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

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 q_i=q_i(s, γ₁, γ₂), i=1,2, where s is the are length of the central ray and γ_j, 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 q_i with respect to γ_j 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 , 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. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 73–93. Translated by M. M. Popov.