sound wave. -46IX. CALCULA'l'ION OF THE EXTINCTION (a) Viscous fluid sphere We must keep in mind that Pnn' T, and vn in (8.09) stand for complex quantities, supplied with the time factor e- iwt • In actually evaluating (8.09) we must replace these terms by their real parts. Consider now two functions, f and g, both of which having an exponential time dependence. Then we can write: Applying this to (8.09) we have: :f = ffn~v;,*dS iRe - fRe [[pc + (~1]TV.y"'clV ll s (9.01) y We now take as our surface a sphere of radius large compared to the obstacle, and concentric with it so that the normal components become the radial components. We saw in VI that both the transverse wave (described by the vector potentialt ) and the conduction wave (described by the scalar potential ~~ ) are absorbed very quickly. Hence, since the radius of our sur- face is large, we can in the surface integral of (9.01) neglect any terms which are due to either.t or:flo • and (6.05): "'" <>~cj>, 2. i,b. Then, from (6.02) fAll I -= - f - 2~ ~ ~ K ~~lcpl-2~ ~ ~ LWfocj>, S -47- since the second term is negligible compared to the first. So that: (9.02) For the volume integral, we obtain from (5.04) and (5.03): But ell and J: are imaginary while oI~ and Jt are real. Thus, the first and last term are imaginary and will not contribute. Hence: But: so that: Ref[~C~4(~~JTV.:£dY=- f.;~i. (J+~)Re. rcp,~!dV (9.03) o.t.i(;je (9.04) -48- Substituting (9.02) (9.03) and (9.04) into (9.01) we obtain: :! = - ifoW ReL f(~i4~')()~ (~!+o/nd¢ + ~;{ (1+ ~J~~ )Rt f~:f!*dV+ 5 i»side + £2~t (1+ !~~)Re. fctL-t~')4':tlV (q.()S,) tlI1si.le since outside the obstacle, we must identify 4', wi th ~i+~' Let n be the number of obstacles per unit volume. Then the total energy loss per unit time per unit VOlumB will be tiE h dt , i f the obstacles take up only a small fraction of the The relative energy loss or the extinction per unit length of path will be 1L ell: where Eo is the average total volume. ED dt energy which the incident wave carries across unit area per unit time. From (9.01) we obtain: Do";/( ~ ED= "2 ~e. f vh = 2: If.e I wfOTitil a~i* ClX ... '2 wgo-R I I In' J Denoting the extinction by ~ we then obtain from (9.05): (9.06) where: - I~ "" ~t.i f(fi+~')~( ~f+~~) d$ ) 'Tv,'" f{e f~:fi*dY J.Y1 '" ~eF1if~')~: dV J 5 (9.07) -49- These integrals are evaluated in Appendix VIII. It is also shown there that the result for the first approximation to the extinction ( ) 9.06, neglecting terms of order t?12. and tll/2. compared to unity is as follows: (9.08) Since we have neglected the absorption of ~L and ~I , rX then represents the additional extinction due to the presence of the obstacles. The first term in (9.08) is identical with Epstein1s result and represents the extinction due to viscosity alone. The remaining two terms come from the additional absorption due to thermal conduction. We shall apply (9.08) to a few special cases in the next section. (b) Elastic solid sphere The only difference between this case and the one just considered lies in the volume integral over the obstacle, as can be seen from (8.14). USing (2.27) and (7.02) we obtain, omitting the primes: Rt rTV'~"V ~ R, fT;~V..!.·JV= i~~ide 7.. =. W RI [( e oll~10*~ '/'1'1,j,'1*-I- oI1.~a'li ~)K "/'afl + "'I-R~n*~qlo/" +o.z4?2.1))/c2.o/J.~2.'II)dV * -50- Since ~I}~A ;z.. and kl are real while k2 is imaginary, the last term will give no contribution, so that: But: since for the frequency range we are interested in,the second term is very much smaller than the first. Also: so that: Re (T\'. E:'t.IV =- ~;' [~~ [rp, ~,'dV + R, rM~ dV ] il\side and hence we obtain by similarity with (9.05): where we have written thermal bulk modulus. ~J]: for {g:)'1).K being the iso- -51- Defining the extinction in the same manner as before, there results: (9.10) where Is, lVI' and IV2 have been defined by (9.07) and: (9.ll) It is shown in Appendix IX that the first approximation to the extinction, neglecting terms of order a and a com- pared to unity is as follows: (9.12) Since we have again neglected the absorption of ~. and ~ , ~ will represent the additional extinction due to the presence of the obstacles. We shall apply (9.12) to a few special cases in the next section. -52- X. APPLICATIONS Since equs. (9.08) and (9.12) are still rather complicated formulas when the expressions for the coefficients ~re substituted, it is best to consider a few special cases which are of practical importance, and for which some further simplifications can be made. (a) Water droQs in air This case is evidently of great importance in connection with the absorption of sound in fogs. Since the average radius of the water drops in fogs is about 10- 3 cm (10), we see that our assumption of GI,'" w! « will permit us to go up to J w"-' 106 sec-l without violating any of oux approximations. The primed quantities will pertain to water and the unprimed ones to air. Since water is much more viscous, much heavier and a much better thermal conductor than air, the paraand X will be very small. E '" .0167 J 8 :: .00117 I The actual values are: x = .02Jij Under these conditions we obtain: .,,36 = "3 ''''', '8, 'i7 '" 0, , (10.01) -53- Even though X is small, we cannot neglect the second term in the denominator of Co and C~ because for very low frequencies, when aa and aZ are also small, we can see from Appendix