Propagation of surface waves through a stochastic inhomogeneous elastic medium (the markovian approximation): PMM vol. 43, no. 4, 1979, pp. 746–752

Abstract

The problem of behavior of an unsteady surface wave in an inhomogeneous, linearly deformable elastic medium is considered. Investigation of the surface wave fronts is based on the ray representations of the wave, regarded as a line of discontinuity of the displacement derivatives propagating along the boundary surface. A system of partial differential equations is reduced using the methods of the theory of discontinuous solutions together with the dynamic, kinematic and geometrical conditions of compatibility, to an ordinary differential equation in terms of the wave intensity, with the velocity of this wave coinciding at every point of the inhomogeneous surface with the Rayleigh velocity. This equation is supplemented by a system of relations characterizing the change in the geometrical parameters of the surface front in the course of the propagation. Specific models of the stochastic media are considered for which the processes under investigation are Markovian and can be described with help of the methods of the theory of multidimensional Markovian stochastic processes. Conditions are established concerning the character of the distribution of the surface inhomogeneity, which admit the application of the Markovian approximation. When the wave propagate through randomly inhomogeneous media, the presence of free boundaries leads to appearance of a number of the boundary phenomena sufficiently well defined in some boundary zone [1–4]. The appearance of waves propagating along the free surface is connected with the possibility of existence of inhomogeneous waves near the boundary [5–7]. The harmonic surface waves in randomly inhomogeneous media were studied in [8] with help of the approximations of the geometrical optics (short waves), and in [1] within the framework of the method of effective parameters (long waves). An approach based on the use of the Markovian approximations was developed in [9] for investigating the processes of propagation of volume waves.