A method for predicting the reflection and refraction of spherical waves across a planar interface

Abstract

An approximate method is presented to find the magnitude of the reflected and refracted waves that result when a spherical wave encounters a planar interface separating two elastic solids. This method relies on: a multipole field solution for spherical waves to describe the compressional and shear waves that emerge from the interface upon reflection and refraction, virtual sources to locate the reflected and refracted wave origins above and below the interface, and a discrete, ray-based approach to solve the complex system of equations that result from the boundary conditions imposed. In addition to the elastic constants, the reflection and refraction coefficients are found to depend on the boundary conditions at the interface, the wavefront curvature, and source frequency. The dependence of reflection coefficients on boundary conditions is significant and has important implications for brittle materials subjected to impulsive (shock) loading, such as a turbine blade containment structure or ballistic ceramic armor. It is shown that a good impedance match between two solids is not sufficient to prevent fragmentation due to a reflected wave. Shear coupling of the interface is necessary to minimize damage. The Matrix Algebra for Reflective/Refractive Systems computer program has been developed to solve the resulting system of equations and illustrate these results.