Propagation of unsteady waves in a layered cylinder

The study of nonstationary vibrations and wave propagation in deformable waveguides is of considerable interest in many fields of science and engineering. This work addresses wave processes in extended multilayer cylindrical bodies. The aim of the study is to investigate the problems of wave propagation in an elastic hollow three-layered cylinder and to develop efficient analytical methods for solving the problem of nonstationary wave propagation in layered cylindrical structures. The problem is formulated and solved in a cylindrical coordinate system. Normal (radial) loads are applied at the free boundaries (either inner or outer) of the cylinder. The solution is constructed using the Laplace integral transform with respect to time, followed by its inversion. The solution in the original (time) domain is presented in a form that is convenient for numerical implementation. This formulation makes it possible to analyze wave propagation in a multilayer cylinder with an arbitrary number of coaxial layers. A spectral boundary value problem is derived for a system consisting of ordinary differential equations and partial differential equations, which is reduced to a system of ordinary differential equations with complex coefficients. The solution in the Laplace domain is expressed in terms of modified Bessel and Neumann functions of arbitrary order. The inverse transformation is carried out in a form free from contour integrals and is represented as a rapidly converging double series of cylindrical functions. It is established that, for large wave numbers, the limiting phase velocity of this mode coincides with the Rayleigh wave speed.