Transverse curvature of the acoustic slowness surface in crystal symmetry planes and associated phonon focusing cusps.

Conditions are derived for the existence of focusing cusps in ballistic phonon intensity patterns for propagation directions in crystal symmetry planes. Line caustics are known to be associated with lines of vanishing Gaussian curvature (parabolic lines) on the acoustic slowness surface, while cusps are associated specifically with points where the direction of vanishing principal curvature is parallel to the parabolic line. A parabolic line meets a crystal symmetry plane sigma at a right angle, and so it is the vanishing of the slowness-surface curvature transverse to sigma that conditions the existence of a cusp. A relation for the transverse curvature is derived and analyzed. It is shown that in an arbitrary symmetry plane sigma there may be up to four pairs of inversion-equivalent cuspidal points for SH (out-of-plane polarized) waves, and up to eight pairs of cuspidal points associated with the in-plane polarized (usually quasi-transverse) waves. In tetragonal crystals, the symmetry planes containing the four-fold axis can have at most two pairs of cusps for the SH waves and up to six pairs of cusps for the in-plane waves. In cubic crystals, the face symmetry planes sigma cannot have cuspidal points for SH waves, as is known, while four pairs of cusps for in-plane waves exist in sigma if and only if the outer-most slowness sheet has a concave region embracing the four-fold axis. The points of vanishing transverse curvature on the slowness surface in symmetry planes of tetragonal and cubic media are identified by concise relations, facilitating their explicit analysis.