Reflection of high-frequency elastic waves from a non-plane boundary surface of the elastic medium

The paper is concerned with a classical problem of reflection of a high-frequency (longitudinal or transverse) wave, generated by a point source located in the elastic medium, by a free non-plane smooth boundary surface of this medium. For the investigation of this problem, we develop a method founded on the estimate of reflection integrals by the two-dimensional stationary phase method. The proposed approach permits derivation of the amplitude of reflected longitudinal and transverse waves in explicit form. The amplitudes of the reflected waves are defined by principal curvatures, by Gaussian curvature of the boundary surface at the point of specular reflection, by the distance from the source and receiver to the point of specular reflection, by direction of the incident wave, and by elastic moduli.