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

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 probl...

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Bibliographic Details
Published in:Acoustical physics 2023-04, Vol.69 (2), p.143-158
Main Authors: Makarov, O. I., Shanin, A. V., Korolkov, A. I.
Format: Article
Language:English
Subjects:
Acoustic waves
Acoustics
Boundary conditions
Classical Problems of Linear Acoustics and Wave Theory
Dirichlet problem
Exact solutions
Far fields
Helmholtz equations
Manifolds
Physics
Physics and Astronomy
Point sources
Wave diffraction
Wiener-Hopf method
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Summary: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.
ISSN:1063-7710
1562-6865
DOI:10.1134/S1063771023600080