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To identify signals passing through solid rock, generated by powerful sources of vibration, in various aspects of practical geodynamics and in solving questions of protection of engineering structures (buildings, pipelines, tunnels, etc.) from seismic waves and pulsed loads, it is very useful to have information on the fundamental properties of the propagation of elastic waves in inhomogeneous and stratified media, composites, and compound structures and systems.One of the simplest models of a compound (stratified) medium is an inffmitely long layer in contact with elastic media. As some limiting cases of this model we can consider the following, a) The media adjoining the upper and lower surfaces of the layer have the same physical characteristics; b) one medium is absent (the corresponding surface of the layer is free from stresses); c) either the layer or one of the media has zero shear strength (an ideal compressible liquid); d) an elastic layer is immersed in a liquid; e) a layer is compressed between absolutely rigid bodies.The problems also differ in the conditions of contact between the layer and the medium-rigid contact (stresses and displacements equal); sliding contact (no shear resistance in the plane of contact); and various types of elastic contact (we have in mind the presence of a linear elastic bond between the layer and the medium).A fairly extensive bibliography has now accumulated on the existence and description of various types of free waves propagated in a system. This work was begun by Bromwich [1] and Love [2], who discussed longwave and short-wave approximations respectively (since the only linear dimension here is the thickness of the layer, the wavelength is referred to it). Among investigations of the dynamic of stratified media, Brekhovskikh [3] gives the dispersion characteristics of the first few modes of a plane homogeneous layer with surfaces free from stresses. Further investigations were devoted to the properties of the dispersion equations and the construction of the phase curves for various wave modes (a fairly complete survey of this work can be found in [4]).in the USSR, various methods of solving dynamic problems of this type have been successfully developed by the Lenin_grad school (G. I. Petrashen', L. A. Molotkov, K. I. Ogurtsov, G. I. Marchuk, E. I. Shemyakin, and others). In a monograph, describes completed investigations of a number of fundamental aspects of the theory of wave propagation in strata, describes the conditions of existence of various groups of waves (mainly surface waves) in relation to the parameters of the strata, and studies wave perturbations in each particular case. He gives references up to 1960.With the advent of high-speed computers, solution of the dispersion equations became a purely technical problem. A great deal of information has now accumulated concerning the properties of the phase curves for various relations between the parameters of the layer and the medium (see, e~., [6][7][8][9][10])o For a layer on a half space (with sliding cont...

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Stresses in a rock bed and dynamic loads applied to rigid inclusions in a percurbar-lon wave caused by an impact or an explosion are estimated from solutions of diffraction problems in a nonstationary statement because the time of rise to ~imum amplitude is a major factor in defining strength and shock resistance criteria.Studies of elastic wave diffraction by variously shaped inclusions are relevant to many practical applications, especially in mining. Quite different engineering processes (from rock bursts and their influence on working stability to formation of cracks and loosening of a bed as a result of stress concentration at boundaries of ore grains and surrounding rocks) can be studied in terms of the same mechanical models and mathematical statements.In the present paper we consider a nonstationary problem of diffraction of a plane elastic compression wave by single inclusions of a circular or rectangular cross-section.We examine the general physical diffraction patterns; the specific type of object is not essential. Among problems with variously shaped obstructions the diffraction of a plane wave by a rigid immobile cylinder has been studied quite thoroughly. A massive inclusion engaged in initial interaction with an incident wave can be viewed as immobile with a certain admissible margin or error (relative to the est~m-te of stress concentration and forces applied to the object). Even in this approximation, an analytic solution of the nonstationary problem cannot be obtained in a closed form. For example, in [i] one has to solve numerically a set of integral Volterra equations of the second kind to compute the elastic displacement potentials at the various points of a cylindrical surface. In [2] asymptotic formulas were derived for the total force acting upon an immobile cylinder, but they hold only at the initial points in time; in [3] an asymptotic solution was constructed (t ~ ~)."In this paper we offer a numerical solution that holds for the entire time interval. A comparison of these results with asymptotic behavior helped determine the applicability of the asymptotic approximation; a comparison of data for circular end rectangular inclusions provided estimates of the integral characteristics of diffraction patterns for obstacles with a convex contour (of a shape intermediate between these two extremes) Figure i presents a scheme of the process and the geometry of inclusions. I. INCLUSION OF A CIRCULAR CROSS SECTIONAn infinite rigid circular cylinder surrounded by an elastic isotropic median is reached by a stepped longitudinal compression wave with a front parallel to the cylinder axis ( where H is a Heaviside function; o 0 is the stress at the wave front spreadin E in the direction z; R is the cylinder radius; c~ is expansion wave velocity; u is Poisson's ratio; and t is time. Total stresses and displacements in the medium are written as the sum of the direct and diffractional components Y = ~c + ~ ~ ~ (or, o0 ' Or0, u, v), where u and v are tangential and radial displacements, respectively.Inst...