Oblique reflection of a nonlinear P wave from the boundary of an elastic half-space

This paper describes mode conversion effects and subsequent waveform distortion arising when a finite amplitude dilatational (P) wave that has already experienced nonlinear distortion is obliquely incident on a stress-free boundary of an isotropic elastic half-space. A two-term perturbation expansion is first employed to identify the dominant nonlinear effects. The understanding of wave interactions obtained from the perturbation analysis is then exploited to derive a successful solution using the method of characteristics for two-dimensional wave. It is shown that the incident and reflected P waves undergo nonlinear amplitude dispersion along their ray paths. The orientation of the rays for the reflected waves are time dependent, being governed by a modified form of Snell’s law, in which the phase speed incorporates the nonlinear correction for the association particle velocity. The reflection coefficients are shown to resemble those of linear theory, except for the dependence on the variable angles of reflection. The nonlinear propagation and reflection laws are solved to determine temporal waveforms for the reflected P and SV (vertically polarized shear) waves. This requires an iterative procedure in order to trace rays arriving at a specified field point at an arbitrary instant back to their source.