The ray method is an effective method for solving problems dealing with the propagation of wave surfaces of strong and weak discontinuities, including problems of dynamic contact interaction. Unsteady vibrations could be initiated by instantaneous loads applied on the plate, resulting in plane waves propagating within an elastic half-space. The solution behind the wave fronts up to the contact boundary is constructed using ray expansions. Unknown functions entering in the ray series coefficients and in the equation of plate motion could be found from the boundary conditions of the contact interaction between the plate and the half-space. “Manual” procedure (without using any mathematical packages) for calculating the ray series coefficients is rather cumbersome, therefore an algorithm to solve this problem using the Maplesoft has been suggested by the authors for different types of contact conditions first for linear problems. In the present research, the ray expansion method and the developed algorithm are applied to analyze the unsteady response of an infinitely long elastic nonlinear classical von Karman plate lying on an elastic isotropic half-space.