In order to accommodate solutions with multiple phases, corresponding to crossing rays, weformulate 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 anassociated 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 ondelta and Heaviside functions and analyze the resultingequations. They form systems of nonlinear conservation laws with source terms. In contrast to the classicaleikonal equation, theseequations will incorporate a "finite" superposition principle in the sense that while the maximum number of phasesis not exceeded a sum of solutions is also a solution. We present numerical results for a varietyof homogeneous and inhomogeneous problems.

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.