On a problem of S.L. Sobolev

TL;DR

This paper introduces a straightforward, computationally feasible method for constructing a fundamental solution to the 3D acoustic equation with variable speed, avoiding restrictive geodesic regularity assumptions.

Contribution

It presents a new simple construction of a fundamental solution for the 3D acoustic equation that does not require regular geodesic lines and is suitable for numerical implementation.

Findings

Construction is explicit and easy to compute.

No assumptions on geodesic line regularity are needed.

Method is applicable for effective numerical solutions.

Abstract

In 1930 Sergey L. Sobolev [7,8] has proposed a construction of the solution of the Cauchy problem for the hyperbolic equation of the second order with variable coefficients in 3-d. Although Sobolev did not construct the fundamental solution, his construction was modified later by Romanov [4,5] to obtain the fundamental solution. However, these works impose a restrictive assumption of the regularity of geodesic lines in a large domain. In addition, it is unclear how to realize those methods numerically. In this paper a simple construction of a function, which is associated in a clear way with the fundamental solution of the acoustic equation with the variable speed in 3-d, is proposed. Conditions on geodesic lines are not imposed. An important feature of this construction is that it lends itself to effective computations.