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.