APL/JHU CM-467 UMH-34' Date: 1 May 1948 ON THE DISPERSION OF SOUND IN HELIUM by C. S. Wang Chang I. Introduction The propagation of plane sound waves through a gas where the mean free path of the molecules is not completely negligible compared with the wave length of sound was first studied by Primakoff(l) and later again by Tsien and Schamberg(2) The starting point of their calculation is the Burnett equations, i.e., the hydrodynamical equations obtained from the velocity distribution function of a non-uniform gas correct up to the second approximation according to the method of development devised by Hilbert and Enskog. In our last report(3)we pointed out how their results were in error because of an error in sign of one of the Chapman's coefficients. After correcting this it has been found that to the second order of approximation and for both the elastic spheres and the Maxwellian molecules-the change of the velocity is about six or seven times larger than the results obtained from Chapman' s original expressions. Furthermore the effect is about twice as large as would be obtained by treating the, Stokes-Navier equations as exact. In fact the effect is of such an order of magnitude that experimental verification seems to be possible. Hence it is thought to be worthwhile to c arry out the calculation for one real gas using the best known knowledge of the force (1) H. Primakoff, J. Acous. Soc. Am. 1_, 14, 1942. t21 H. S. Tsien and R. Schamberg, J. Acous. Scc. Am. $, 334, 1946. "On the Transport Phenomena in Rarified Gases" by C. S. Wang Chang and G. E. Uhlenbecko APL/JHU CM-143, Feb. 20, 1948,

-2 -law. This task has been made much easier because of the work of Dr. Hirschfelder* and his co-workers who calculated all the collision integrals ll(n) (4) necessary for the determination of the coefficients in the pressure tensor and the heat flux vector. The dispersion formula derived in this way by using the revised BurnettChapman pressure tensor and heat flux vector holds true only for a monoatomic gas where there are no internal degrees of freedom. For polyatomic gases, the additional absorption and dispersion of sound due to each internal degrees of freedom are characterized by the relaxation times, ', the time required for the ith internal degree of freedom to attain equilibrium with the translational motiqn.Only when the t / are known does one know whether this correction to the velocity is negligible compared to the contributions due to the viscosity p. and the heat conduction coefficient %. From the formula for the elastic spheres and the Maxwell molecules it is seen that the latter effect will be appreciable only when (f/p)2 105 where f is the frequency in megacycles per second and p is the pressure in atmospheres; under atmospheric pressure this corresponds to a value of l/f of the order of 10'9 seconds. On the other hand experiments seem to indicate that the rotational relaxation time for most gases are of the order 10 8 -10-9 seconds. Thus in most cases the relaxation effect will 1nsk the effects due to JL and j. We there. fore choose to study the affect in a monoatomic gas and especially helium because its interatomic force has been most extensively studied. (4) See Chapman and Cowling, "The Mathematical Theory of Non-uniform Gases". * We like to take this opportunity to express our gratitude to Dr. Hirschfelder for sending us their 'reports and permitting us to make free use of them.

We will limit ourselves to classical calculations. To avoid making the necessary quantum corrections we will calculate the effect for the temperature range T a 200 to 5000K. This temperature range has the further advantage that the non-ideal behavior of helium can be neglected. In the next section we will give the general expression for the coefficients W and 0, in the second order pressure tensor and heat flux vector, and show how they can be written formally as proportional to /X a, uX. j or *9. In section 3 we will give the derivation for the velocity of sound up to the second approximation. The last section will be devoted to the application to helium. II, Lhe Second Order Pressure Tensor and Heat Flux Vector* the Coefficients..-... -'W and. The expression for the pressure tensor has been given by both Burnett and Chapman, and in his book, Chapman also derived the heat flux vector. In our report(3)we have presented them in tensor notation. There the values of the Q's and a) 's are given for elastic spheres and for Maxwell molecules.* The aim of this section is mainly to give general expression for Q and wo and to show how they are connected with the collision cross-sections. In the problem of sound propagation one usually limits oneself to small amplitudes so that only linear terms need be considered. To save writing we will restrict ourselves therefore to the linear terms. The heat flux vector and *Some of the numbers in Table II on p. 13 of the previous report are in error, The third column shbuld read: 03 Multiplied by 1.030 Multiplied by 1.039 @+ Multiplied by 0.806 instead of Multiplied by 0.814 ~c*) P'.obId- -'* 'i. " " o.i50)

