COMMENTS s J. E. Drummond, R. A. Gerwin, and B. G. Springer, model is that the charge is not conserved instan- J. Nucl. Energy C2, 98 (1961). taneously, but only on the average over a cycle. • t D. E. Baldwin, J. Plasma Phys. 1, 289 (1967). 10 E. P. Gross, Phys. Rev. 82, 232 (1951). It should be pointed out that with this collision P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. model, the collision frequency v appears only in 94, 11 511 the time-varying part, but not in the time-independ- u T. F. Bell and 0. Buneman, Phys. Rev. 133A, 1300 (1964). ent part of the Boltzmann equation. Consequently, Eq. (1) cannot be derived directly from this ap- proach. However, there is no difficulty . in deriving Comments on Supraluminous Waves the ac conductivity from the time-varying parts and the Power Spectrum of an of Boltzmann and Maxwell equations. Therefore, Isotropic, Homogeneous for the system considered, the time-independent Plasma current density, J 0 , and electric field intensity, E 0 , are not necessarily governed by Eq. (1). The NoRM:AN RosTOKER consistency of analysis must. not be judged solely Department of Applied Physics, Cornell University on the basis of Eq. (1) (Ohm's law). Ithaca, New York If Scharer's objection is that of the usage of the (Received 28 January 1969) particular collision model because it would not lead to the simple Ohm's law, then I can well understand. Calculations of the power spectrum of fluctuations However, the usage of Ohm's law has its limitations 8 in a plasma have previously been based on the time- too. Next, I would like to show that there is no asymptotic solution of a testrparticle problem. difficulty or inconsistency in satisfying the time- Certain poles corresponding to waves with phase independent part of the Maxwell's field equations velocities w/ k ~ c have been omitted in these by taking J 0 • = 0 and E 0 • = const; static field calculations. In a recent paper, these poles have been considered by Lerche who finds that they lead equations are to terms that persist in time. When included, they (2) lead to significantly different results for the power V > <Ho = Jo· spectrum. It is shown that these persistent terms The first equation is obviously satisfied. The second are caused by the impulsive acceleration of the test equation yields particle that is inherent in the mathematical model. and are, therefore, physically unacceptable. aH~ _ aH. _ J We shall. begin our consiqerations with the solution OX oy- O•• of the testrparticle problem as given by Lerchel Since a one-dimensional analysis is being considered, liv+oo i.e., o/ox = o/ ()y = 0, the leftrhand side of Eq. (3) E(x, t) = ( 11')4 dk iv- ' dUJ vanishes so that J 0 • must also be zero. Finally, I agree that if the electrostatic field Eo. ·exp [i(k·x - wt)]E(k, w), was not sufficiently weak, then a shifted Maxwellian E(k, w) = EL(k, w) ET(k, w), should be used for the time-independent electron distribution function . However, this form of dis- 411'qk exp ( - tl!: • Xo) tribution function would only approximately satr DL(k, w)EL(k, w) = kzr k ) \w - •Vo isfy the time-independent part of the Boltzman equation. The dispersion equation then would have DT(k, w)ET(k, w) to be modified accordingly. Furthermore, for ap- __ 47rqw f(k xv 0) xk] exp (- ik·Xo). propriate conditions when electron streaming occurs c 2 k4· (w - k·vo) in plasma, wave-plasma interaction would be expected. Following Lerche we can approximate D L (k, w) and J 1 I. P. Shkarofsky, T. W. Johnston, and M. P. Bach~ki, The Particle Kinetics uf Plasma (Addison-Wesley Publishing Company, Inc., Reading, Mass., 1966), Chap. 1, p. 20. 2 K. V. N. Rao, J. T. Verdeyen, and I. Goldstein, Proc. IRE 49, 1877 (1961). a M. A. Heald and C. B. Wharton, Plasma Diagrwstics tui.th Micruwaves (John Wiley & Sons, Inc., New York, 1965), ChaR. 1.1.p. 12. 4 N. v. Gerson, Radw Wave Absorpti.on in the l<nWsphere (Pergamon Press, Inc., New York, 1962), Chap. 1, p. 2. 6 W. P. Allis, Hand.buck der Physik, S. FlUgge, Ed. (Sprin- ger-Verlag, Berlin, 1956), Vol. 21, p. 383. s P. Molmud, Phys. Rev. 114, 29 (1959). ' H. C. Hsieh, Phys. Fluids 11, 1497 (1968). DT(k, w) as follows: uz DT(k w)::: 1 - TI - ck ck We shall further approximate these functions by assuming n = n. = 0, i.e., we shall neglect the plasma density completely and still obtain supra.- luminous waves in their most transparent form.