Approximate near-front ray solutions in the axisymmetric strain dynamics of a linear elastic half-space

The dynamics of axisymmetric two-dimensional strains in a linear elastic half-space bounded by a smooth surface of revolution with positive Gaussian curvature is considered. An approximate solution of the initial-boundary value problem is constructed on the basis of ray series with expansion in a time-like variable. The limited number of terms of the ray series is used for near-front domains of curvilinear waves of strong discontinuities. The coefficients of this series are the discontinuities of the derivatives of displacements with respect to time (starting from the first derivative). It is shown that it is necessary to take into account the ray series components up to the $(k + 1)$st order inclusive at the $k$th step of the ray method for a two-dimensional type of the deformation process.