Diffraction of waves by inhomogeneous obstacle (Q1106397)

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).