Transmission of progressing waves across a moving interface

We describe global time existence nd uniqueness results for the wave equations with boundary conditions of Dirichlet type on a characteresitc cone and either Dirichlet or Neumann type on a timelike tube. We find that the solution is in general only half as regular as the data and we provide estimates which describe the differing differentiabilities of the solution in directions which are either tangent or transvers to the characteristic cone.