III that the first term will be proportional to second term will be proportional to )I~ rz; while the so that as w~ 0, the second term will eventually predominate even though x«/ • Substituting (10.01) into (9.08) we obtain, after some slight rearranging: } But: 1 J A rJ..j c.,l. "0/100»1 ) Rence, we can approximate the bracketed factor in the second term by: since S(~) t« } , in our case. ThiS essentially means that we could have neglected the term arising from the integration inside the drop. USing (5.07) we then obtain: -54- 1. ex '<: )i1i 31At { - Re (i Ga) + _1)(I-1-k) Re. . i ~; j, (a1)AJa,.) ,. (8 01." cl. t\ I . r I, j tl . ( I) 0 ( l thf'la.l D{a,,)- j( lJO Q].. All &1,,) } (10.02) We must now investigate what happens to ~l if ~ are small. andS Under these conditions, (6.22) becomes: Here, again, we must keep Gel and GIl even though 8«) ,be- cause for very low frequencies, band b' will both be small, so that the first and third tarm in the denominator of (6.15) will be proportional to ~ while the second and fourth term (those involving GOI and GIl) will be proportional to 1~ ,so that as w ~ 0, the GOI and GIl terms will eventually predominate even though 6«1 . USing Appendix III and the calculations of Appendix VI we obtain, for the case that € and S are small: (10.03) The difference between this and the resulu of Appendix VI consists of the fact that here g is small whereas in Appendix V~ -55- it was identically zero since we wanted to go back to the case considered in V. Substituting (10.03) into (10.02) we obtain the rather complicated expression: + (10.04) It would be very cumbersome to evaluate these real parts exactly, but fortunately this is not necessary in our case. Due to the smallness of b and X we see that the second term in each denominator will be of importance only when b 2 is smaller or of the same order as der as X. S , and when a~~ is smaller or of the same ~r­ We can therefore replace the second term in the de- nominators by their assymptotic expressions for the case of b, b I , a2, a~ small. This will of course not hold for intermedi- ate or high frequenc~es, but then the whole second term is so small anyway, that it is immaterial what form we assume for it. Subject to these approximations, the re~l parts in (10.04) can then be easily evaluated and the calculations are indicated in -56- Appendix X. c<::: .ii1/'I\R2.{ C The resulting expression for (10.04) is: 3/2. /+ (~r fetAl [~+ Ivuy ] + (g-J){I+ £f,.) [J!... + r~LUI]J R.).. J-+ (3XCle )2. I ~2wt ~! (10.05) :1f The first term represents the extinction due to viscosity and the second term is the additional extinction due to thermal conduction. As 1t~0, only the first term remains and our re- suI t becomes identical with Epstein I s'~. If in addition, the frequency is high enough so that (~~:)4« I , then we are left with Sewellls result. We see that (10.05) gives zero extinction for zero frequency which eliminates a bothersome point in Sewellls theory. Since in Sewell IS case, S was not small, but identically zero, the denominator of the viscous term was unity and hence Sewell came out with a finite attenuation at zero frequency. This difference in behavior, as pointed out by Epstein, is due to the oscillations of the droplets, which were not taken into account by Sewell. These oscillations will become more im- portant at the low frequency end, and less important the heavier the droplets are compared to the surrounding medium. clusions are borne out by (10.05). Since xtre'=t1fS c... c,J These conwe see that the deonominator of the conduction term in (10.05) is also due to the finite denSity of drops and is therefore connected with their oscillations rather than with thermal conduction. *Epstein considered frequencies that were high enough so that ~OY/Rl~)~<~J ,in obtaining his equ. (S9) from (SO), and thus Came out with Sewellls result for the special case of fogs. -57- We note that ~~ has the dimensions of an area, and it represents the absorption cross-section per drop_ Denoting it by ~a we obtain from (10.05): 4iR [ 0:/l\ " 'C.- I f we now plot (10.06) ~ .Jj~R against R2w we obtain a uni;versal plot for all R, which is quite an advantage. This is done on Graph I, where we have also plotted Sewell's and Epstein's resul ts for comparison. Vie see that for R2w 9:" 10-3cm2sec-l, the inclusion of thermal effects just about doubles the extinction that would have been obtained if viscosity alone had been taken into account. We also observe that for R2w~ 10- 2 (which corresponds to wI{ 104sec- l if R;v 10-3 cm ) the absorption is much more strongly frequency dependent than would appear from Sewell's result. We shall compare (10.05) with experimental findings in the next section. (b) Air bubbles in water A great deal of experimental work was done during the war in connection with the absorption of sound in water due to air bubbles because of its bearing on subma~ine detection. Recent- ly, a paper by Loye and Arndt appeared on the acoustical insulation of a submarine repair dock by an air bubble screen a- -58cross the open end of the dock (11), and it was found to be extremely effective. It would therefore be of interest to ap- ply our equations to this particular case. Here we have the opPosite situation to the one just considered. The primed quantities refer to air and the unprimed ones to water. If E1g (;'$) 6"8)::: f (' _ I" oo-=-c) D) and X become large then, from (6.22): 01) 2.19~ ) Substituting this into (6.15) and using the relations of Appendix III, we obtain! 