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Diffraction of waves by inhomogeneous obstacle. (English) Zbl 0651.35020

Motivated by the study of wave diffraction in elastic material with a localized imbedded inhomogeneity as considered in earthquake engineering and seismology the authors investigate a mixed boundary value and transmission problem for the Helmholtz equation in the half-plane. Imbedded in the half plane is a bounded domain R with a common piece of boundary with the half plane (a valley or cavity) for which a different wave number is assumed to describe the properties of the medium. On the boundary of the half plane a Neumann boundary condition is assumed. Transmission conditions are imposed on the solution and its normal derivative across the interface between R and the rest of the half plane. The radiation condition controls the behaviour at infinity. The first observation is that by reflection the problem can be considered as a pure transmission problem. This is then solved by considering the corresponding boundary integral equation in a Hilbert space H (space on the “jumps” completed with respect to a suitable norm) of traces of the solution on the interface. The solution is obtained by a Galerkin method in H. Corresponding uniqueness, existence and approximation results are obtained (numerical results are published elsewhere).
Reviewer: R.Picard

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R05 PDEs with low regular coefficients and/or low regular data
41A30 Approximation by other special function classes
41A63 Multidimensional problems
86A15 Seismology (including tsunami modeling), earthquakes
78A45 Diffraction, scattering
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