Properties of S waves near a kiss singularity: a comparison of exact and ray solutions

We have studied the properties of S waves generated by a point source in a homogeneous, transversely isotropic, elastic medium, propagating in directions close to that of a kiss singularity, which coincides with the symmetry axis of the medium. We have proved analytically as well as numerically that the ray solution can describe the S waves correctly far from the source in all directions, including that of the kiss singularity. We have found that, in contrast to the far-field P wave, which can be reproduced satisfactorily by the zeroth-order ray approximation in all directions from the source, the far-field S waves can be reproduced satisfactorily by the zeroth-order ray approximation only in directions far from the kiss singularity. In directions near the kiss singularity, the first-order ray approximation must also be considered, because the zeroth- order ray approximation yields distorted results. The first-order ray approximation can be of high frequency and can be detected in the far field.