13:; - l.ia' t.e:R,{l)- JJ,..{t) I 3 I GJ,l_ CJI,,(J) - j),2 (.B) (10.07) Also from (6.16) (6.18) and (6.20) we obtain, if S and X are large: (10.8) If we now substitute (10.07) and (10.08) into (9.08) and make use of the fact that \' ", {C\~ .. 0»1 cl ' it can be shown that the term in (9.08) arising from the volume integral over the outer region is negligible. We then obtain: -59- Closer analysis reveals that for the case we are considering, the first term is negligible compared to the second, • sin~e This essentially means that the main energy loss occurs inside the bubble and is due to thermal conduction. We then obtain: (lO.OS) The real part is evaluated in Appendix XI and the final result is: (10.10) where: ~~Rtd 2~' Thus, the absorption cross-section per bubble turns out to be: ~ '" ~R . A. f (:) (lo.n) where: (10.12) (10.13) so that if we plot c 6"~ ::nrRA versal curve for all R. against 2!- we obtain again a uni- This is done on Graph II. It can be shown that for ~t< I :i.73 ' F(t) is proporti_onal to w'2. while for ,F(::) is proportional to fi'. Thus, as in the previous case, we have a strong frequency dependence for low frequencies. -60- Whereas in the case of fogs, the extinction due to viscosity and thermal conduction are of the same order of magni"tude, for the case of air bubbles in water the extinction due to thermal conduction is the dominant term, which is a very interesting result. Assuming reasonable values for R, we see that the extinction calculated by (10.10) will be very large, indicating that an air bubble screen for purposes of acoustic insulation would be very effective. We shall compare (10.10) with experimental measurements in the next section. fe) Elastic solid obstacles suspended in air. In this case S will be evidently small, and since most solids are much better thermal cqnductors than air, X will also be small. By analogy with part (a) of this section, the second term in (9.12) (arising from the integral over the volume of the obstacle) will be negligible, so that we come out with a result identical with (10.05). Hence, to o~r approxima- tion, the influenoe of elastioity is imperoeptible. -61- XI. COMPARISON WITH EXPERIMENT The absorption of sound in fogs has been determined by many observers, but in most cases an analysis of the water drops as to distribution in size and number was not attempted. This, of course, makes it very difficult to compare the measurements with theory. Recently, however, Knudsen (5) made some new measurements, using the reverberation chamber technique and he also determined the size-number relationship for the droplets used. If Vs represents the total volume occupied by the obstaoles per unit volume (~is thus dimensionless) then (10,,05) becomes: (11.01) Now, Knudsen split his drops into five groups according to size and determined the number of drops in eaoh group. The quantity ~(pertaining to all the drops) was also determined. We Can then calculate the extinction for each of the five radii with the same Vs for eaoh and then take a weighted average aooording to volume. The resulting curve is shown on Graph III where we have plotted the extinction (in db/m) against frequency (in cycles per second). indieated. Knudsen's experimental values are also We see that the agreement i-8 rather good at the higher frequencies, but the pOints for frequencies of 500 and 1000 cps fall slightly below the calculated curve. This may be -62- due to the fact that at the low frequency end the errors in measurement tend to increase. The dotted curve on Graph III indicates the extinction we would have obtained if thermal effects had been neglected, which would give values too small by about a factor of two. Thus, the inclusion of thermal conduo- tion seems definitely to be a step in the right direotion towam a better understanding of the absorption of sound in fogs. The data taken by toye and Arndt (11) does not lend itself readily to comparison with our theo~J since no analysis of the bubbles as to size and number was made. The bubbles were produced by compressed air being passed through a hollow pipe with holes drilled in it at regular intervals. If the airflow is kept fixed then n will be roughly inversely proportional to R3 so that the factor in front of F(i:) in (10.10) will be inversely proportional to R2. Now if ~ I"\.. 3 or greater, F(i:) is proportional to R so that the extinotion will decrease inversely as R. It would thus be advantageous for purposes of acoustic insulation to use a larger number of small holes, rather than a smaller number of big ones. This effect was actually observed, but undoubtedly it was also partly due to t he fact that the bubbles made less noise upon emergence when they were small. the other hand, when %:"'"" On t or less, then F(i:) is proportional to R4 so that the extinction will be proportional to R2. It therefore appears that for a given air flow there exists an optimum size for the holes for maximum absorption. Hartmann and Fecke (4) measured the extinction of super- -63- sonics in aqueous suspensions of lycopodium particles of density~J = 1.00 gx/cm2 • In this case 8 = 1 so that fxom (7.07) BI = 0 and thus the