02764-14-T Copy /6 THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory 2764-14-T- RL-2081 THE DETERMINATION OF THE ELECTRON DENSITY PERTURBATIONS RESULTING FROM THE MIXING OF TWO DIFFERENT PLASMAS. by F. V. Schultz May 1962 The work described in this report was partially supported by the ADVANCED RESEARCH PROJECTS AGENCY, ARPA Order Nr. 120-61, Project Code Nr. 7400. ARPA Order Nr. 120-61, Project Code Nr. 7400 Contract DA 36-039 SC-75041 Department of the Army Project Nr. 3A99-23-001-01 Contract With: U. S. Army Signal Research and Development Laboratories Fort Monmouth, New Jersey Administered through: OFFICE OF RESEARCH ADMINISTRATION * ANN ARBOR

THE UNIVERSITY OF MICHIGAN 2764-14-T THE DETERMINATION OF THE ELECTRON DENSITY PERTURBATIONS RESULTING FROM THE MIXING OF TWO DIFFERENT PLASMAS by F. V. Schultz 2764-14-T June 1962 ARPA Order Nr. 120-61, Project Code Nr. 7400 Contract Nr. DA 36-039 SC-75041 Department of the Arniy Project Nr. 3A99-23-001-01 The work described in this report was partially supported by the ADVANCED RESEARCH PROJECTS AGENCY ARPA Order Nr. 120-61, Project Code Nr. 7400 Prepared for THE ADVANCED RESEARCH PROJECTS AGENCY and the U. S. Army Signal Research and Development Laboratories Ft. Monmouth, N. J.

THE UNIVERSITY OF MICHIGAN 2764-14-T ASTIA AVAILABILITY NOTICE Qualified requestors may obtain copies of this report from ASTIA m

THE UNIVERSITY OF MICHIGAN 2764-14-T ABSTRACT The time-dependent perturbation of electron density arising from the mixing of two collisionless plasmas, characterized initially by unequal densities and equal electron temperatures, in a field-free region is investigated. It is assumed that changes in electron gas pressure, taken to be a scalar, occur adiabatically and that, for the time interval considered, the ions are immobile. Viscosity and heat conduction are neglected. It is found that the following phenomena occur in the more-dense medium within a distance of about 1. 5 Debye lengths from the interface: (1) The electron density at the interface immediately assumes a value about midway between the two unperturbed electron densities; (2) a rarefaction wave of increasing amplitude propagates into the more-dense medium with a velocity equal to the adiabatic acoustical velocity; (3) after the passage of this wave, rapidly damped electron density oscillations at the two plasma frequencies occur; and (4) after the oscillations die out, the electron density varies smoothly from a value of about one-half the difference in the unperturbed densities at the interface to the unperturbed value far from the interface. Similar phenomena are expected to occur for the less-dense medium. iii

I STATEMENT OF THE PROBLEM The problem under consideration is that of determining the mixing of two plasmas, originally of different densities. It is assumed that, for t(time) < 0, the half-space z < 0 (Region 1) is filled with a plasma having electron and ion densities each equal to n10 particles per cubic meter, and the half-space z ~> 0 (Region 2) is filled with a plasma having electron and ion densities each equal to n20 particles per cubic meter. Here n10 and n20 are constants, independent of x, y, z, and t. For t 3 0 the two plasmas are allowed to mix and the problem is to determine how the perturbations in electron densities, nll in Region 1 and n21 in Region 2, vary with t and z. The time interval considered is assumed to be short enough that the ion densities in the two regions remain essentially unchanged. Other assumptions are these: (a) The mean free path is assumed to be very large so (b) (c) that collisions can be neglected. The initial electron temperatures (T ) are equal in the o two regions. The perturbations in electron densities, nll and n21, the z-directed electron streaming velocities, ul and u2, and the induced electric field intensities, E1 and E2, are small enough that second-order, and higher, products of these 1

terms can be neglected. This is also assumed to be true of the derivatives of these quantities. (d) There are no externally impressed electric, magnetic, or gravitational fields. (e) The plasmas obey the perfect gas law. The electron gas pressure is, rather arbitrarily but necessarily, assumed to be a scalar given by p = nkT. (f) The various physical quantities vary only with z and t, there being no variations in the x and y directions. (g) Pressure changes occur adiabatically, so that, for the electrons, VP=k ap kT a where 'Y is the ratio of specific heats and k is a unit vector in the z-direction. Our working equations for the electrons, then, are these, written in two dimensions, z and t. MKS units are used. m n a + YkT a + n e E = 0 (momentum equation) (1) a + a (n u) = 0 (continuity equation) (2) at az - = e (n - n) (Poisson equation) (3) z k T (equation of state) (4) p = n k T (equation of state) (4) 2

There are five unknowns, p, u, n, T and E, in these four equations, plus the equation under (g), above. Here: m = mass of electron n = number density of electrons n = number density of ions O p = pressure of electron gas -19 e =1.602 x 10 coulomb E = electric field intensity (z-directed) u = z-directed streaming velocity of electrons -12 E= 8. 854 x 10, permittivity of free space -23 k =1. 380 x 10 joule per degree K T = temperature in degrees, Kelvin. Now assume that, in Region 1, for t > 0, the electron density can be written n n10 + nll(z, t), where nl1 4 n10 and n10 is a constant. Similarly for Region 2: n2 20+ n21(z, t). 3

