Elastic waves in nonhomogeneous media under 2D conditions: I. Fundamental solutions

The purpose of this work is to present three methods of analysis for elasti c waves propagating in two dimensional, elastic nonhomogeneous media. The f irst step, common to all methods, is a transformation of the governing equa tions of motion so that derivatives with respect to the material parameters no longer appear in the differential operator. This procedure, however, re stricts analysis to a very specific class of nonhomogeneous media, namely t hose for which Poisson's ratio is equal to 0.25 and the elastic parameters are quadratic functions of position. Subsequently, fundamental solutions ar e evaluated by: (i) conformal mapping in conjunction with wave decompositio n, which in principle allows for both vertical and lateral heterogeneities; (ii) wave decomposition into pseudo-dilatational and pseudo-rotational com ponents, which results in an Euler-type equation for the transformed soluti on if medium heterogeneity is a function of one coordinate only; and (iii) Fourier transformation followed by a first order differential equation syst em solution, where the final step involving inverse transformation from the wavenumber domain is accomplished numerically. Finally, in the companion p aper numerical examples serve to illustrate the above methodologies and to delineate their range of applicability. (C) 1998 Elsevier Science Ltd. All rights reserved.