first order extinction due to viscosity will vanish, but the extinction due to thermal conductivity will not. Since in Epstein1s case the additional term due to thermal conductivity did not appear, he calculated the second ordex terms in the extinction due to viscosity and added the additional enexgy loss due to scattexing but came out with a xesult that was still about 10 times too small. Since in our case the first order extinction due to thexmal conduction still remains, it may be that this additional texm will increase the calculated values by about the right amount. Unfortunately, however, cextain constants of lycopodium which we must know fox numerical computation have not been determined, so that it is not possible to compaxe oux results with these paxticular measurements. -65- PART II This part contains the various mathematical appendices referred to in part I. for numbering equations. We shall here adopt the same system If reference is made to an equation of part II, the number will be followed by the letter A (suCh as 3.04A). If the letter A does not appear, it is to be un- derstood that the reference is made to an equation in part I. -66APPENDIX I The solution of (3.02) is: (1.01) Let: A= jlA?y+w(Je C = .If /Jf£02.y } 3 :13= c.2. J ]I '" -4Gfwcl. if Then we can write (1.02A) as follows: -i"", 211,}'{,"A-"B) [I ± (1+ C-t i1) )1ll] (iA-"'JJ)" C + C;D It turns out that as long as w{ 107sec- l , then we can write with suffiCient accuracy: since A « B, C« D. (See Table I for actual values in the case of air and water). Since in our later applications we shall not go to ~requencies beyond w = 106sec-l, we can ex- pand the square root and, taking the lower Sign, we obtain: (I. 03) using (1.02A) this becomes: .$.4~ Lo2.[I+ ~ (1V+tf- ~)] • I C'I. c}, 3 if -67- Since the imaginary term is small, we have: 11 r;; 11.) y + cJf (1- .l.»)] !d[I+ iw c. c.'1. (2:. 3 3 (1.04) 2 Combining (1.03A) with (3.01) there results In (1.03A) we had to keep terms of order A~ and ~J3~since they were the only imaginary terms we had (indicating absorption) but for k~ this is not necessary. Hence: so that: (1.05) (1.04A) and (1.05A) are the desired equations. -68APPENDIX II The solution of (3.06) is: This can be written as: , / {~iW-ae(7.(fl+2,,«) .;2iWarpD M ~2.:: 'dfr"WJb+,'W-{rl-t-fu+ M) ± ILItl~+~+M) 1+ {t\+0%'+1'1? -(Jl+r~M)1.. 1~(M7) (2.01) We can estimate the magnitude of M in the following way: we know that: =_ ..!-(dV) ,y (~) = dX ( ~) OlT Y dT' ;)V " T f Y where X is the isothermal bulk modulus. oI"A-}O-FoC I ) X-IOI2.dYhes-cm4 For most solids: (~)Y ~ 10' , so that Since for most solids A and~ are of order lOll, and 0v is of order 10 8 ergs - 0- 1 - gr- l we see from the definition of M that M is of order 10 7 , so that M «A or)'. Now consider the terms inside the square root. Since ~ and f are of order = 1 for most solidS, then at w 10 6 the terms inside the square root are of order 10- 5 , 10- 9 , 10- 10 respectively, so that we can expand the square root. [ ~1.", 2~(~+f<) -I[ ( Then, (2.01A) becomes: J,v'l.afp.{iH2jA) J ~f.lAl+iW tl+}k+M)± i~(tl+}I'HI1)~ ri+}A-I-M r > ± (2.02) -69- Taking the lower sign, this becomes: Since 11 <.:. A or/ ' we have: so that: Since the imaginary term is small, we obtain: which is (3.07). Taking the upper sign in (2.02A), we get: Since the second and third terms are negligible compared to the first, we obtain: which is (3.08). -70APPEND IX I II We shall find the following properties of the spherical Bessel and Hankel Functions very useful: (3.01) I. )" j (y..)= t ll - I L i;e ( 1;. lMI • ~ X II (IH)!J!2x -r ()I.1. IHl • L -t ..... + 1.(11 )/'11) .L l! ==: f') 1. n!(2x) (h-2)!21 (2x) (3.02) Also: xj~[x)-= hj"()()-xJ~-tJ}() - f (3.03) j" (x) -t- Xf~-l x) ( (3.04) (2114J)fll (X)= Xjh..,(X) + Xj'h_JX) (3.05) x:J.pJ (;<) -2xjll.. Ji<) = - )(2jl! (x)- n (n-I) j" ()() (3.06) X ~ (x) ,.. - (n+, ) and identioal expressions in whioh f~ fx) is replaced by lJx) If x is small, so that we can neglect x 2 compared to unity, then: (3.07) (3.08) (3.09) -71- (3.10) In the following appendices we shall find occasion to use the two xe1ations above vexy fxequently. -72- APPENDIX IV The coefficient Bn is given by: where, from (5.09): ;j = BIl -a,jUa,) Q2~:{aZ) -}J(n-t))j~ (-0) -J~(o.,) A:Jaz ) -~~I{f)-),h I.e) d,£l'J~ (tAl) .) ~a.z.):hJ (GIl) D ~~ (0.,) a2~~ raJ -'tl{h+J)jJ~) ~~ / a,) j~ (0. 2 ) -~~~J(-e)-Jh (-f,) Il, /j.." d,~,j~ (G,) el2 a),,' (0. 2) 0 Hence: ~p":: ¥l {>H I) ~J{l) [ ol2«l2jh(a')~~(QJ ~ c<) ~lf~(al)J.(a2)J- (C(~-oI,)QIAlP(Ot)j,.)(o..l) [j~~'(4)-+~l1(i)] and using (3.07A) and (3.08A) this becomes: I l\J3~ ~ ~(\'1-j-I»).Jg) til Q~{l> (4,)j~) [~2) - ,,(, YI j. (tl,)~" {~J]­ - (oIl -e{,) U2 11j. (a,)j~ (a l ) r.G~! (~) -T j» (.g)] -73- The two terms in the first bracket are of the same order except for oil and 0(2,' But since I:: ]« I I, we have very nearly: 4B~ -;; J2,a"J~(a,)jhr&)j,~J{(j2.) (h""+I)_Y1";,:m - 11) h~ and using (3.03A) this becomes: (4.01) For 4 we obtain: _4 ~ - n(11-t ))~,,(.g) I,(~ 1" (a,) j,/ (fA,,) - ot. a, ib' (a,)~ (0,,>]+ 0.2 + (o/~ ~cf,) a,o.2. j,~ (a, )1,,) (01.) [ il~ (i) + J» (.g) ] Again neglecting oll compared to 0/2. and using (3. OSA), we get: if. ]l ~ ::-- 01, a.i: l"yI, (~) {'I'4 I) i, (0,) +('41)1, ra,) [1+ ;~ and using (3.04A): (4.02) Combining (4.01A) and (4.02A), Bn becomes: -74- For On we have: where: ali~(Q,) - a'f'1 (Il,) - h(l1+I)A rg) ~I> (Q,) - j_{ a,) -.e.g"1 (~)-JII(~) = e<,a,J~(Q,) -eX,I.I'J.")