Also, in the same way, P1 =P10 + P11 P2 -P20 + P21' The unchanging ion densities are n10 and n20 in the two regions. Making use of the assumptions mentioned above, one obtains the following forms of the four basic equations for Region 1, with a similar set for Region 2: au an a11 m n10 at + y kTa +n1leE =0, (la) an au n11 1_ at + n10 = 0, (2a) aE 1 e z - n11' (3a) 0 o 11 =n1 k T1 + nll kT (4a) To obtain a single partial differential equation in nl1 alone, one may differentiate (la) with respect to z, differentiate (2a) with respect to t, and use this differentiated form of (2a), and (3a), in (la). There results: 2 2 an 1 ykT n T.1 - + n11w =0, (5) 2 m Oz2 pl 11 at az 4

where 2 2 e n10 2 1= (6) P1 n E o is the electron plasma angular frequency squared in Region 1. In order to solve (5) for nll(z, t) one may take the Laplace transform, obtaining ano 2 kT a2 s N1l(z, s) - s nll(Z+0) ~ + (zs) = m + plN1(Zs)=0 2 p1 11 t=+0 z (7) Here Nl (z, s) is the Laplace transform of n (z, t). We are assuming that both anll n11 (z +0) and are zero so (7) reduces to t =+0 2 a N 2 11 2. 2. c 12 (s+w ) N11 =0 (8) az2 Pi 11 where c = YkTo/m (9) is the adiabatic acoustic velocity in the electron gas. The solution of (8) is z 22 z 2 2 — s + s A N (z, s) =F(s)e c + F(s) ec (10) where Fl(s) and F2(s) are undetermined functions of s, only. By a completely similar process the following equation is obtained for Region 2:

+ z 2+2 N (z, s) =F3(s) e p + F4(s) e p. (11) In order to take the inverse Laplace transforms of (10) and (11) for obtaining n11(z, t) and n21(z, t), one must find the undetermined functions F1(s),... F4(s). This is done by applying boundary conditions. 6

II BOUNDARY CONDITIONS Boundary conditions on n 1(z, t) and n21(z, t) must be satisfied at z = 0 and at z = + oD. These last conditions are: lim n21(z, t) = 0, (12) Z -+OD lim n1 (z,t) = 0. (13) Z —~ - 00 The boundary conditions at the interface (z = 0) are obtained by using the conservation equations for mass, momentum, and energy, together with the following assumptions: (a) Viscous effects disappear because of the uniform unidirectional drift motion of the particles. (b) Second, and higher, order terms are neglected. (c) Heat conduction is negligible, because of the very low density of the gas. The resultant boundary conditions at z =0, then, can be shown to be these: T, =T (14) u1 =u2 (15) nI n2 (16) 7

From (16) one obtains (for z = 0) n21 = nl + (nl0-n20) (17) which shows that our results will be valid only for values of (n10-n20) of the same order of magnitude as nll and n21. 11J 2\ 8

III CALCULATION OF ELECTRON DENSITY PERTURBATIONS We are now in a position to apply the boundary conditions listed above and thus to determine F (s),... F4(s) of (10) and (11). It is necessary to use also the conservation of momentum equation and (12) and (13), in the process. When this is done the following relations are obtained: F1(s)=0 (18) (n -n ) s+w 2 F2(s) =, (19) 220 2 2 2 n10 H p p2 J F(s) = F2(s) + (n1- n20) (20) 3 2 10 20s F4(s) =0. (21) We may now determine n11(z,t) and n21(z,t) by using the results of the last paragraph and then taking the inverse Laplace transforms of (10) and (11). We obtain 'y+ i3 st nll(zt) = lim - N1(z, s)e ds, (22) and a similar equation for n21(z,t). 9

By using the expression for F (s) given in (19), we obtain z 2+ 2 Y +i 2 2 C pl st n -n s+ 2 e e 20-nl 10 p2 n (z' t) = - lim — ds. ' 2iri ( —w n20 2+2 2 2 s -+- s + s5+ j P-i3 p 10 p2 (23) An examination of the integrand of this integral shows that there is a simple pole at the origin, branch points at + i w and + i w 2, and only these. Branch pi p2 ' cuts can be chosen as shown in Figure 1. It turns out that the Riemann surface of the integrand has four sheets. On the branch cuts between i w and i w ), and p1 between - iu 1 and -i u -2' the first and second sheets are connected and the third pl p and fourth sheets are connected. Between i o and - ic 2 the first and third sheets p2 p2 are connected and the second and fourth sheets are connected. One may, as is customary, alter the path of integration of (23), as is indicated in Figure 1, in order to facilitate the integration process. In choosing the alternate path we must make sure that, in traversing this path, we remain on the same Riermann surface as is used for Path 1 from Y - i P to y+ i 3, and either that no poles are enclosed between the two paths, or that the values of the residues at the poles are properly taken into account. In order to make a proper choice of alternate path, let us rewrite (23) as follows: 10

/ / / cL_ _ - -- -- B- - — C _ - \ iw +iP 13 I \ 112 I', p2 I | 11 i,% 1 8 Path 1 S{t MP2 1 I - i1~ LI ~ 'p1 A I / / X Branch Points 0 Simple Pole. - - - - -Branch Cut - -- - Paths of Integration FIGURE 1: S-PLANE FOR INTEGRAND OF EQUATION (23) 11

