We present an analysis of the radiation produced when an electrically short transiently excited dipole antenna radiates from within a cavity in a medium. We show that, if one formulates the problem in terms of a Debye series in the Laplace domain, complete results may be obtained analytically in the lossless case. When loss is present in the surrounding medium, some numerical evaluations are required to obtain the complete solution. As an example we treat the problem of communication between a transmitting antenna located beneath a metal craft on the surface of a lake and a receiver some distance below the surface.