(iG,) fl.c ,.. l"l h 0 and using (3.03A): iJ. CII I ~ n(n -+ I) eX} tA, ~Il (&) f~{a,) ,t-tJ(.I,) - J~-tJQ')~l"l {a,,] But from (3.09A) and (3.l0A) we see that the first term is of order YQ~ and the second of order unity so that we can retain I only the first term. Hence: no.,J(" (Q.)~~-+I fa,) -K~ ('I,) Using (3.09A), (3.l0A) and (3.04A) this becomes: (4.04) Also: il:D.= Il aJ~ (0.,) a.)~ (a~) - QF(a,) j~{Q,) J (Q~) - Jft (a,) d,J,~{Q,)tI, It til tl2 in) (Oo .} l ) -01, a,j" (Q,) -75- making the same approximations as before; we obtain: Hence; (4.05) When n = 0 we must solve the following set of equations: o,,-edfo.,Y'Bo + ~~;(4~)Co ~ - Q.jd(Cl.) C<I Q.jd {a,fPo + cI"a,;.1j(f).1.)Cp = - cllo,ljj[Q.) from which: L\~ (~-d,) Q,Q,.i';rtl,)A,/(Q.d Ll at;=: - (cl~-ct.)a'~f~I(Q.)j.j{t~2) Aco "" 0 so that: (4.06) (4.07) Equs. (4.03A)-(4.07A) are the desired solutions. -7SAPPENDIX V Consider at first the case when n = 1. We shall have to evaluate determinants of the form:: k1 A2 A3 A4 A5 AS Bl B2 B3 B4 B5 BS °1 Dl °2 D2 °3 D3 °4 D4 0 0 0 0 El E2 E3 E4 E5 ES Fl F2 F3 F4 F5 FS (This notation will cause no confusion with the coefficients in the expansions of the potentials). Let us now investigate the determinant of the coefficients of the unknown in equs. (S.08)-(S.13). Since al and ai are small we obtain, using the relations of Appendix III: ~ .. - j.(a,) • o (o,l) } A:; :: Gz.-1t/ A~:: - 1).2 Jl ra;l As'" -2i.('&) } A:: 2{Jt·) 13'1 j, (Q2') 1>" ~ "'£, (a,) I . I ') =: - J ']fi~-,g),/{~)-.1.f,t) , 13'~J/(~V-rfl{~0 c)=d,i,la,) } Y1 ~-2xcl,iJQI) C.2,:-oI}JJfl,') I (3'" "ll,(~2) I J 7)2-;;; -oI/j/a,~ J EI~-3G1,(QIL E2~ 0 7)?i =: )( cIz.tJ.l. el{ =-dljJaoz') -C:(Q:I.) } ~ ... -4, DJj F(Ill) E3=:-Glll.J;l(£I~)J £.,,,,p,lj),(~j) J £. =: 1-£1 4 ,,/ (.e)) £ =: _ f .g1.,J /)(-8) 5 ::2. I • r; ~ Jl -77- Since I~~}<.t;1 J=:) ~J we can, in view of the results of Ap- and pendix IV, neglect all terms involving ~ or ~i in the determinant. = 02 = D1 = D2 = O. This means that we can set 01 Un- der these conditions the determinant becomes: 1I '" (C{J)1j ~ C.,J>3) I('»~As-'B.rA, )£) + (A,G',,Flo A.r E,) ('bJ ~ -13.1 F,) - - (As ~ - A,Fs- )"Ba E)+ (""35 f6 - ".3,F5' ) (AI Ii - AJ 0) - ('"B~~ -135"~) f\f J -r - + (f5'~ - E/"s- ) (A/i3z - A~,)] (5.01) Now: ElF)..." 3G&II;{IJ./)).,(a,) , 'B/~-~Z~:: [S(.8 2-1 2)-A/ ]f,(iIIi)iJa l) 1,. £/ "" 3~ j"Q/)),fa,) }t/G - I ~£l "" 3€jJa/jJ.,fIl,)} 2 A;F; ..., [l: (gl_IZ)+ 2BI'1.] jl (A,I).RJR)) AtBl - A2~) ':. 3jJo.U"cJ4,) 1,Ar-~S'A, => 2 U!BI).U)B)-Jlf~[lI)j.rS)] AEs- Ark{g = Ib- .8 1 2._ 1 E(,-.P) J,/(,)/ (ljl) -+ ~ E.gjl {-ClUle)-.-CIJ, (,f)J~(.B~] f Ar~- A, F;- -;: 8 I3('-I) jJtl)J, ("')-1- f(.e)J~( J)- Gj,(.$~ijl>r.e)] -78- "1>5£" -13'£5" = i [,_.gJ2._ E(£_.!Z)] P-S,),g/&) + (l-~)iJ fiJI)jJ&~ + I (' -+ ~ _-8,1)_ J ]~.r&).gJo(I~ + [~-i(' _-&11)] jl ($)'&.£,).1) 13(,rS- -'BrF=G ;: -4 [(G-J)3jJ~J)iJt) + (f-I)-CJ~(.BI)J-KD(.G)+ (I-3l;J.KjorC'J1,(J)+ +(3-6 )J-AD (I) jl (.$~] £.r~ - ~~ = 2, {3(B1.-~J")4/t)j.flJ) +J.&I[-GIjJr~)}(.$Q-JiD(8~JI!I)]J Substituting these expressions into (5.0lA) and collecting the different combinations of Bessel functions, we obtain: A'" /Jai)l,{a ,)(C~~ -['17)3) [ADo .t.&J~($I).iD{.t) + [:.01'£ jA.B11JeJ-r +~JO if; (JI)lor-l ) -+ L\)l jJg~jJ.&) ] (5.02) where: 11f)() = (J- E) (6.8 2+2-&12.) /:!.DI = ~B2 [q('-l)- i-C1G:- .((1.] i A1G ,.. flzL?J(G-I) oj. -BIL J-+ ~12 (.,&12_ ')+ 'E./lk- (5.03) All ~ 3e.B l [Q(J-E;)4- i (S,cl-_ .//1.)] The coefficient Bl is given by ~ where As} is the determinant A obtained by replacing the first column of 6 by the terms on the right sides of (6.08)-(6.13). Hence, now: -79- and all other elements are the same, so that (5.01) becomes: 'Where we have made use of the fact that Al = Bl' Aa = B2' Again collecting the different combinations of Bessel functions, we obtain: (5.04) where: L\~I"'I-G } Ll:'= !(,_A1.)-3 ) A.~)~ 3E- ~.r£-,g(2-) ,1~) '" ~ [6- -&,2 - G(fg- .g~)J (5.05) Dividing (5.04A) by (5.0aA), dividing numerator and denominator )1.. 0;: 01'rh.. S',(A,y "'-' 1 . ; by b ,and making use of the fact that 0"0: Gll and /(J(,I .: JIIlI AI, A,) we obtain (6.15) and (6.22). To solve for Bi we must evaluate A~! which is the deter- minant obtained by replacing the second column of A with the right hand terms of (6.08)-(6.13). Comparing ~~ } wi th A rie 1 see that to our approximation they differ only in so far as al ) -"2.; in f:. is replaced by ell in tlS,' , and that the b) in the Fa ele- ment of /). is replaced by £.s2. in the F2 element of del • J we obtain: £/ S. = :',"'I1. ·' (fA,) J l (tI,) / -:B,s.--:B. . F, "" -}~&jJQ,)j.rQ,) '132.];;J :0 AJ~ - A F; :. 3£ (-8 2-.lJJpa,).E/o,,) S 3~ I, (a,).R,{a,) A2 E, ~ 3Gj.fu.,)j,Jo.,) A/13J. - Aa"B) "" 3S~ (Q,)},Ja,) Hence, -80Otherwise everything is the same as in the caloulation of A and we obtain: !:lBI ::: {I (Q,)~,{/l,) (C3')4 - CY))3) [~~ -C£JD(~~.£O{,g) + !J!J1J~{.g~lJB)-j­ +a!J jj,(8~jo{$)+b~/i' (-&~iJ (I)] (5.06) where: A!::. 3,~Z.{I_,)) A:/= ~-&~[r{s-I)- tEgz] }f1!: = ,i2. [9(E-J)+~£~z] ,1~ =- £~ [27{I-S)+ ! (E8~- Jil-) 7 (5.07) ',l.. Dividing (5.06A) by (5.02A), factoring out a b l in the numera- tor and denominator, and making use of the faot that 3-8,2", G/1and that { Po.1~,J' a ~ ().ftQ,) we obtain (6.17). l' ) Having found BI and Bt, we can det.ermine 01 and 01 from (6.10) and 6.11) since Dl and D~ do not appear in those equations. We oan write (6.10) and (6.11) in the form: C,tXl.,J.Jo,l) - C/r:l.l f,{aJ) ~ (5.08) =. CJ ~Xil1..i,)[t.I.. ) - C/clli11f~(41J -= 6i} (5.09) where: 1'1'= -1J:ol! j.