7) + n20-nl0 l n (z, t) =2 — lim - 11 2 i if 7- ip3 s(t+ 1+ ) i13 2 2 s ) s2- e ds Ns p2 s -- s +1 +Is + -l %- -j (24) remembering that z < 0. If we assume that (t +- ) - 0, c t e IzI/c for the real part of s very large, we see that which means that (25) lim nll(z, t) =. (26) Re s — Ioo Consequently, since no singular points of the integrand are located on Paths 1 or 3, or within the region bounded by these two paths, we see that lim ( - /)lim =0. (27) Path 1 Path 3 Path Therefore, n(z, t) =0 for t < zl/c. For the case when (t + - ) > O, or c (28) t lzl/c, (29) we have devised Path 2 of Figure 1, since in this case the integrand becomes infinite on Path 3, as Re s-ro. Path 2 has been so chosen that one remains on one sheet by crossing no branch cuts. Also, no poles or other singularities are 12

enclosed between Path 1 and Path 2. Therefore, the value of the integral as determined along Path 2 is equal to the value determined by following Path 1. In order to carry out the integration it was found necessary to make some simplifying approximatioes in the integrand, which limit the validity of the results to values of z less than about 1. 5 Debye lengths. Then for Iz |.1.5 Debye lengths, t >Izl/c the following approximate result was obtained: w (cosh w ) eW 2+~ L nS i(o t) nl (z t) =(n 20-n 0) [ P + Z Si( n20 7r P n10 pi p2 1 z p2 z - sin - -- cos 2 2 2 c p2 27r w t c p2 27r pl Pi 16 nf2Tl W p2 1 + 22 (cos t + coCs t) 2 n ) P.1 p2 7r Wp2 (n20-nlO)(1pl02) p + 2 - (sin w t - sin p2t) + 2 pp p2 2-pl t (Eqn. continued on next page) 13

4n Aw B C 1- + -+ - ) cos t 2 f5 t P r 2 (n20-n )( + t A B C D E + (- + 2+ _ ) cos p2t+( -+ )sinp w t t5 t3 tp2 4 2 p D E 2 +( + - ) sinwt (30) t4 t2 p Here A1, B1,..o E2 are functions of wl and p2. 1 1 p p It is obvious that the first four terms in the expression for n l(z, t), given by (30) go to zero as (n20-n10) —0, as they must. The remaining terms in this expression also go to zero as (n20-n10 )- 0, since they arise from evaluating integrals between the limits of wp2 and wp 1 and these limits are equal for n20= -n10 The expression for nll(z,t) given by (30) was obtained by assuming that n10 n n20. In order to complete the analysis it is necessary to either compute the perturbation n21(z, t) in Region 2 for n10 > n20 or to recalculate nll for n10 n20. The latter procedure was followed and the following result obtained (for n10 - n20) 14

n1(z, = n2n10 F (jP) cosh ( &w~ n~20 ~+L nlo p p2 7T C p1 1 22r p1 Sinl (w t) + 1 z 27t c Cos (w 1t + 16 n2 w 3w 7 2(n - )(W + w p2 ~2O lO p1 p2 5(o~ +Cswt (cs t+ cos2) 2 + 3 t (w P2-w~i (sin w 1t -sinw 2t) 4 n2 (.0 20p1 AI -z P -' B1 t3 +-) Cos wpt t p 7TfW2 (n p2 20 n 0)(w1 +w 2 lOp1 p2 A I + ( -p2 t 2 B' + 2 t Cos w t+ ( Dt 1-)sin w 2 + ) sinw t t2 P1J ti (31) Here AI B'Ipa E'I are functions of w and w 1' 1'** 2 Pi p2' 15

It is now in order to investigate the physical significance of the two different expressions for nl(z, t): equation (30) for n10 > n20 and equation (31) for n1o n20. This is done only for the case of n10 > n20. It will be remembered that nll(z, t) = 0 for Izl/c > t. This means, of course, that the electron density perturbation nll has a front which is propagating in the direction of decreasing z with a velocity c, the adiabatic acoustic velocity in the electron gas. This is characteristic of weak discontinuities, as discussed in Section 93 of [1. We are, of course, dealing with a weak discontinuity since the linearization of the original partial differential equation, (1), is valid only for nll/nlO (n10-n20)/n10 0. 1. Another restriction on the solutions is that z must not exceed about 1. 5 Debye lengths. The following are the values of A1,..... E which appear in (30): 2 1 24a /co 2 B1 =2a (5- 2 )+4b W1 pl C1 =0 24a + 6b Di 1 w pi 16

22a+b 2 2 E- ( - W2 1 wp ( p p2 A 24a/w 2 p1 2 2 W2 B - 10a -2 + 6b E +2c) 2 2 6bP1 plp C =O C2 0 D = 1 (24a -p2 +6b) 2 o w pl pi E - 2 a( 2 )3+b( p2 )2 +c( 2 pi pi pi Here a = -90.1 b =169. 5 c =-79.3 a + b + c = 0 (exactly), these being numerics arising in a curve-fitting procedure used as an approximation in evaluating the integral of (23). If we now consider the following numerical values: 12 n =10 particles per cubic meter, 12 n20 = 0. 9 x 10 particles per cubic meter, To =1000 degrees, Kelvin, 17

we find that -3 h = 2.18 x 10 meter, (Debye length) c = 1. 590 x 105 meters per second, (adiabatic acoustic velocity of the electron gas) f = 8. 98 megacycles per second, (plasma frequency of Region 1) f = 8. 52 megacycles per second, (plasma frequency of Region 2) 2 7 wp = 5. 64 x 10 radians per second = 21r fl p1 7 w = 5. 35 x 10 radians per second = 2ir f2, p2 2 p2/p = 0. 9486, (p 1w) _ 0.900. p2 pl If these values are used in (30), there results, after collecting terms: nll (z, t) 7 - = - 0. 512 cosh 355z - 113 z Si(5. 35 x 10 t) n l- n+ 5.03 x 1013 + 5.52 x 10 1 + 2.62x1025 sin(5. 35 x 107t) t2 t3 241 + [- 0.949 x 10- 6 + 0.802 x 10 - - 1.003x10-19 z t t2 t3 + 9. 74 x 10 -33 cos(5. 35 x 107t)+L8.65 x 10-13 - 5. 52 x lo-21 1 t t - 2. 91 xlO25 sin( 5. 64 x 10t)- 0. 802 x 10-14 + 8. 60 x 10-19 t t2 2 t3 -9.74 x 10 -- cos(5.64x 10 7t). (32) t 18

