Clifford O. Bloom, "High frequency asymptotic solutions of the reduced wave equation on infinite regions with non-convex boundaries", Mathematical Problems in Engineering, vol. 2, Article ID 806908, 33 pages, 1996. https://doi.org/10.1155/S1024123X96000385
High frequency asymptotic solutions of the reduced wave equation on infinite regions with non-convex boundaries
The asymptotic behavior as of the function that satisfies the reduced wave equation on an infinite 3-dimensional region, a Dirichlet condition on , and an outgoing radiation condition is investigated. A function is constructed that is a global approximate solution as of the problem satisfied by . An estimate for on is obtained, which implies that is a uniform asymptotic approximation of as , with an error that tends to zero as rapidly as . This is done by applying a priori estimates of the function in terms of its boundary values, and the norm of on . It is assumed that , , and the boundary data are smooth, that and tend to zero algebraically fast as , and finally that and are slowly varying; may be finite or infinite.The solution can be interpreted as a scalar potential of a high frequency acoustic or electromagnetic field radiating from the boundary of an impenetrable object of general shape. The energy of the field propagates through an inhomogeneous, anisotropic medium; the rays along which it propagates may form caustics. The approximate solution (potential) derived in this paper is defined on and in a neighborhood of any such caustic, and can be used to connect local “geometrical optics” type approximate solutions that hold on caustic free subsets of .The result of this paper generalizes previous work of Bloom and Kazarinoff [C. O. BLOOM and N. D. KAZARINOFF, Short Wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions, SPRINGER VERLAG, NEW YORK, NY, 1976].
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