4. L. Debnath and S. Rosenblat, "The ultimate approach to the steady state in the generation of waves on a running stream," Quart. J. Mech. Appl. Math., 22, No. 2 (1969). 5. L. V. Cherkesov, Surface and Internal Waves [in Russian], Naukova Dumka, Kiev (1973). 6. A. K. Pramanik, "Waves due to a moving oscillatory surface pressure in a stratified fluid," Trans. ASME, Ser. E. J. Appl. Mech., 41, No. 3 (1974). 7. A. K. Pramanik, "Generation of internal gravity waves in a stratified ocean," Bull. Cal. Math. Soc., 75, No. 1 (1983). 8. F. Tricomi, Differential Equations [Russian translation], IL, Moscow (1962). 9. V. V. Goncharov, "Some features of internal waves in the ocean," in: Tsunami and Internal Waves [in Russian], MGI, Sevastopol' (1976). i0. N.A. Zavol'skii, "Features of the propagation of linear internal waves in a continuously stratified liquid," Izv. Akad. Nauk SSSR, MZhG, No. 1 (1987). ii. R. G. Rehm and H. S. Radt, "Internal waves generated by a translation oscillating body," J. Fluid Mech., 68, No. 2 (1975). PROPAGATION OF A BOUNDARY DISTURBANCE IN A STRATIFIED GAS FOR ARBITRARY KNUDSEN NUMBER D. A. Vereshchagin, S. B. Leble, and A. K. Shchekin UDC 533.72:551.511 Introduction. A systematic treatment of wave disturbances in rarefied gases should be based on the Boltzmann kinetic equation or its standard approximations [i, 2]. The purpose of the present paper is to use the kinetic equation to study the forced vibrations of a ver- tically stratified gas in a gravitational field for given types of excitation on the bound- aries. Defining the Knudsen number Kn of the problem as the ratio of the mean free path of the gas molecules to the scale of the inhomogeneities due to the propagating waves, we find that Kn increases with height because of the height dependence of the mean free path in the stratified gas. Hence it is required to determine the motion of the gas for arbitrary Kn. In many respects this problem is similar to the well-known problem of propagation of ultrasound in a uniform gas. Interest in the latter problem from the point of view of the kinetic theory of gases was stimulated by the work of Van Chan and Uhlenbeck [2]. Important results in this field have been obtained for the linearized Boltzman equation and for ap- proximate kinetic equations using analytic continuation of dispersion relations [3], the Wiener-Hopf method [4], reduction to a Riemann-Hilbert problem [5, 6], and numerical inte- gration along the characteristics [7]. These results suggest that the wave nature of dis- turbances persists in a gas with Kn ~ ;. In this case the phase velocity and absorption co- efficient of acoustic waves calculated with the help of the BGK kinetic equation are found to be in good agreement with experiment. The BGK equation can also be used to analyze the propagation of wave disturbances in a stratified gas. Physically, the stratification of the gas leads to internal waves, together with the usual acoustic waves. The dispersion relation for internal waves is quite different from the dispersion relation for acoustic waves and the study of kinetic effects on the propagation of internal waves is of interest in the physics of the upper atmosphere [8]. However, the presence of an external field and the stratifica- tion of the gas complicates the problem, since theresult is an equation with variable coef- ficients. Hence the usual methods of finding the solution for sound in a uniform gas are no longer applicable, since they rely on separation of Variables with the help of the Fourier transform. The method of integration along the characteristics has to be modified to take into account nonlinear characteristics. To describe the propagation of boundary disturbances in a stratified gas for arbitrary Knudsen number we reduce the integrodifferential BGK equation to a closed system of integral equations for the first five moments of the distribution function. A general integral kinetic equation including the boundary conditions on the surface of a body in a flowing gas was ob- tained earlier in [9]. This equation was solved in [i0] by transforming to a system of Kaliningrad. Translated from'Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 70-79, September-October, 1993. Original article submitted April 24, i992; revision sub- mitted November ii, 1992. 660 0021-8944/93/3405-0660~12.50 9 1994 Plenum Publishing Corporation