6. 7. 8. 9. t. D. Landau and E. M. Lifshits, Mechanics [in Russian], Nauka, Moscow (1973). N. N. Bogolyubov and Yu. A. Mitropol'skii~ Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka~ Moscow (1974). V. V. Migu!in, V. I. Medvedev, and E. R. Mustel', Fundamentals in Oscillation Theory [in Russian], Nauka, Moscow (1978). E, Yanke, F. Emde, and F. Lesh, Special Functions [in Russian], Nauka, Moscow (1978). S. S. Abdullaev and G. M. Zaslavskii, Zh. Eksp. Teor. ~iz., 80, No. 2, 524 (1981)~ REFLECTION OF ONE-DIMENSIONAL WAVE FIELD FROM A HALF SPACE WITH CONDUCTIVITY FLUCTUATIONS N. N. Zernov UDC 538.566 The distribution function of the modulus of the coefficient of reflection of a plane wave from a one-dimensional half space with conductivity fluctuations is constructed. This function is used for analyzing the effect of conductivity fluctuations of ionosphere on the modulus of the reflection coefficient of a wage with frequency in tens of kiloherhz range. A large number of investigations of wave propagation in a one-dimensional random-inhomo- geneous medium [I-8] have been devoted to the situation, when the real part of the dielectric constant e of the medium undergoes fluctuations. Thus, at sufficiently low frequencies the relative complex dielectric constant of the ionospheric plasma disregarding the effect of the earth's magnetic field can be expressed in the form e" m = 1--}-i ( e= N ~z) /eorno3v (,z) ), (1) where e and m are, respectively, the charge and mass of electron, e 0 is the dielectric constant of vacuum, w is the angular frequency of the field, add N(z) and ~(z) are, respectively, the height distribution of the electron density and the number of collisions of electrons with other particles in the ionosphere. According to (i) the simplest model of ionospheric plasma in the low-frequency range is a half space with dielectric constant given by the formula ~ = I+i~+i~(z), (2) where ~ is a constant whose order of magnitude will be determined below, and g(z) is a random correction with zero mean characterizing the conductivity fluctuations of the ionos- phere over height. It is known [2, 4, 8] that in the investigation of reflection in the approximation of diffusion Markoy process the determination of the distribution function of the coefficient of reflection from an absorbing halfspace with fluctuations is equivalent to constructing a stationary solution of Einstein-Fokker equation (E~E) for an absorbing layer of finite thickness. Therefore we shall construct EFE for a layer of finite thickness with dielectric constant (2) and then find its stationary solution. We shall consider a plane wave of horizontal polarization propagating at an angle % to the normal to the layer with conductivity fluctuations along the height. Then function w(z), related to field E(x, z) through the formula E(x, z) = w(z)exp(ikx sing), is determined from the equation d~w +~2[cos 2 ~+i~+i~(z) ] w =0, dz---S- (3) Leningrad State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 29, No. 12, pp. 1425-1430, December, 1986. Original article submitted May 6, 1985; revision submitted October 22, 1985. 0033-8443/86/2912-1065512.50 9 1987 Plenum Publishing Corporation 1065