Some new results in multiphase geometrical optics

Abstract

In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta and Heaviside functions and analyze the resulting equations. They form systems of nonlinear conservation laws with source terms. In contrast to the classical eikonal equation, these equations will incorporate a finite superposition principle in the sense that while the maximum number of phases is not exceeded a sum of solutions is also a solution. We present numerical results for a variety of homogeneous and inhomogeneous problems.