the pressure tensor can then be written as: pv 3 xix. -s ) Thus among the e's and the 0's only 92, 4, GO 2, and C33 will be of interest to us. These are given by: 4 \5w~Yo \ t _:kT t ~)? (2) T )Ato c A and B are functions of c, the dimensionless molecular velocity, and the solutions of the integral equations: nI ( s a lina )c or di b y: ~I -C _ )3 i = CLC-3 j) (3) I is a linear and isotropic operator defined by: I= 51 4 Idn TI (, A) - ( V

where g is the relative velocity, I(g, 9) is the collision cross-section, and 4', ~ are the values of 4 and c4l after collision. The integral equations (3) can not be solved in closed form. One solves them by expanding A and B in sets of orthogonal functions, the most convenient sets being the Sonine polynomials. One writes:() oo - (r) x(,..;.,... b (- an.,Substituting (4) into (3), multiplying by 5$ (c~)c~and S 'c=)(C 3;j, respectively, contracting and integratinr over d;, one obtains for both ai and bi an infinite set of -linear equations of the form: where. Irm ~ brm' 5s 'Yh ( (5) See Chapsn and Cowling, p. 123 for the definitions of the Sonine polynomials SI (X); we will follow as much as possible the notation of Chapman and Cowling.

-6 -Both are eight ftld integrals and for centrally symmetric molecules, six of these integrals can be carried out leaving only the collision integrals of the type: m -a 1... _ 7- 5... Chapman has calculated arm and bra for r and s up to r s 3. ): The linerar equations (5) are solved bysuccessive approximations, To the first approximation one stops at the first term in the sum obtaining: a,- bIn general to the mth approximation one stops at r s m, and (-1) _ A1() where A is the determinant lars, and is, the minor of the element ali. The a and b are functions of the temperature and they depend on the force law. One can show that to all order of approximations the following relations always hold true: a- a kT (6) (6) See p. 161, Chapman and Cowling.

-7. and to simplify later writing we have written: -5 f %/ where / has the same dimension as A. In fact, it is exactly JA, for Maxwell molecules. For other spherically symmetric molecular models S is a dimensionless temperature dependent quantity. It is different for different torce laws but it is.never very much different from unity. To the first approximation s is 1 for all these. models. Furthermore, one can show that for molecules interacting with the inverse power law r' all the a's and b'e have the same temperature dependence, so that their ratios are independent of the temperature. It is therefore convenient to write: 02.= Elg - Tats, - () = p(er) s)3 (7) where the coefficients F, 8z, and 3. are dimensionless constants independent of T for the rn law molecules. This way of writing is only formal. For other molecular models the t's are still temperature dependent. III, The VelociLty of Sound. In t-his section we will derive the expression for the velocity of sound from the Burnett equations. This will, of course, be mainly a repetition of known work, but to get uniformity in notation it seems not too superfluous to do it briefly again. Making use of (7) and (1), and considering only one dimensional motion one obtains for the total pressure tensor and the total

heat flux vector up to the second order the following expressions: I: c aS 2t _ t 3 _T < P (8) F- I-~ -~, -t - ~,- x The linear one dimensional Burnett ~equations are: ap - a ' 0? dt ax.t.. + — 1t _3 - 3 (9) 3 aT k IT ' T;k EZ Lka3 T' I- 3 --- aT ~at: 4' T T< 3* '7p?~ 3 'J 3 Eliminating u from these equations and expression T in terms of p and 5 by the equation of state and finally introducing L ( o":- t.t) one obtains two homogeneous equations for - o and ro. Equating the determinant to sero leads then to the dispersion equation: 2. - ' 2at. I _ 4(10 () ~- pf' +'-~' p- (.-3 )L... eo

