Mathematical simulation of acoustic wave refraction near a caustic

A. V. Baev

M. V. Lomonosov Moscow State University, Faculty of Physics

Abstract: Issues related to the computation of wave fields in an acoustic medium near caustics are considered. A boundary condition on a caustic is established, and the Green’s function of a boundary value problem for the general case of a varying speed of sound is constructed. For this purpose, an auxiliary Goursat problem is considered and a system of its particular solutions is constructed using hypergeometric functions. A Volterra integral equation for the Green’s function is obtained, and an algorithm for its expansion with respect to smoothness is described. A finite difference scheme approximating the solution of the differential problem with an unbounded coefficient is proposed. Numerical results are presented.

Key words: acoustic wave equation, caustic, Goursat problem, hypergeometric functions, Volterra equation, Green’s function, finite difference scheme, numerical simulation.

UDC: 519.63, 517.958:533.77

Received: 09.09.2011

DOI: 10.7868/S0044466913070053

 English version: Computational Mathematics and Mathematical Physics, 2013, 53:7, 947–961

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