This paper considers the finite amplitude wave propagation that results when a layered continuum consisting of concentric isotropic compressible hyperelastic layers surrounding a cylindrical, or spherical cavity is subjected to a sudden, spatially uniform application of pressure at the cavity’s surface. The layers have different elastic constants, and wave reflection at the cavity’s boundary is considered as is wave reflection and transmission at the interface between the layers. Governing equations for the above problems are identical, except for a constant multiplier and are expressed as a system of first order partial differential equations in conservation form. This system of equations is strictly hyperbolic, for the class of strain energy functions used here, with three families of characteristics. The numerical scheme used to obtain the solutions presented here is a finite difference predictor-corrector method which uses the relation along straight characteristics parallel to the t axis in the (R, t) plane. It is a shock finding scheme so that the jump relations are not required for implementation of the scheme. In order to present this method we discuss the finite amplitude cylindrically symmetric wave propagation in a compressible hyperelastic layered solid. Nonlinear elastodynamic solutions for similar homogeneous problems have been addressed in the previous paper [1], in which the hybrid characteristic method−finite difference scheme was introduced. The results shown here are arrived at by the extension of this technique and represent the case of radially varying nonhomogeneity.