An algorithm for computation of the second term of the ray method series in an inhomogeneous isotropic medium

N. Ya. Kirpichnikova ; M. M. Popov ; I. Pšenčik

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 23, Tome 210 (1994), pp. 73-93

Résumé

A computational algorithm for the second term of the ray method series in the case of elastic inhomogeneous isotropic media is proposed. Main idea of an approach to the problem can be formulated as follows. Let the central or support ray of the ray tube be known. We introduce the ray centered coordinate $s,q_1,q_2$ in a vicinity of the central ray, then the rays from the ray tube can be described by functions $q_i=q_i(s,\gamma_1,\gamma_2)$, $i=1,2$, where $s$ is arc length of the central ray and $\gamma_j$, $j=1,2$, are ray parameters. We show from one side that integrand of the second term of the ray method series can be expressed via derivatives of the functions ft with respect to $\gamma_j$ first, second and third orders. From the other side, additional differential equations for the derivativies as function of $s$ can be obtained from Eulier's equations for the rays. The paper contains also initial conditions for the derivatives in case of point source. Thus we obtain the algorithm involving additional differential equations for the derivatives $\frac{\partial q_i}{\partial\gamma_k}$, $\frac{\partial^2q_i}{\partial\gamma_k\partial\gamma_l}$, $\frac{\partial^3q_i}{\partial\gamma_k\partial\gamma_l\partial\gamma_m}$ and initial conditions for them in a source. The algorithm is elaborated in details for calculation of the admixture component of displacement vector. Bibliography: 14 titles.