If one now attempts to use this expression to determine numerically the rvalues of n. on the wavefront progressing in the direction of decreasing z, it is found that, because of the very small values of t involved, it is very difficult to ob tait accura.e results. Consequently the integral of (23) was re-evaluated under tihe assumlption that t was small enough to allow us to replace sin rt by rt, and cos rt by unity. The resulting required integration is straightforward (after making other sinmplifying assumptions as before) and the final result is (for nLo n20 and very small t): z 2 n (z,t) w (cosh - w. W t22 p2 Z 7T c 6 2 110 + 16 n20 J p1 p2L -4 n10 o- Co 2 3- n20li + p) co -p 2 7r n2 cJ (n0- n20) (W p + Wp2) p2 6 5 4 22 b c P -1 - - ) (- ) i 6F 6 5 5 )4 4 pl pi pl 4 3 2 2 -a b Co C2 w (1)+-(1 + 1 -2 (33) p2 4 4 3 3 2 2(1 Wpl 1 pl pl 19

If we now use (33) to calculate nll/n 10-n20 ), as a function of z for t = Iz l/c, we get information concerning the amplitude of the wavefront of the electron density perturbation as it moves in the direction of decreasing z. The results are as follows: TABLE I z (mm) i1 /(nl -n20) 0.0 -0.510 -0.1 -0. 510 -0. 2 -0. 560 -0. 3 -0. 643 -0.4 -0.759 -0. 5 -0. 908 -0.6 -1. 091 -0.7 -1. 306 -0.8 -1. 555 -0. 9 -1. 837 -1.0 -2.153 Another result of physical interest is the variation of n l/(n0O-n20) as a function of z for values of t large enough that the oscillatory terms of the electron density perturbation have essentially damped out, but t small enough that the ions have not yet started to move. This result is easily calculated by using only the first two terms of (32) and one obtains the following values: 20

TABLE II z (mm) n11/(n10-n20) 0.0 -0.512 -0. 1 -0. 494 -0.2 -0.477 -0. 3 -0. 461 -0.4 -0.445 -0.5 -0.431 -0.6 -0.418 -0.7 -0.404 -0.8 -0.391 -0.9 -0.379 -1.0 -0. 368 -1.2 -0. 347 -1.4 -0.328 -1.6 -0.313 -1.8 -0. 301 -2.0 -0.293 It also is of interest to obtain an indication of the rapidity with which the electron density oscillations, as shown by (32), damp out. The t term will damp out the most slowly. Our results are valid only out to z 2 millimeters and -8 it takes the perturbation about 10 second to reach this distance. Consequently, -7 -1 at the end of 10 second this t term will damp out to about 10 percent of its original value. Since the frequency of oscillation is about 8.52 megacycles, only a cycle, or so, of this oscillation would occur. The Si(5.35 x 10t) term is also a damped oscillating term, and a plot of it shows that several cycles of this exist. It is considered unnecessary to make a detailed analysis of the behavior of n21 since the equation for it is very similar to that for nll, so very similar phenomena will occur. 21

IV CONCLUSIONS From (32), and Tables I and II, one can conclude that as t increases from t = 0 the following phenomena occur: 1. Immediately the electron density at z = 0 assumes a value approximately midway between n10 and n20. 2. A rarefaction wave of electron density moves in the direction of decreasing z, with a velocity c and with an increasing amplitude. 3. At each point of the medium, after the passage of the rarefaction wave, the electron density oscillates at two different frequencies, one corresponding to w and the other to w. These are damped pi p1 oscillations consisting of only a few cycles. 4. After the oscillations die out the electron density varies smoothly from a value of about 0.5 (n10 -n20n ) at z = 0 to a value of n1 for large negative values of z. 22

V A CKNOWLEDGEMENTS The writer wishes to acknowledge the helpfulness of discussions with several members of the staff of the Radiation Laboratory, particularly Dr. R. J. Leite and Dr. R.K.Ritt, and with Dr. R. A. Holmes of Purdue University. 23

APPENDIX A DETERMINATION OF BOUNDARY CONDITIONS AT INTERFACE The boundary conditions at the interface, z = 0, are determined by first using the applicable form of the momentum equation, at - k \ xk - n a (n V. ) + E ( xk kxk where V. and V are components of the peculiar, or thermal, velocity of a par1 k ticle, and q is its charge, plus the continuity equation, =-, a (A-2) 3t ^ ^ * ka t a xk where p = mn, the mass density of the fluid. To obtain useful results from (A-l) and (A-2), one uses these equations in connection with the equation a auu. p at ( pu) = at + u (A-3) By substituting (A-l) and (A-2) for the right-hand terms of (A-3) we find that -_ (pu.) - Ukk 'xk k at ( i - k a x n ax (nV Vk) qp - (Puk +- E - (A-4' m -ii i m - a( xk k Xk In our case of uniform unidirectional drift motion of the particles, viscous 24

effects disappear, with the result that V.V 6 V (Ai k ik i k where 6ik is the Kronecker delta. Then t = ~- 4- (nv.)= J Aa6a - 1 ap ~1 6(Aa vk x i k ax. 1 m ax. m ik axk By using this result in (A-4) we obtain a (puiuk) = P, i axk au ( P Uk) a - t ( Pu) = -P U - u u. ik &x +qn E + (Aat k L axk xk k Equation (A-7) can be made more compact and its physica significance m more evident by using this result: a a(i puk) axk (p uk) PUk axk i 8xk Then (A-7) becomes - P = -^ a (pu Uk+ 6~kP) + q n Ei (AT i To get (A-8) into its final useful form, one expresses the electric volume force in terms of the Maxwell stress tensor, by pages 95-97 of [2]: qn Ei = ax (Ak k 5) -6).7) ade -8) -9) 25

