Eeflection of space-time ray amplitudes from arbitrary moving boundary

The problem of finding of the geometrical spreading for the space-time ray method in three-dimensional inhomogeneous medium with arbitary moving reflecting boundary is considered. A syatem of linear differential equations and initial conditions on reflecting boundary for geometrical spreading is obtained. It is proved that the problem can be solved under the natural conditions
i) reflecting boundary is timelike,
ii) incident ray doesa't touch the boundary.