Some new results in multiphase geometrical optics

In order to accommodate solutions with multiple phases, corresponding to cr ossing rays, we formulate geometrical optics for the scalar wave equation a s 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 so lution from an associated system of moment equations, closed by an assumpti on on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta and Heaviside functions a nd analyze the resulting equations. They form systems of nonlinear conserva tion 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 sol utions is also a solution. We present numerical results for a variety of ho mogeneous and inhomogeneous problems.