Diffraction of Elastic Waves by a Sphere in the Semi-Bounded Region

The author considers scattering of plane elastic waves by a rigid sphere near a plane rigid boundary, which causes multiply re-reflected dilation and shear waves and generates strong oscillations of the wave field. The problem is formulated and reduced to finding scalar functions and for shear waves also as a consequence of symmetry. Approximate formulas for the field in the far zone and in case of the Rayleigh long wavelength approximation are presented. Estimates of the construction of multiply re-reflected waves by the image method are obtained. Calculations of scattered wave fields presented in the form of scattering diagrams are carried out.