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The paper reviews recent work of a Leningrad group of geophysicists on the development of the ray method for arbitrary anisotropic elastic media. The methods and algorithms can be applied for descriptions of both the body and head waves.

In quantitatively estimating interference wave fields by methods of function theory, the initial integration path is subjected to deformation in the complex (() For the convenience of the reader, here we outline general information (borrowed from [1, w on the location of the roots ( = (~,(x) of the dispersion equations of the problems from [2, 3] on the principal sheet of the (() plane. The main purpose of this section is to study the roots of the above equations on the second sheet of the complex (() plane whose parts adjacent to the cuts often prove to be essential to implementing the method of contour integration when we quantitatively estimate the spectral functions of displacement fields.1. We shall write dispersion equations (12) and (20) when we drew the cuts from the branch points of the radicals to infinity along the real axis without intersecting the point ( = 0. In what follows, we need some results from [11, although the main problem is to obtain necessary information about the location of the roots of the dispersion equations on the second sheet of the complex plane, the passage to which is carried out by intersecting the cut (7, c~ and by bypassing only one branch point ( = 7 of the radical/3((). hi this connection, the paper is subdivided into three parts. In the first part, we discuss the reasons that stimulated us to direct our attention to the study of the roots of tile dispersion equation not only on the

A historical and procedural outline is presented of several methods of solving problems of elastic wave propagation from impulsive sources. The methods include a summary of the formalism of the geometrical theory of diffraction, eigenvalue methods, and many others, all of which are used to solve the boundary value problems of elastic wave propagation. The mediums considered are homogeneous or inhomogenous in composition; the geometries are plane‐parallel and multilayered, or spherical (cylindrical) in analogy with the earth.

The problem of quantitatively studyin 9 wave fields in a free elastic layer ezcited by a nonstationar~ point source is discussed. Much attention is given to the low-frequency part of the Fourier spectrum. In the present paper, we continue the study of interference wave fields in layered homogeneous elastic media within the framework of general methods whose elaboration was begun in [1]. We consider "lowfrequency wave fields" in an unbounded elastic plate excited by basic point sources applied to its "lower" and "upper" faces. The representations of displacement fields in the form of the so-called Lamb method were chosen as a basis for analysis. We touch briefly on the methods of obtaining such representations that apply the solution of problems on wave propagation in layered homogeneous elastic media with planeparallel interfaces by the method of contour integrals, which actually is in use in the literature (see [2]) if not in its canonical form [3]. The roots ff = ff,,(x) of dispersion equations (in the form of the Lamb method) are studied extensively, first, by logical-analytical and, then, by numerical methods. On this basis, the displacement field in the plate can be represented in theoretical seismograms in the form of sums of damped and undamped modes, i.e., of Fourier integrals with respect to t (over fixed frequency bands 0 < w < w0) of the residues of the integrands of inner integrals (of the representations of the fields in the form of the Lamb method) at the roots of respective dispersion equations. The physical consequences obtained here are discussed, in particular, as applied to some problems concerning plate ultrasonic seismic modeling.The paper is divided into five sections. Section 1: Introductory discussion (statements of the problems; approaches to their solutions; some auxiliary questions and results).Section 2: Study of the roots of dispersion equations of a plate by logical-analytical methods. Behavior of the roots in the low-frequency band of the Fourier spectrum.Sectior~ 3: Quantitative study of low-frequency wave fields in a plate. The representation of fields as a superposition of space-time modes.Section 4: Properties of a principal mode in low-frequency bands of the Fourier spectrum and the problems of plate seismic modeling.Section 5: Study of the wave fields of space-time modes by numerical methods. Some consequences and problems.It remains to make two short remarks. First, it is necessary to emphasize that the paper is not brought to a logical completion, despite its length. Therefore, the authors hope to continue the study of more interesting wave processes in a plate and the development of ideas related to plate seismic modeling in the near future. Second, it is pertinent to note that V. S. Pivovarov (a post-graduate student of the Physics Department of the St. Petersburg State University since 1995) did not join this work from the outset. Therefore, the paper has been written by the first coauthor using numerous (intermediate and final) numerical results of the second coauthor. T...