On calculating the second term of the series of the ray method for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media

Abstract

This paper continues the series of papers [1][2][3][4][5][6] devoted to the development of algorithms for computing the second and subsequent terms of the ray series for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media. For many years the ray method has remained a classical method for finding the principal term of an expansion of the solution in acoustic, electromagnetic, elastic, and other media. However, an evaluation of even the principal term of the ray series for nonhomogeneous media requires cumbersome analytic calculations (see, e.g., [7,8]). In calculating the subsequent terms of the ray asymptotic expansion in a nonhomogeneous three-dimensional medium, these difficulties increase many times. Thus, we arrive at the problem of finding algorithms such that their comparison could provide an opportunity for simplifying the solution process in problems related to wave propagation in different media. In this paper, for determining the second term of the ray expansion of the displacement vector in a three-dimensional nonhomogeneons elastic medium, we suggest an algorithm similar to that of [1,2]. Simultaneously, we compare the methods of [1,2] and [3,4]. In the latter papers, the need for finding the second term of the ray series for the solutions of problems on wave propagation in different media was thoroughly justified.Consider a nonhomogeneous isotropic elastic medium with the Lam~ parameters A(~ ), #(Z ) and density p(~, ). The functions A,/~, and p are assumed to be smooth enough as functions of the point ~. The ray expansion of the vector of displacements U is considered in the formwhere w is the frequency, v is the eikonal, t is time, and s = (x, y, z). The principal (first) term of the ray series (0.1) corresponds to k = 0, whereas the correction (second) term, which is the subject of our considerations, corresponds to k = 1. We analyze the case of longitudinal vibrations, for which the recurrent relations of the ray method are known (see [9,10]). These relations were used in [3,4] as a starting point for determining the second (k = 1) term of the ray expansion (0.1) of the displacement vector U. For the convenience of comparing the algorithm for finding tl presented in this paper with that in [3,4], we recall these relations: t0 = ~0Vr, q~0 = ~b0(V1,72 9 (0.2)Here, VT" is the gradient of the eikonal r; 71,7~ are the coordinates specifying the ray; ~bo is a constant along the ray that describes the initial data for t0 and is dependent upon the type of source in question; a = (()~ + 2#)/p) 1/2 is the velocity of longitudinal waves; J is the geometrical spreading of the ray tube equal to I J= 77(7.71,72 "