In our case, the Maxwell stress tensor is 1k T - E^ E? S E(A-10) since we are dealing with a plasma in otherwise empty space, with a constant electric permittivity E O By using (A-9) and (A-10) in (A-8) we obtain a[ — (pu) =P - P U-uk p- + o Ei Ek ik A-11) By pages 12-15 of1 ], we can write aT a,, ak at (PUi) - where Tik is the momentum flux density tensor, given in our case by k 1 Uik k- E Ek + E2 (A-12) Now ik isth Now k n k is the flux of the i component of momentum through a unit surface area having the unit vector n along its outward normal. The flux of each component of momentum must be continuous at an interface between a Region 1 and a Region 2, so (T i) nk - (Ti k)2 k i =1,2,3. (A-13) k k 26

At the interface z = 0, we have k = 3 in (A-13) with the following three resulting scalar equations for i = 1, 2, 3, respectively. pu u -e E E =pu u -E (A-14) P lUlx z o lx 2z 2U2x 2z o 2x 2z4) p lyulz - oElyEl z o 22y 2z (A-15) p1 + 2 - e + E2 = P2 + U2 c- E2 +1 E2 (A-16) Because our regions are unbounded and homogeneous in the x and y directions, the x and y components of u and E are zero in both Region 1 and Region 2. Also, we know that E =E2 Therefore, only (A-16) gives a useful result: Pl + pl 2+P2 z U2 or +2 = (P +p ) + 2 + p2 )u2 (P10 p11) + 10 l = (20 +21)(p20 P21U 2z Dropping terms of second, and higher, order gives PI = P10 +p1 = P2 = P20 +P21 (A-17) A second boundary condition at z = 0 is given by the fact that the normal flux of electrons must be continuous at the surface z = 0. That is, u. n_ = u- n_. (A-18) 27

We obtain a third boundary condition here by recognizing that the normal component of energy flux must be continuous at this plane interface between the two regions. By sections 51 and 53 of [3] we have for the conservation of energy, in our case, 3(1 2 1p1 a-t - Pu2 + pe + - e E2 + 2 H2) = -V pu( 2 u2 +w) -u * I' - qVT+E X - - -V -. (A- 9' Here: e = internal energy per unit mass, w = enthalpy per unit mass = e + p / p, r' = viscosity stress tensor, = thermal conductivity. As mentioned previously, viscous effects do not enter in our case, so the term u * a' in (A-19) is dropped. Also, we are assuming very long mean free paths so collisions can be neglected, allowing us to drop the heat conduction termn t~VT. Because we are dealing with longitudinal oscillations, there is no magnetic field and the Poynting vector term disappears. By Section 85 of [1i we have for the enthalpy per unit mass of a perfect gas. w = p- = P -, (A-20P 7-1 y-1 p p ' 28

whence 1 e = 7-1 p p (A-21) Now if these expressions are used in (A-19), together with the results of the last paragraph above, we obtain 1 2 + 7 P! 1 1 1 2 1 y-l mn1 2 2 2 2 By using the facts that 7-1 P2 +y-1 mn2 y-l in2 (A-22) p1 =P2 and u1 n = U2 n2 given by (A-17) and (A-18), respectively, (A-22) above becomes u2 + 2 1 P 1 1 2 y-1 mn 2 2 7-1 m +y-1 mn2 (A-23) 1 2 and 1 u2 in (A-23), we obtain By dropping the second-order terms, ~ u2 and u2 in (A-23), we obtain 2 1 2 2' n = n2 (for z = O) (A-24) By using (A-24) in (A-17) and (A-18), we get T = T2, (z= ) (A-25) 29

U1 = u2 (z = 0 ). (A-26) From (A-24) we have n10 +nll = n20 +n21 21 nl+ n10-n20)' (z =) (A-27) This equation tells us that our results will be valid only for values of (n10-n20) of the same order of magnitude as nll and n21 Taking the Laplace transform of (A-27) gives n10 -n20 N2 (0,s) = Nll(0, s) + 20 (A-28) A second relation involving N1 (0, s) and N21 (0 s) can be obtained by taking the Laplace transform of (la), obtaining 8N (z, s) inn10 [sU (iZ, )- (Z l + ykT -1 +n10e ( (z,s) = 0, (A-29) rn!0 Ul1z 1 0 Z z 0 1 where Ul(z, s) and l(z, s) are the transforms of u (z, t) and El(z,t), respectively. One of our initial conditions is that u1(z,0) = u2(z,0) = 0 Consequently (A-29) can be rewritten e kT10 aN 11 (z s) U1(z,s) i (Z.S) - (A-30) Ms mn s 1 Oz 30

with a similar equation for U2 (z, s). Because of (A-25) and (A-26), we get from (A-30): 3N21 ( 20 11 (') = -- - (A-31) z ~ n10 z where we have made use of the fact that El(0, t) = E2(0,t). We can now use (12), (13), (A-28), and (A-31) to determine F (s),... F (s) in (10) and (11). From (13), lim N (z,s) = 0. (A-32) Z - ) - 00 Now F1 (s) and F (s) are not functions of z, so (A-32), in conjunction with (10), shows that it is necessary that F (s) = 0. Similarly, it can be shown that F4(s) = 0. Lastly, we can determine F2(s) and F3(s) by using (A-28) and (A-31). This is routine algebra and the results are: (n -n ) n s2+ w2 F (s) F () +1 p20 (A-34) 3 2 s 31