-9 -where Vo is the velocity at zero frequency. Equation (10) relates the angu' lar frequency cO to the Cr, which is in general complex oC-= oC C+ The real part (t is related to the velocity of sound while the imaginary part Oz gives the absorption coefficient. Equation (10) can be solved for Cd by successive approximations by treating "p2" and "/" as small quantities. This is obviously consistent With the Hilbert-Enskog development. 1) Zeroth order: Euler V P Vd 0 2) First order: Stokes-Navier V - V0 3) Second order: Burnett V32 t5 p V (3 q ih Vo* (n) x- F'3 V3 A-O V: Formally one can go on and on, the equation (10) can even be solved exactly. But for a consistent theory one should stop at the second approximation because we have not taken the terms of the third order in, and 9i into account in the pressure tensor and the heat flux vector. Thus the influence

-10 -of the Burnett terms on the absorption coefficienrt will not be seen till one (3) (3) ineludes in the equations p and q(3. In the present paper we will limit ourselves to the second approximation and therefore will calculate only the velocity of sound. The second order effect due to the Stokes-Navier terms alone can be obtained from (11) by putting the I's equal to zero. IV. APication to Helium. For the interaction between helium atoms we take the Lennard-Jones potentialh ( )=4~[ g4E(l 6 where S is the minimum for V(r) and ro is the value of the interatomic idistance for which V = O. Dr. Hirschfelder and his co-workers have made numerical evaluations of the collision integrals Qi(in) that are necessary for the calculation of ars and bre up to r - e - 3 for different values of kT/. Using de Boer and Michels' values for ~ and ro we have computed up to the third approximation the a's and b's for kT/. 20, 30, 140, 50, corresponding to temperatures 203.4, 3051, 1406.S, and 508.50K It is found that for these temperatures and within the accuracy of the available values of QQ(~) the ratios ai/al and bi/bl are independent of T. The result is shown in Table I. For comparison we give in Table I also the values for Maxwell molecules and the elastic spheres. (7) J. de Boer and Michels, Physic 2, 945, 1938; i, 409, 1939.

Table I a2/a1 a3/a1 b2/bl b3 /bl Maxwell 0 O O O Lennard-Jones 0.0672 0.0127 0.0435 0.0061 Elastic spheres 0.094 0.O0194 0.0613 0.0094 This shows that within this temperature range the helium gas behaves as simple repulsive centers of the power law 5 -n 4 oo. In Table II are the values of the F 's and 31/ for the different models. To exhibit the convergence we give in Table III the values of $ in the successive approximations. Table II Maxwell de Boer and Michels Elastic spheres F__. -45/8 -(45/8)xl.008 -t(45/8)xl.017 e_ 3 3x0.855 3x0o800* F,3 2 2x1.007 2x1.014 1.000 1.004 1.009 Table III First Approx. Second Approx. Third Approx. F, -45/8 -(45/8)xl.,007 -(45/8)xl.008 3 3x0.862 3xO.855 2 2x1.006 2x1.007 Putting these values for ' into equation (11) we find for the velocity of sound,~~ ~~~~~~~~.....;e,,,-.r,:.......... ~- I.... ~The difference between this value and the value given on p. 12 of (3) is because of the ratio ~"/,a.

-12 -or la V a 3-= 4.4c4 (12) where A is in c.g.s. units, f in megacycles per secondil and p is in atmospheres. Treating Stokes-Navier as exact would give: tVc - 7 2 ht ($p ) (13) In Table IV are the computed AV/V0 from equations (12) and (i3). The values of jl in the second column are the theoretical values. In this temperature range they agree to within 1% of the-values calculated from the empirical formula /t = o. 023 T~ 7 x c.s (8) given by Keesom (8) W. H. Keesom: Helium p. 107. A detailed discussion on the comparison of the theoretical and experimental results, will be published in a paper by Dr. Hirschfelder in the Journal of Chemical Physics.