(Q:)- oilI'Jo.,) -'1J)oi,i.(o.,) (5.10) &. I "" "B/ el! l.{ltD - Xcl l J; (o.,) 4-1."'8IXol,)1 [al) (5.11) Making use of the faot that the arguments are small, we obtain, using (6.15) and (6.17): ~ Q- ~a,d, r~ r;. )-/- {J-8)G"',... ] ,xl 8, -81- Q~ ~a,tX, [~G,&,'-X -t l.X(I- 8)G~,J Hence, making use of (3.04A) we obtain: 11 = - c(zc<,.la; ,.. -t1;.~) PD.;).f,/o.,,) -t cil.~);( o.;j. flal) i/ (llz) -::: [1 (X-I)j('\i)iJo;.)-l- a;~o [ai)J,(a,j- X~7.~l"1) jJQ~)] (5.12) I =-~oIyJla, b'l~)Ptx- ~~:'Ga:-t.l{l-x)()-8)~~J J-tAi j.(~D[~';~: -J-O-S)G.,] (5.13) llc! == ~J~j,ra:t) -~x~nJi(a~):: ~ - i "",a, {'(.r"J[i"2x)~':....' - 3xJ~ .,.i,I'')l( [1+ (I-SJG..- :-: ~7) (5.14) Combining (5.12A) (5.13A) and (5.14A), we obtain (6.19) and (6.21). I Since we are not interested in Dl and Dl , this com- pletes the case for n = 1. When n = 0, we must solve the following set of equations (where we have already Simplified the terms, due to the smallness I of al and al): -~Do.tJtl,)+7>!Q/J;ra,0- co Q1.1,(fJ;L)-l- C; IJlj;{an == 'll}{t{J) (5.15) 'Sot(1),,(o.,) - 'B;cl,'jo(al) -t CDcl21D(~) - C/cl.1. jo (al) = -cl'5~ (R,) (5.16) -loxcl.,a,£JQ,)4-1>~~: o./pla.') - C.Xc(il~~J~)-+ C)c:Jj~Pad) "'X~tl,~[o.l) ~oG [.g2:$o (Q,)-4a,.,Il,fal >T- --:B~ ~'f~(().:J + COb I.c zpz.J/a1.)-21\1."£o'J ((l;z,)]- (5.17) -Col [.c/ 2fo1 J~(lll)-2Qtp(aJ.)] '" - ECJ~{o.,) (5.18) -82- Labeling the elements of the determinant as before, we obtain I~ )« J} I~}« ) ( since ): 1 ~ (A"D,. -A;t1>,) (1?3(1j - "'Bl!C3 ):; . . Ia,~.ra,) .glfo (I(:) - a/j, (o.:)€ (~~D (Q,) -4a,j, (D.,) )]('Yl., -1\(3)~ "" .,g,'" (1S/1j - 'B.t/3) [a,~,((;l.) jolAi) - $a:~t){AI)fl (£1:)] = ,J12"(' B3 CIj -"&.,c a) a\~, (A,) jo (Il/) (5.19) 12. since the second term is of order al Making the same approximations in ~ ~ compared to the first. and ~~I we obtain: 0 l\~o ~ - Jil. IQ,s', (14.)jD (t~,,) - 64:S~( Q,)j, (IA:)] (1'3 c" - ~l~) (5.20) 1I'3~ ~ (5.21) 0. J eg'-~, (fAl)J~ (D.,) ('all! -13./:; ) so ~hat: (5.22) (5.23) which are equs. (6.14) and (6~16) respectively. From (5.16A) and (5.17A) we obtain: Cocl.)~o(a.J - c~ eli J~ (aj) = 1; jl -CoX.el1.{;t2-RICa.l.) + Col eld. ~~ {Ai) ... {(o (5.24) (5.25) -83- where: /j'" == - al, (. (a,) - ~o!X,lD (~.) -l- c<. }'Do)f~ (0.:) Qo = Xci,Ll.;".(a.) + --p,flXd,a.,g,fa,)-"'B~C{/D.!J;(a/) Making use of the fact that the arguments are small, and using (5.Z2A) and (5.23A) we obtain: ~ ~ ell (b~:1 - }) ~o ~ Jd,t>.,2.g (x- £1) J so that, from (5.24A) and (5.25A): fA= d2c1,.' [aJ f; (A~)),o[f).1.)- X (al)l, ra,J] !leo ~ d2. c(J al j)a1) J ().ljo ('b :;' - J) !leI ~ titcl,J( "' • .Ria].) (8 ~ - I) o (5.26) (5.2?) (5.28) &it Dividing (5.27A) and (5.28A) by (5.26A) we obtain (6.18) and (6.20) respectively. This completes the calculations for the viscous fluid sphere. -84- APPENDIX VI To compare the results of VI with those of V we must consider the case when t:: = 6 =: 0) }\\ .... co . B = 0, we have from (6.14): --0 .3 -Do -W - .3 La, When I . which is identical with (5.10). Also, when ~-=- 8 ... 0 we get from (6.22): G.'S,"",- I 00 I G0,-a. =-3 I so that, from (6.15): ., ,g~j~($')iD(g)-3.gJ~(8')i,{g)+H'--Bl) [-.,gjJg')lof~)+3jJg')~{J)] = G 2.£.g)J~(.$'),go(-C) + (,&,1_ b)J.j.{II)Jo(.8) '5, 6,£_0 "'< JSp(l? {K.RoW-31.(i)] 4- i (,_-£.2.)} (i') [-ii.m-t3JJt)] _ 21 io{$) [.8jO(,&I)- 3j, (CI)] + .c)~Jf (C~,po(J) - llat-t) f-JI(O (iY t 3l.I~I) -iil'-,; (~I)] --,~2~~~~~o(B-)[r}--4-}-U~')~-i-~~'J-i(~~I)~]~----: _ 1 (t) [i t4 rg,)- ±.g'Ji (~')] - ,2go[8) F.lJ:.rKI)+ $'S; [1N] 2 i 1,.[/) = - 2.ED (,G) so that: which is identical with (5.10) specialized for the case of n = 1. 8:::. 0 , We see immediately from (6.16) and (6.17) that i f -85- then a~ = 0 and also BJ. = O. since ~) .. 3 S~'&I mains finite. It is also evident from (6.18), that i f then 0 0 3=-0J)(=d~ = 0 because of the presence of)( in the denominator. For 01 we obtain: ... '!'Q ~ .,tr~)+~{.g) D I".. (2-i,{0z) _ f).2. ~D(Q~») )..o{C) :: =0 lJ-t) a,d, cl2 (2~,(4>.) - a.. ~,,{a;.») .$J,o{fs) where we have used (3.05A). This is identical with (5.11) specialized for the case of n = 1. Also, from (6.20) and (6.21): But from (5.06) we have: el, -'" DI~ '" d,. :::: oi, dJi c". -.L - c/2. O<fg'CJ X cl. eli so that both O~ and C{ will go to zero as 8-"0 and X~ 0 '. This agrees with V where the acoustic fLeld inside the obstacle vanished. -86- APPENDIX VII From (7.05), the coefficient of Bl can be put into the following form: (7.01) If we now again set up the six equations, then, as was pOinted out in VII, the first five equations are identical with the ones of VI. The sixth equation (continuity of Prr) differs , , only in the coefficients of Bl and 01' But to our approXima- I tion the coefficient of 01 ( which is the F4 element in the determinants) does not appear when the determinants are evaluated, so that this difference is immaterial". , We see from (7. OlA) that as far as the present coefficient of Bl is concerned, it differs from the one in VI by the additional term-- -4a;5~{a,~, But since this term is of order al~ , it Will, as can be seen from (5.01A), give us additional terms of order ar~' while the other terms will be of order ~ and we can thus neglect it. It is true that the first term in (7.0lA) is of the same order as the second (since b 1 is of the same order as a 1 ) , but we shall retain it temporarily because then, in order to evaluate I Bl' Ell that we have to do is to take over the results and Bl" from Appendix V and let 6 and b''1. become small. -87- Hence, neglecting terms involving ~ and b l 2- we obtain from (6.22): G1J I,. J D'--''3 &11.',.._3 G?