APPENDIX B EVALUATION OF CONTOUR INTEGRAL In order to evaluate the integral in (23) we use Path 2 of Figure 1 and rewrite the integral thus n20 - n10 n1l (z, t) = 2i (0 + ) pp2e 11 12) Path 2 7 ei 1 P12 e2 012)+ 01 2 (_+ zT 11 where (B-re) iOzi s-iW 2=2l ip12 S + %! = P12e i022 PSp2 P22e i0 s = re, and -7/2 < 0mn (3/2). It is to be noted that y > 0, but y can be a small quantity. Figure B-1 illustrates the geometry. The value of the contour integral evaluated on Path 2 is, of course, equal to the sum of the integrals evaluated along the various separate paths which make up Path 2, as shown in Figure 1. We now evaluate our integral along the various separate paths. 32

S i "wp -i -i WI FIGURE 1>1.: POLAR REPRESENTATION, OF s + iw - p Q 9

First, we note that our integrand is analytic along the negative axis of reals, so the integrals along paths B-O and O-C are equal in magnitude and opposite in sign. Also, then, the integrals from 1 to 2, and from A' to D' can be evaluated along continuous paths between these points. Next, let us consider circular arc A-A'. On this path r = r and r -Oo, p oo, 0 — -2r/2, and 0n- -vr/2 as 3- aco. Then 21+022) 2 Pl i 11+12 -7r/2 jp 2 1p2e e 11 1 et id0 lim I A,_ lim 3 A- a P - oo 20 2 (011+012) 021+ 22 0 ~21+ 22 L0n —/2 [n 11'12 P +p21P22e ] 040 (B-2) -7r/2 i t'+it/3 l lim lim e d0 =0 /3- A-A (n20/nl0)+ 1 3 — 00 ---7r /2 0 'o?=po Similarly, lim ID-D =0 (B-3) 3 —.Goo 34

On the circular arc, A' - D', we use the form of the integral given by (24), remembering that (t + c ) > 0. In the second and third quadrants (Re s)< 0, and as 13 -oo, (Re s) — - oo, so lim IA'-D =0 (B-4) 3-* A c D We next consider the integrals over the paths 1-2 and 9-10. Here r= ro, a cnnstant during the integration, and Pll = Upl P21 p2 lim r — v 0 o P22 p2 P12 pi On the path 1-2: 3 011=i 2 3 lim 021 2 lira r -O - 0 1 o 022 2 0 = 1 7r r12 2 For the path 9-10, =- 2 7r 011 -2 021 =lim 1 1 roo 022 -2 r 012= 2 35

Then I1-2 = lim r — O0 o 0 E -h0 3 37r -E =2 +E / -z o tr e1p c pl o i2 e e d p2 (B-5) n20 ]4 I1-2 1-2 -z - Z-w1 p2 10pl p2 10 lim r — 0 o E — 0 3 r -t - E =2 C rot ei0 e -(B-6) z c " pl 7r i w%2e n20 n10 pl p2 For the path 9-10, I9-10 - lim r -s-0 o E- 0 z — to c pl e __2op2 z e rot e ei d0 tP%2 ~p2 (B-7) itp2 19-10 l n20 n10 lim r -O- 0 e -~ 0 / 2 rot e0 / e do, 0= -E+ w t +t 2 pi p2 36

i7 eC P1 I p2 1910 =n~2 120 27rilw cosh -z p2 c pl (B-8) (B-9) 1, I1-2 + 9-10 n20 -pi) + CL2 n10 for t >I z1/c. For the circular path 3-4 we let p -*- 0 and -7+ E -- e3-t-E n 22 2 2 then let E-~ 0. On this path as p 2 - 0 — 0 LL)w +(4 1ll P1 p2 =::2w p21 p2 1 1 2 3 r21 =2 / 12~ 2 w -w 1l2 P1 p2 s =- iw P2 + p22 eiO22 ds =1 p e022 do dsi22e 2 Then lim I34= lim p 22 )-0 E - F-0 z 2 2 7r02)-~j wp- w2 e i5022 i (-iw2~+P22 e )t i22 e ip 22e 2 (-ito +P2 1 io2 n20 2 2 - 22)[ lpl p2+ 2w 2p22e (B-101 37

Similarly, lrn P22 O-0 p22 0 lrn urn~lim2 I3- =0 (B-li) I7-8 =0. (B-12) I11-12 ~:0I (B-13) I15-16:: (B-14) On the path 5-5', we let p12-* 2 n 7 3! - 01 7 w. Here., as p ---0 p11 =2 w p1 2l p2 p1 P22 = pl - WJp2 p1 3 3 021 =2'I 0 3-/ r22 2 'o 12 e 1012 e d1 012' ds =1i p12 38

Then we obtain P12 2 2 E >~ 0 ~-(* 7r +012 2 2 1 e '012 2 (-iw 1 +p1 z w e (IW +p 2e e )t '01 2 p 12e 1.2 e L0 12 n0 L 0 - 2w7 12 2wNo ~2 2 p1 p2 I4 (B-iS) (B-16) lrn I 51= 0. p12-~plo 50 Similarly lrn I51 6=Oj ' (B-17) well. lirn I 1 14= 0 (B-l8) We now consider the following four integrals as a group, since they combine 0 z 2 2 2 2 c Pi -rirt ( p2- re e dr I16=1 Fn02 2 2 2 (B-19) rw ri 2 2r + 2w r= p2 Lnl p1 P2j I 39

p2 2 2 p2 I2 -rnf20 4 r = 0 r - z W2 2 c P' e-irt d e I (B-20) 2- r2 + w2 -r2 0 2 2-r2 eC P1 -irt rw2r-~- ro -r + to n2z 2 2 2 P2 nloto -r p2 2 2 c pl irt p w- r e e dr p2 10-11 top12 2-r2j =0 r n I- r +to-2ombining these four integrals gives 2 16-1 + 2-3 + 8-9 +10-11 (B-21) (B-22) (B-23) I =4 2P r =0 2 2 tow - r p2 (sirnh z w - r2 )(sin rt) dr F "20 2 2 2 2 r - to -r + w - r L "10 P1 p2 J (B-24) 40

