Determination of the shape of a cavity in an orthotropic half-plane given the wave field at the boundary

Abstract

The shape of a cavity in an elastic orthotropic half-plane, in which vibrations are excited by a point source situated on the surface, is determined from the vertical field of displacements on the surface. The problem is reduced to a system of three non-linear operator equations, and a linearized formulation is proposed for convex cavities. In the final analysis the problem is reduced to solving (1) a system of two boundary integral equations over the boundary of a known cavity, which is done by the boundary element method, and (2) a linear integral equation of the first kind with a smooth kernel. As the problem thus obtained is ill-posed, it is treated by regularization in Tikhonov's sense [1]. The effect of the initial approximation and the vibration frequency on the efficiency of the proposed algorithm is also investigated.