Summation of Gaussian Beams in 3D Problems of Radiation and Scattering of Elastic Waves

Abstract

The method of Gaussian Beams Summation is applied to the two important problems of the theory of elastic waves — the scattering of compressional wave from a planar crack embedded into a homogeneous and isotropic elastic medium and time-harmonic radiation of a normal transducer of arbitrary shape directly coupled to a homogeneous and isotropic elastic solid. The problems are studied in the case of high-frequency approximation. Moreover, the radiating near zone of transducer and the near zone of the field scattered from crack is analyzed. The radiated and scattered fields have the ray structure of main beam and edge di racted rays. A family of the edge diffracted rays is singular near to caustics. A well-known ray asymptotic solution of the Geometrical Theory of Diffraction (GTD) is not valid near to caustics. Application of the method of Gaussian Beams Summation to both problems in the neighborhood of caustics has proved to be effcient from the point of view of asymptotic and computational analysis.