A computational algorithm for the second term of the series of the ray method in an inhomogeneous isotropic elastic medium

Abstract

An algorithm for computing the second term of the series of the ray method in the case of elastic inhomogeneous isotropic media is proposed. The main idea of the approach to the problem can be formulated as follows. Let a central (or support) ray of a ray tube be known. If we inroduce the ray-centered coordinates s, q1, q2 in the vicinity of the central ray, then the rays of the ray tube can be described by functions qi=qi(s, γ1, γ2), i=1,2, where s is the are length of the central ray and γj, j=1, 2, are the ray parameters. On the one hand, we show that the integrand of the second term of the series of the ray method can be expressed via the derivatives of the functions qi with respect to γj of the first, second, and third orders. On the other hand, additional differential equations for the derivatives as functions of s can be obtained from Euler's equations for the rays. This paper also contains initial conditions for the derivatives in the case of a point source. Thus, we obtain an algorithm involving additional differential equations for the derivatives\(\frac{{\partial q_i }}{{\partial \gamma _j }},\frac{{\partial ^2 q_i }}{{\partial \gamma _j \gamma _k }},\frac{{\partial ^3 q_i }}{{\partial \gamma _j \gamma _k \gamma _1 }}\), and the initial conditions for them at the source. The algorithm for calculating the mixed components of a vector of displacement is elaborated in detail. Unfortunately, the Russian version of this paper contains some errors, which have been corrected in the English translation of the paper.