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We consider a layered-homogeneous elastic medium involving a half-space E < H (q = 1), a thin layer < E < /~ + h (q = 2), and a lower half-space E > -~ + h (q = 3) that are in welded contact with each other and that are characterized by real (and also relative) constant parameters = /z~ (q = 1,2,3),7 --v3 ' vl #2 /~2used for the description of elastic waves of SH type. It is assumed that a given point source is located at the point O = (0, 0, 0) of the Cartesian {E, y, z;; ;, k} or cylindrical {E, 0, E; F1,0"l, ~:l } coordinate systems. In the medium, it excites a plane or axially symmetric fieldof SH type that is a result of reflection-refraction of the initial field generated by the source under consideration by the thin layer q = 2 separating the half-spaces q = 1 and q = 3 in an unlimited elastic medium with parameters from (1) for q = 1. Since this problem is simple, we do not dwell on the mathematical statement and solution of it and restrict ourselves to the indication of the expression of the field of the incident wave and representations of the fields Ul and us of the perturbations reflected and refracted into half-spaces q = 1 and q = 3. Considering such fields, we illustrate the efficiency of the methods applied to the quantitative study of low-frequency (interference) wave fields. 1. For a plane source of the form0 turned on at t = 0, the initial wave is incident from the half-space q = 1 on the thin layer q = 2. This wave is expressed in terms of repeated integrals of the Lamb method (,x)

Simple algorithms for finding a numerical solution on a complex plane of dispersion equations dependent on a real parameter are described. Algorithms for numerical integration along stationary contours on a complez plane in the case where the integrand possesses singularities of the pole and branch-point type are also presented. These algorithms are applied to compute the spectral functions of wave fields propagating in layered media. Bibliography: 3 titles.In the present paper, an interference wave field u(t) (in the case of chosen models of media and points of observation) is represented (see w167 3 in [1]) by the iterated integrals On the original Lamb integration contour A, the integrand in (2) is always highly oscillating, which makes the direct numerical evaluation of the integral quite complicated. Because of this, it turns out to be more advantageous to evaluate the spectral function by using methods of contour integration. These methods start by carrying out a deformation of the original contour A to the stationary lines 10 of steepest ascent of the phase function f(() that are defined by the equations which must be compensated for by adding (with appropriate signs) the residues of the integrand from (2) at all roots that are intersected by the contour A (see w Secs. 4, 5 from [1]) to the integral under transformation. In addition, in the process of deformation the contour A may come across the branch points r = 7 of the factor at the exponent, which is conventionally referred to in what follows as "the slowly varying part" of the integrand from (2). In such a case, the cut (7, oo) is preliminarily deformed, starting with the point ( = 7, into the half-plane ~ < 0 up to its coincidence with the stationary line F~ of steepest ascent of the phase function with origin at the point ~ = 7. The original contour )~ bypasses the cut F~ along a loop that can be reduced, by continuous deformation, to one of its boards.

The head interference wave associated with the propagation of the P-wave in an elastic half-space is studied by using as an ezample the propagation of pressure waves in a liquid layer covering an elastic half-space. As is known, the mass density p and the velocity v of the propagation of a wave are real parameters of an ideal liquid. A deformed state of it is determined by the velocity vector of the displacement of particles of the liquid if(F, t) and by the pressure p(f', t) that are connected by the relations 0~ p~-= gradp, c0pot = -v2pdivff"For p = const and v --const, it follows thatThe boundary conditions of the contact of the liquid (z < h, the index t, is equal to 1) and the elastic medium (z > h, the index u is equal to 2) involve the pressure p and the component (Ct)z of the velocity vector of displacements. The conditions on the boundary (for z = h) are as follows:t~ + p = 0, t ~2~. = 0, ~ = (~)~1).Thus, it is assumed that the tangent component of the stress in the elastic medium vanishes on the interface z = h; moreover, the tangent components are not subject to any additional condition both in the elastic medium and in the liquid.1. We assume that to the boundary z = 0 of a medium composed of a liquid layer (0 < z < h, u = 1) and a homogeneous isotropic elastic half-space (z > h, u = 2), an external action is applied in the form of the pressureturned on at the moment t = 0, which is spatially concentrated in the form of the Dirac function 5(x) as ~ --"r 0.The velocity of the displacement in the liquid, if(l) = _ grad~1(x,z,t),