We can simplify this integral by considering the first term in the denominator n20 2 2 20 2 20 2 w -r = o -- n10 J pl n10 p2 n10 Following (17), we noted that our results will be valid only for (n10-n20) of the same order of magnitude as nll and n21, and we have restricted our work to the case of nll<< n10 and n21, n20 Therefore we must restrict n20/n10 as follows: and we can say that Then we can write This integral still n2 /nl 0 9 20/10 0.95, n w -- =. 95 2 ^20 2' 2 | 20 1O p2 n10 p2 - z 2 2 / P"(2 r (sinr rt) dr 1 2f 2i -—. (B-25) 2 r r=0 is untractable so we must look for further simplifications. Now 3 5 x x sinh x =x+ - + - +... We can use only the first term with fair accuracy if 3 x 10 41 (B-26)

or if x z 0.774. (B-27) z 2 2 z Sinh - - has its maximum value, sinh u, when r = 0. Therefore, to c pl c pl represent sinh z w1 by only the first term in the corresponding series we must c pi satisfy the relation Z - W 00.774. c pi By using the numerical values for w and c, which are given just before equation p1 (32), we find that our approximation for sinh- W 1 will be valid for c P z ~-3 millimeters, (B-28) which is of the order of a Debye length. We then assume that z12, z | 2 2 z i 2 sinh - w -r -- -r (B-29) c 1p c pl' Our last simplifying assumption is this, 2 2z 2i1 r (1 - (B-30) pi pl In our integral of (B-25) the upper limit on r is p2 = 0. 948 w P For this value of p2 pil r we have: 1- (r/w ) 1 - (W ) = 0.316, 2 (1- ) =0o550. 2 2 pl1 42

Our approximation (B-30) is thus seen to be only fair at this limit, however, it is much better for smaller values of r. By using the above-discussed approximation in (B-25) we get t wp2t I 2i l / ) d(rt) - 2 / (rt)sin(rt)d(rt), 2 C(rt) c Prt= O (B-31) ( 2) (B-32) p2t Here pe 24.4 x t) radns x (B-33) p2I x 0 In order to obtain some idea of the validity of (B-32), we have numerically evaluated y/2i, as given by 3(B-25), for the following values of the various parameters: z =1.0 millimeter, c = 2 x 10 meters per second 7 t = 5 x 10 radians per second, t = 4.74 x 10 radians per second, and for t 2t taking the values 0.5, 1.0, 1.5,..., 20. 0 radians. Then I2/2i was calculated by using (B-32). The results of the two different calculations are shown 43

in Figure B-2. The numerical integration of I2 was carried out by using Simpson's rule, with the number of subdivisions increased at w2 t = 5, 10, and 15. This p2 accounts for the jumps on the curve of y1 at these points. Our last integral to evaluate is I3 I5+ I6 7+ I12 13+I14 15 ' 3 4-5 6-7 12-13 14-15 (B-34) where pl I4-5 = r =( p2 P2 I6-7 r =w pl p. 112-13 - r = p2 z 2 2 r 2 2 - c pl-r -irt ir-p2e e dr p2 rn20 i2 2 2 2 i r - e e-irt dr r - -- p l-r + i r - p n10 z 2 2 2 r-to e e dr i p2 I22 n20 2 2 2 10 P P 44 44 (B-35) (B-36) p (B-37)

0.5 0.4 - \ c 0.3 - ------—.. 0.3 0.2 0.1 4 - - - - - - - - 0 5 10 15 20 up2t (radians) (sinh w r )(sin rt) d(rt) FIGURE B-2: PLOTS OF Yl = / ( rt ) t(rt) rt=O AND z 1 Y2 Cpl si 2 2. t2 pi - sin(w t)+ c p2 2c Pit - cos( pt) c p2' FOR 310, c05, Z10, c=2x10 7 5 10 7 w 5 x 10, = 474 x 10. p1 p2 45

z 2 2 U P2 2 c pl r irt p2 ir e dr r p2 I14-15 =. (B-38) r = r — + 1 r + - w Pi l n10 Pl I - P2 After considerable algebraic manipulation, we find that P1 ( r2 2 )(2 2 z[~2 - 4i p2 )(W - r )(r cosh c - r )(cos rt) dr 13=4i n20 p i c 3 no-n20 p (r W2 )(sinh- W -r )(sin rt) dr 4i L P2 c Pi n20 1 0O 20 2 2 p2 1n "O / p2 In order to handle these integrals we make use of the fact that our range of integration is very small, since ( 2 = 0.948 U. p2 pil First, we consider the denominator of the two integrals. 1/r is nearly constant throughout the range of integration, so we replace it by its value at the midpoint of the integration range: 12 - - = -------- '. (B-40) r (plW p2 / 2 pl +p2 PI P2 Pi p2 46

Next we consider the other factor in the denominator, at the end points of the range of integration nl+ "20 2 2 n20 2 r ( )r- W p2 nlo p2 n10 p2 n0o n20 2 2 n10 2 r p=w )r - w p1 nlO p2 n20 p2 2 Since n20 =0.9 n0, the above factor is very close to w2 throughout the range of 2p2 integration, and will be taken equal to w 2 p2' The first of the integrals in (B-39) can then be written: W 51 C pl - (rw 2)(w -r )(cosh w -r2) (cos rt)dr 3 n1n20 W 1+l 2 (c2 os r t p2 r tp p2 (B-41 In order to handle this integral one must resort to further simplifications. The cosh term will be equal to unity at the upper limit of integration and equal to z 2 2 (cosh l - ~2 ) at the lower limit. At this lower limit, the cosh term will be c pi p2 equal to 1.1 if z 2 - = 0.445 (B-42) c p1 p2 By using the previously listed numerical values of c, Ut, and wt2, one finds that the relation of (B-42) will be satisfied if z = 5. 56 millimeters. Therefore, one is 2 2 justified in replacing cosh - j - r by unity, if z -=-5. 56 millimeters. c p1 47

