The aim of the article is to review the literature devoted to the solution of wave dynamics problems resulting in the propagation of nonstationary surfaces (volume waves) or lines (surface waves) of strong and weak discontinuities. In so doing one-term and multiple-term ray expansions, which are truncated power series with variable coefficients, are used. Jumps in all kinds of order of the time-derivatives of the functions to be desired serve as the coefficients. With their help the solutions are constructed behind the wave fronts up to the boundary of the wave motion domain. If the domains of the wave motion existence are extended, then the truncated ray series approximating the solution are not always uniformly valid over these domains. In this article, the methods for regularization of the truncated ray series are discussed, and in particular a new method which has been called by the authors as the method of “forward-area regularization”. These methods have gained recently wide application to solve the boundary-value problems connected with the wave propagation in bars, beams, plates, shells, 3D bodies, as well as with the dynamic contact interaction. In so doing, linear and nonlinear elastic, isotropic and anisotropic, thermoelastic, elastoplastic, and elastoviscoplastic media are used.

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