Method of iterated kernels in problems of wave propagation in heterogeneous media

Abstract

AThe approximated solution of the wave propagation problem in smoothly heterogeneous medium on the basis of the method of iterated kernels is proposed in this article. The solution is obtained by the method of successive approximations to an integral equation, which is equivalent to the Helmholtz scalar equation. Since the exact calculation of iterated kernels is impossible for arbitrary spatial dependence of medium dielectric permittivity, approximate estimation is used applying several first Taylor expansion terms. In purpose of exact calculating of the double series for resolvent a method, based on identifying of coefficients of a power series with orthogonal polynomial, which is calculated by Rodrig's generalized formula, will be applied. The final solution has a compact form and unites the advantages of Born scattering and short-wave asymptotic methods. The proposed solution requires smoothness of medium heterogeneities changes, scilicet the smallness of first and second derivatives of the dielectric permittivity, but not of the dielectric permittivity itself.