2 2 2 2 Last, we consider the factor ] (r - W 2 )(w - r ) in the integrand of p2 p1 (B-41). To handle this radical we do the following: 1 2 = 2 p2 - p2 p2 2 2 1-^^ -^pJ2%1 - r w r = plr +rr 2wp w-r so p p1 p1 so 2 2 2 22 r-r,l(r2-p)(o2 - ) plp2 W (rp2 )(p — It p2 pi pirP2 p "2 -' (B-43) Now the function i (r-w 2)(w p-r) is zero at each limit of integration, has a maximum at r = (w 1+w 2)/2, and is symmetrical about this maximum. We replace the radical by a parabola which passes through these three points. The parabolic function is 2 y p2 ) ( lp2 ( plp21 (B-44) To check the validity of this substitution, a normalized numerical check was carried out to compare Y (r-)(. 0548- r) Y! = l (r-l)(l. 0548 - r) with = -36. 5 (r-1. 0274) + 0. 0274 = 48

as r varies from 1.00 to 1. 0274. The results are shown below: r Y1 Y2 1.000 1. 005 1.010 1.015 1.020 1.025 1.0274 0. 0000 0. 0158 0. 0212 0.0244 0.0264 0.0273 0.0274 0. 0000 0. 0091 0.0163 0. 0218 0. 0254 0. 0272 0.0274 These values of yl and Y2 check quite well. One feels fairly well justified, then, in replacing the radical by (B-44). We finally arrive at nl 1 D, - 8 i n20 i = (- % 2)(r-pp2)2+( pl -wp2 ] pl-Wp2 2 2 1 L - ~ P2 ~ cos rt dr. (B-45) The integration is straightforward now, and we obtain 32i n2Wplp2 i 1 3 (nlon0)( pl2) - t2 )(cos it+pi 2 cos w t) + t(sinw 2t-sinw t) I P2pi p2 Pi p2. (B-46) It is to be noted that I1 =O if w =w 3 p2 pli 49

By using the simplifications discussed between (B-39) and (B-41), we can write the second integral of (B-39) thus: ) 1 p1 4i n10 2 1 2 2 2 2 I" n 2 / (r2 )(sinh - r)(sin rt) dr. 3 n10-n20 W pl+ W2 2 t P2 r = p2 (B-47) By using the same reasoning as was used in obtaining (B-29), we find that, in the integrand of (B-47), we can use this approximation, z 2 2 - z 2 2 sinh pli -r = u, -r C pi C) pi if z is not greater than one centimeter. With this simplification the integral still is intractable because of the factor 2 2 2 t - r = 1- (r/) ). 2 The series approximation of (B-30) is not satisfactory here, because, throughout the range of integration in this case, r/tW is pi1 close to unity. Because of this, the radical was approximated by the following polynomial: 1 - (r/W 1 = a(r/w1) + b(r/ )+ c, (B-48) where the constants of the polynomial were determined by matching the two functions near the end points and center point of the range of (r/p 1). Following are the values of a, b, and c for our particular range of values of r/pl: 50

a =-90.13 b =169.46 c =-79.32 The goodness of the fit of the parabola for these values of a, b and c is shown by the following table: r/wp - (r/w )2 a(r/Wp) +b(r/ )+ c Pi p1 I ~ ~ ~ ~ ~ ~ ~....I......... L 0.95 0.96 0.97 0.98 0.99 1.00 0.3122 0. 2800 0.2431 0. 1990 0. 1411 0.0000 0. 3150 0. 2881 0. 2431 0.1801 0. 0991 0. 0000 With these simplifications, the integral of (B-47) becomes:, ~ 8i nlo I3 2 Wp2 10 20)(pl1W p2) z c a(r/w )2 + b(r/Wp) + c (sin rt) dr. Pi1 Pi1 i,., Wp p2 P2 2 2 (r -W ) p2 (B-49) 51

This integration is straightforward, giving 0pi z a [14 4 4 3 (wnl-n2 2 COS Pi 1 t2 p2 p2 l~~ ~p2 tp(Jl'J2 l3sin wo t)- 12(Csw t- w2 cosw t) +-24(w oiw t -w sinco t) lp1 P1 3p p2 P PIp2 p2 P1 P1 +-24(Cos o t-Cos w t)j +b[i((3 Cos w ttw3 cosw t) t5 p2 P1 w 2 p2 P1 P1 wsn w t- w siu t- 6wCos w t-w Cos w t) +6-(sin w t- sin w t) 2 )F1a (2 Cos w t-w Cos w0 w sinw t- wo sin wo t) w2 t2 p2 P1 p 2 p2 p2 P1 P1 2 (Cos o 2t-cos o t) -b I '(t Cos w t- to cosw t)3 2 P1 2 Lt p2 p2 p1 p1 jt(snto2 p1 p2 Lt csp2t osP1Ii (B-50) 52

REFERENCES 1. Landau, L.D., and E. M. Lifshitz, Fluid Mechanics, Addison-Wesley, 1959. 2. Panofsky, W. K. H., and M. Phillips, Classical Electricity and Magnetism, Addison-Wesley, 1955. 3. Landau, L.D. and E. M. Lifshitz, Electrodynamics of Continuous Media, Addison-Wesley, 1960. 53