>I- J G(} /0 J c;.Pk q II c; "" 0-1-.2, (;OJ .. -Q6, G-Jo "'-}(61 2 ) I GJ, =27b 100 so that: -8JS~{{N.Ro(J) - 3t J;{.&I)~{i) - 3&,(; (Nit/i) + ~c'J [~'J1tJ-&) G ;: a (o~2.)-&.£o(t) UJ~{~J)-3jl(B~]- qS '~J~) [.gJJ~(.&I)- 3J'I(~')] J .flliS?[lJo {4) - 3i,UJ] - 35(SV [11,/8)- 3il (S)] = 1 (J·Jr2J8~(}($) [4If~{.&~- 3f~ (B~J - q£1, (~) [.BJ·D{~~- 3S~ ($)] ~ -.&jiWJiiz(.&)+3j;r.&VJ.l.rSJ. = ___1~'.l2.~(.&.!....)_ _ -O+l),&1,o(&)Jf~(.$') q81r,f&j.$i;(41 q&-«Im - (8tl)~D(8) oj. 1..(t) so that: whiCh is (7.07). Since: G..J = 3S~'3 J , we also obtain (7.09~. "I We cannot use the same method for Bo ' B~, Co' and C~ since G and b l 1. do not appear in the equations of VI for these quantities. If n = 0, then in view of what was said before, we must solve the following four equations: - -"Bool,.e,la,) -+ 'ld aUJQ:) - CpfAz.J,[Q~) + cj a; j){ a~) '" a1j,(A)) ~O",,~orQ.l) - raJd/jora.') + C()ei.],.l,D(o.1.) - C}i:/f~ (Q}} ~ -c{)fO (a (7.03) l ) (7.04) -88- -13.Xcll().l~da,) -r'BJo</ai} (a! ) - Col(~a2. -RJa1.) + C/~I 0.1 pal) = (7.05) == )<.,(IQ.jtfa l) 1'oE (~l-Eo(Ql)-2jal4:.(al») -1>; (~IJ~(lln -ita: fl [tli~ ~ Co(; (,glf)o{A,J-2a:jolJ(~V + -r CoI (1~QA,1.,j»(al)+lj1,12So. I) lal I )1) ') = -~'6th.Ja' ( a,) (7.06) Labelling the elements in the determinant as before, then Ici;l. Ii! I,(.c I sinoe I ,we obtain: 0<,1 .("'} oIj I:. ~ (A ;]).1 - A,.1>, ) (13:/" -13" C,3) := - (,&.c" -1>, c,) [ ~.1.1.,) (-g, 'j. IQ") - ~.:vall) - Q,'j. (.,.)€ (J'J!.r.,Ha,J,I.,))] = I sinoe a l , al' b' and ~ are small. tions for 113 and AIS' II • Making the same approxima- there results: 0 ~BII = (~jCq - 'B'tC~) [- QIi' (a,) (.f,'Z.5~ ('11)- 2Jll: I' (~i)) -I- a/ {I (~O ~gl.J~ (a,)]=: 5") t.I.'t =- - (-al., -~C3) i-a~.gI2. (J- ~ ~t - S OJ i) .g' '" L\&J :: C~3C1j -"B,/3) [ (l,li.rD.I)G.glJ~{QI) - alJ:ra,l E- (.g~/al)- 2ja l Jall )] == i ~ - (133[lj-'B~C3) .i SilIl, so that: ;;]Do -- s - - bo -=== --..! L\ La 3 I' - 3 I (I - -89- But: and henoe: (7.07) "]1 ~ 3$ A~3;u 3t\-l-~ o (7.08) From (7.04A) and (7.05A) we obtain: Cj)o('J, ~o( a~) - C: e<'l f~ (I).i) :: 1! - Co~C<JA~.c(\a)" Cj~ilAl jl[elD:: 6(0 where: '1'0 = - O<ljl (~.) - 'lotX){l/ (tal) -+ ~~ rX,ISD (an &.0 ~ /(1)('. 14 5' (a,) +'Dt>xc<,o.,iJa,) - '8~o(,'ai l.(an 1 , USing (7.07A) and (7.08A), we easily obtain (sinoe aI' al are small) : so that: -90- Consequently: (7.09) (7.10) which are (7.10) and (7.11) of part I. -91APPENDIX VIII From (9.07), we have: I~:: Re ~ )({~~+tfl)~ (fiJil-rf,*)dS = Re. Lf(~. ~l + ~l~t -t o/~ ~* -l- ,j,1 ~ )d$ I db. eMgIL 'f '(M.. 5 j Since kl is real, the first term will give no contribution. For laxge r, we get: J~(X) ~ ~ CtJ5 [x- (n"'I)± ] ...glt(x)~ te.ifX-(~+I)!] Then, since the xadius of our sphere is laxge, we obtain from (6.01) : m..l Re. t f{Llt-l'A(2h"J)~)14-+))1;,1'rn [- :;... e (-A,il.-S,,)Si,... (~)A.-JJtI)ii' Is i n,l!! But: so that: " -92- =. ~ ~ (211+ I) ['],,,€. i(~.1l. - S,,) (_ i s"'- (.£,IL _$,,) -+ COS (~\I!.-S,,») + +1': e- i (~,l/.. -2») ( i s~ l'&,lt- Sh) + cos (J.k-S~)) + 2'B11--S:] !!I =:, ~ (11H I) (1)11 -t~:-l- 21>"~: ) so that: Keeping only the term for n = 0 and 1, we get: But Bo is imaginary and BoBg, BIB! are negligible compared to Bl' Hence: Is ~ Also: I~ = R~ ii~/ ~/. 4],f2e. (?J~.) (8.01) )1.".- D ,z7i4.l.si Yl edlloiB '" RT 0 [f ~ ~e L:. i .-. (lH,)(lM4I)1l; C:' j.li!')j! ljNr.1'.·']j.~s..ad,,-,IB~ b,ik DO. f{ ::: ljJjfe. L)2Yl+ h l)'"b~ C~ *[jt1 (.fl!Jl.) j: (~; h.) i/.'- oIlZ.. 0 -93- Since: we can write: Iv, ~ ~t (~~.ryc P- 0 ,)5; Ci' f:r...(~.) ~'l. (-Ii"") hdiL (2)>> (8.02) /) For IV2 we obtain: rJ) Iv2. =AlI~tL (,Z11+J) r[j"('~/l)-+-:Bn~"{~JA)] C:Jn"lj«(~Jl.)A?·dJL '= ~ " In order to evaluate these integrals, consider the following: let u and v be two solutions of Bessel's equation such that: Multiply the first equation by v and the second by u, and integrate from, say a to b: e ~ (~.>-"';) [!t1l1TA ~ - f [11 ~ it ~)- u1. {A. i) JoI~ A A subtra~ -94- Integrating the right side by parts we obtain: This does not make use of the form of u and v. They can be any solutions of Bessel's equations, in particular u and v can be linear compinations of I n and Yn • Applying (8.04A) to (8.02A), we get: ~ J~tj~:,t) ~.*J~tlz)hJh. == j"-' -P,)*l. (o.~*J /o./)in+",Jo.~*)- o.Ft"-+i. (a;,*)J") (QD) = h+;" ~ 0;1:>. I - Z. where we have used (3.03A). h -+1 Substituting this into (8.02A) we get: It will be sufficient for our purposes to maintain only the n = 0 term. Then: -95The first term is negligible compared to the second so that: Tv) ~ J.,1iR' Rq, But since f{ai)~} 'vc,:I< ~*~ S',{D.1*)jo(&I;) and B~ is real, we have: I ~ ~ .LniR 3 RR 1J~(1 J', (ad) a' (8.05) ). Using (8.04A) in (8.03A), we obtain: 1 , [J,~ (.!) - i ~;/a:)] -~, [J.i~f)-i~,l':)][{"':1l')J.;i (',). ill.Y,:/.,)J = 3 )' (21f<+l)C/'{ ~ Nt r' 0] ::'-6jIIR 12t4' o.;J'-GJ1- A2.,-'11" (aa) J»(o.,)~"bI1;nl1rQI) J,- -~l-K~*(Q,.) [jn'ffl,)4-'Jh ~~ au] } Using (3.03A) and neglecting af o@mpared to a;2 in the denominator, there results: T - -Va: Lnr'i<3 T<t~ \ ' (2~-tdC!