The Sommerfeld Integral in Problems of Simulating the Diffraction of Acoustic Waves Using a Triangular Lattice

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Abstract

Analytical solutions are obtained for two problems for a discrete analogue of the Helmholtz equation on a triangular lattice: (1) the problem of radiation from a point source on a plane; (2) the problem of diffraction on a half-line with Dirichlet boundary conditions. It is shown that in these problems, the complete field can be represented as an integral of an algebraic function over a family of contours located on some complex manifold. The solution to the first problem is found as an integral of some differential form over this manifold, and the asymptotics of the far field for this solution is obtained. The second problem is solved using an analogue of the Sommerfeld integral. It is checked that the obtained solution coincides with the solution of this problem by the Wiener–Hopf method.

About the authors

O I. Makarov

Moscow State University, 119991, Moscow, Russia

Email: olegmakarovlip@gmail.com
Россия, 119991, Москва, ГСП-1, Ленинские горы

A. V. Shanin

Moscow State University, 119991, Moscow, Russia

Email: olegmakarovlip@gmail.com
Россия, 119991, Москва, ГСП-1, Ленинские горы

A. I. Korolkov

Moscow State University, 119991, Moscow, Russia

Author for correspondence.
Email: olegmakarovlip@gmail.com
Россия, 119991, Москва, ГСП-1, Ленинские горы

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