Multidimensional Reflection and Refraction of Finite Amplitude Stress Waves in Elastic Solids.

In small-signal (linear) theory, oblique incidence of either a plane dilatational (P) wave or vertically polarized shear (SV) wave at a plane interface between two elastic media results in reflection and transmission of both types of waves. The direction of propagation of each wave is constant and is governed by Snell's law. If the incident wave arrives at the boundary in a distorted manner as a result of nonlinearity, the reflection and transmission pattern becomes intricate due to coupling effects between dilatational and shear waves. The first study of this problem addresses a special case--the reflection of an initially sinusoidal, finite amplitude plane P wave from a plane stress-free boundary of an elastic half-space. A second-order perturbation expansion successfully discloses the most significant nonlinear effects, but it ultimately encounters difficulty for making the results uniformly valid. This shortcoming is corrected by an analysis using the method of characteristics for two-dimensional waves. Allowing the incident and reflected waves to undergo nonlinear distortion along ray paths having variable propagation direction leads to finite amplitude forms of Snell's law and the reflection coefficients of the outgoing P and SV waves at each instant. A numerical algorithm is developed to calculate the waveforms of the reflected P and SV wave received at a specified point. The physical and mathematical insight provided from the analysis leads directly to a generalized solution for the case of an interface between arbitrary elastic and/or fluid media, in which critical angles may exist.