Table IV T a-th'eory V/Vo in % V/Vo in % OK c.g.s. Stokes-Navier Burnett 203.4. 1.58x10- o0 67x10-4('/p)2 1. 1O104-L,(./p)2 305.1 2.05 " " 1.86 406*8 2.47 " 1.66 " 2.71 508.5 2.85 " 2.2 3.61 Thus we see that the effect is really not too small. For avalue of f/p = 335 corresponding to, say, p - 10-2 atm. and f = 3.35 me and at? = 305.1~K in contrast to the 3% obtained by Tsien and Schamberg. Of course, for aV/Vo so large the next term in the developnmst will not be negligible. To get the next term in the velocity one would have to calculate p() and q(4).

DISTRIBUTION: Chief, Bureau of@dnance Chief, Bureau of Aeronautica (35 copies) Attn: Capt. B. Brown, Re9(11) Attn: M. K. Ray Comdr. L. Pooler, Re9g Lt. Col. H. C. Beaman (AMC) R. E. Gibson Lt. Col. Jo A. White (AGF) G. R. Tatum Office of Chief of Ordnance, H. H. Porter Fort Bliss, Texas A. G. Ennis Attn: Major James P. Hamill E. A. Bonney Bendix Aviation Corporation C. W. Besserer (Dr. Harner Selvidge) D. T. Sigley (A. Hoeckley) W. A. Good (R. M. Russell) G. B. Harris (H. G. Tarter) G. C. Munro Consolidated Vultee Aircraft E. R. Sanders Corporation A. Kossiakoff (A. W. Abels) (3 copies) H. S. Morton (J. E. Arnold) (1 copy) W. H. Goss Cornell Aeronautical Laboratory J, E. Cook (R. Schedvin) (5 copies) R. K. Dahlstrom Curtiss-Wright Corporation T. Davis (B. G. Eaton) (2 copies) E. S. Franklin Experiment Incorporated W; G. Berl (J. W. Mullen, II) T. W. Sheppard Farnsworth Television and Radio W. J. Mayo-Wells Company W. B. Snow (J. D. Schantz) W. F. Hilton Johns Hopkins University F. K. Hill (F. H. Clauser) A. R. Eaton Kellex Corporation N. Edmonson (G. R. Graham) H W. Bishop Massachusetts Institute of Technology C. E. Swartz (H. G. Stever) (5 copies) R.R Po Petersen Michigan, University of J. A. Van Allen (A. M. Kuethe) R. C. Herman (W. C. Nelson) Bertha Jacot New Mexico School of Mines T. Fo Ball (E. J. Workman) H. A. Tellman New Mexico A. and M. C. N. Warfield (George Gardiner) AC Office North American Aviation, Inc. (E. F. Smellie) (D. H. Mason) H. M. Nicholson (BAD) Princeton University J. B. Green (NOL) (Harry Ashworth, Sr.) Naval Research Laboratory Radio Corporation of America Attn: Code 1100 (N. I. Korman) Lt. Comdr. T. Stanwick (NODU/ Submarine Signal Company BuOrd) (Ho M. Hart) Commanding Officer, NOTS, Inyokern (3 copies)

DISTRIBUTION (Continued) Texas, University of (Dr. C. P. Boner) (2 copies) Virginia, University of (Dr. J. W. Beams) Washington,' University of (Dr. J. E. Henderson) Wisconsin, University of (J. 0. Iirshfelder) BB Reportq;(A.1B.C.) (3 copies) BB Reports (F.GS,.) (5 copies) BB Reserve (APL) APL Contract Representative (L. I. Barbier)