{ ~* [, 0] a t ' l l ,,(0.1,) Jb,JA,)-+'1)l!.j\»~Jal) - -Q!(~,I•.) [i' (a.) -+ 13.-£,I•• )J] Again, retaining only the n = 0 term, we get: Iy~ ~ .liiiR3(lQ. ~!*,- [a, 1/(0...)IfIA,) ~'B()iJQ,)l-~,...r.(a,) [ria.)~1>,.1. (.,J] ] -962 The first, second and fourth term are of order a l oompared to the third and can hence be discarded. What remains, yields: Substituting (8.0lA), (8.05A) and (8.06A) into (9.06) we obtain: which is equ. (9.08). -97APPENDIX IX In this case Is and IV2 are identical with (8.01A) and (8.06A). In view of the definition of the primed potentials, we see from (7.01) and (8.05A) that now: 47ft< s 0 ~()ICj 'I I) he. - IJ az - .1,/ :: YJ aJ.. fAJ" (9.01) and furthermore! Ru J..~ = ~ L i~-l'A{2}H1){2l'A"")Dtj~~ [IJ~ (.~:A)jM [j/\)1,'1l>]o1.1rA.:2. S~f)rJll- Je "'" 3 t..:J ~'" V 0 I R ~ ~~~ (AI1+1) I~l r- ~~{~/Il)]~~A II Neglecting terms of order al-2.. compared to one, this becomes: {l. J- V3 ~ lliTw~ Jldj2. jA~2.dlJ.. :: LI7i~?J"3JI'l.. .... o 3w (9.02) 0 Substituting (8.01A), (8.0BA), (9.01A) and (9.02A) into (9.10), there results: which is (9.12). We were able to replace rs~{j/lJ..il2.. in (9.02A) by unity since the argument of the Bessel function was small over the entire range of integration. -98- APPENDIX X In view of the discussion in X we can in the second term of the denominator of the first real part consider b and b l to be small. Then: so that: Now, let: Then: . -K,f&) '" - 1+ . 7i '3 3~ 1--;--+ 0-1>. (&) .g -i)~ '=: - 'J + -3 ::::. -3 (J+ 1.) - i (j-I.3- ) -= . '3 ( ~ + - )-\ 2; .2t 2~.ll and hence: . J+~ 3 - ( 1+ .2d £ 3 J- NT J.) «t ---,~;!.... j J_ 3i8 ~t . 2~ -99- The real part differs appreciably from unity only when y « 1, so that we can write: (10.01) Similarly, in the seoond term of the denominator of the second I real part we can consider a2 and a2 to be small. So that: Hence: Let: liJ (/+ i) Q~ =~(/-I- I);. « aJ. ,., l/{H- t') :: If{ / 2(JR' ~ (J+ i) Then: . -100and hence: Li. ~e.i ),.ft~ilJ _ _ ~ J... (1+2.) ~t J- ~ 4')O(a,J 1_ ~;( D''" -.... No~when ~) is small, c 2i: 1+ 3iX 2 • .2lr is even smaller, so that since the real part differs appreciably from unity only when:' is small, we obtain: =.!. (..:!... + (7) _ I_ R p.",~ /1:;j H (3 XiXl)," (10.02) Rlwi Substituting (lO.OlA) and (lO.02A) into (10.04), we get! which is (10.05). -101- APpENDIX XI We have: so that: {(I {aD = L a;jD(al) Let: Then: COS ::ecoshiil: - i sinr5il'lh r I L coSr(J.+i) I ({I (aj = ---':::;.---Sinr cosh%.- -+ iC05.i!sil1hz. 2%,1- ~(I ... i) sil'\~ (14 L) ).;e,.... 2:1(,Ii 5~ (ol) and taking the real part we obtain: (cOHcosh-i SillHil'lhr)(sil'1>l: cosh x- -l COSi-sinhi-) = (11.01) Substituting this into (10.09) we obtain! -102which is (10.10). If .z. <<. I ,then we can write: ~i~h2~+ siVl2;;; --;;0- cosh1.~ - COS';" 2. =- 1+ ~z.~ J. 1-+ ~ %'1 4s" so that: (11. 02) I f :») ,then: so that: (11.03) We see from Graph II that (11.02) is actually vaid up to ~~i and (11.03) begins to hold beyond ;;",3 • -103REFERENCES 1. Lord Rayleigh: Theory of Sound.- 2. C.J.T. Sewell: The extinction of Sound in a Viscous Atmosphere by Small Obstacles of Oylindrical and Spherical Form; Phil. Trans. Roy. Soc. Lond. (1910), (A)210, p. 239. 3. P.S. Epstein: On the Absorption of Sound Waves in Suspen- sions and Emulsions; Theodore von Karman Anniversary volame (1941) • 4. G.K. Hartmann & A.B. Fecke: Absorption of Supersonic Waves in Water and Aqueous Suspensions; Phys. Rev., (1940), 57, p. 221. 5. V.O. Knudson: To be published shortly in Jour. Acoust. Soc. 6. G. Kirchhoff: Vorlesungen uber Mathematische PhYSik, vol. IV. 7. S. Chapman & T.G. Cowling: The Mathematical Theory of Non- II Uniform Gases; pp. 265-269. 8. A.E.H. Love: A Treatise on the Mathematical Theorv of Elasticity; ~t.he~.__:Q~ _1Q.~.• 9. W.R. Smythe: Static & Dynamic ElectriCity; 1st ed. p.394,456. 10. Hann-Suring: I.ehrbuch del' MeteoFologie; 5th ed., vo1.1,p.385. 11. D.P. Loye & W.F. Arndt: A Sheet of Ail' Bubbles as ~Acoustic Screen for Underwater Noise; Jour. Acoust. Soc. (1948) 20, p. 143. -104- TABLE I Ail' w' seof:'l 10 3 105 Water 107 103 105 107 C em sec- l 3.30104 1.45105 cm- 3 1.2910- 3 1.00 g cm- l sec- 1 1.8210- 4 1.1010- 2 )} cm2 sec- 1 .141 .011 5.8010- 5 1. 4310- g .204 1.4310- 3 1.61103 1.61105 ? ~ cal cm- 1 sec- l deg- l ~ cm2 sec- 1 CT A cm2 sec- l B cm2 sec- 2 a cm4 sec- 4 D cm4 sec- 4 D/B2 3.9210 2 1.09109 3.9210 4 1.0910 9 3.9210 6 16.1 2.101010 2.101010 2.101010 2.101010 1.5310 5 1.53109 1. 531013 83.8 8.3810 5 8.3810 9 6.36-10;1.1 6.3610 13 6.361015 1.201011 1.201013 1.201015 5.3710- 7 5.3710- 5 5.3710- 3 2. 721cr lU 2.7210- 8 2.7210- 6 -105- TABLE II Air Water w see- 1 2.4110 8 lL2 em 4.1510 11 4.1510 7 4.1510 3 1.7110- 2 1.7110- 3 1.7110- 4 1.6910- 3 1.6910- 4 1.6910- 5 em 1.6810- 2 1.6810- 3 1.6810- 4 4.6910- 3 4.6910- 4 4.6910- 5 lL 1 IT .em· 2.41104 2.41 1.0 Graph I . V , / Plot of oaCj4lfR vs. R2w .9 for Water Drops in Air II / A - Sewell B- Epstein C - [quo 10.06 .~ 7 II II .7 .6 , -=--v , It! v, ~ E ~ .5 ul~ Vc bl::l / V V t,...- V vI,.. v V / V V V J / ) .4 I !/ , . . II ,< • • • --, J .3 A • .2 ) V L- -/ / if J V .1 I---' ~ .00001 v vi-' II V .0001 V 1I 1I II I-L--I- 1-1- v V v vv ~i-' / V1 o (1) V I ........ V /B V / V .01 .001 R w (ctn1sec- l) 2 .1 L_~ _ 1.0 o -fI -"f~ r ' ";'~I,I'-~, -'. LII I ' I ' -'I,' '" 2.