Analytical determination of parabolic points on slowness surface and swallowtail points on wave surface of cubic crystals

Abstract

Slowness surface for bulk wave propagation in anisotropic media can be divided into concave, saddle and convex regions by the parabolic lines. When a parabolic line crosses a symmetry plane, it leaves either an inflection point or a parabolic point. Surface normal at these points is associated with cuspidal point and swallowtail point, respectively, on the wave surface and in phonon focusing patterns. By examining the degeneracies in the Stroh eigenvalue equation, we have calculated the cuspidal points in cubic crystals analytically. In this work, the parabolic point and its surface normal are discussed. The main idea is to establish a connection between the parabolic point and the extraordinary transonic state that is related to a degeneracy with a multiplicity of four in the Stroh eigenvalue equation. Such a connection yields a series of simple expressions, which determine the locations of parabolic points and the corresponding swallowtail points. The result is demonstrated using phonon focusing patterns of cubic crystals, and the method also provides a tool for general discussion of the slowness surface geometry.

Structure of phonon focusing patterns in tetragonal crystals

“…[12][13][14] Recently, the present author developed an analytical scheme for finding locations of various caustic points in the symmetry planes in cubic crystals. 15,16 The scheme is based on the so-called Stroh formalism for two-dimensional elastodynamics, where the zero-curvature points on the slowness surface are treated in terms of highly degenerated Stroh eigenvalues. The main idea for the scheme will be adopted in this investigation.…”