Integral transform techniques are utilized in order to analyze the propagation of spherical waves in bispherical elastic media. A typical model consists of a cavity of radius r0 located at the center of a spherical material (medium 1) which in turn is embedded in an infinite medium of different properties (medium 2). The inner surface of the cavity is loaded with a time dependent explosionlike pressure. The resulting transformed stresses and displacements cannot be inverted to give exact analytic solutions. Exact solutions are only possible to obtain for the special case of a single material and also for the arrival time (wave-front) regions. Solutions of the general bispherical model are obtained by inverting the transforms numerically. It is demonstrated that the maximum stress attained in the outer material is independent of the interface location r1. It is also demonstrated that the maximum stress attained in the inner medium as well as the entire time history of the stress in both materials depend upon r1. The wave-front analysis results are found to form a skeleton of the general solutions which well predict their maximum stresses and their arrival times.