Nonlinear nonstationary vibrations of an infinite elastic isotropic cylindrical shell of constant thickness in welded or smooth contact with a continuous elastic isotropic medium are considered. Nonstationary vibra-tions are excited by snap-action loads, resulting in two types of cylindrical shock waves propagating in the medium. Behind the wavefronts, the solution for the desired functions is constructed along the rays in terms of a power series (ray series), whose coefficients are the discontinuities in various orders of partial derivatives of the functions to be found with respect to time. A variable value is the time needed for a disturbance to propagate along the ray from the point under consideration up to the wavefront. In so doing, the power of the variable value corresponds to the order of partial time derivative of the desired function.