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 pho non intensity patterns for propagation directions in crystal symmetry plane s. Line caustics are known to be associated with lines of vanishing Gaussia n curvature (parabolic lines) on the acoustic slowness surface, while cusps are associated specifically with points where the direction of vanishing p rincipal curvature is parallel to the parabolic line. A parabolic line meet s a crystal symmetry plane sigma at a right angle, and so it is the vanishi ng of the slowness-surface curvature transverse to sigma that conditions th e existence of a cusp. A relation for the transverse curvature is derived a nd analyzed. It is shown that in an arbitrary symmetry plane sigma there ma y be up to four pairs of inversion-equivalent cuspidal points for SH (out-o f-plane polarized) waves, and up to eight pairs of cuspidal points associat ed with the in-plane polarized (usually quasi-transverse) waves. In tetrago nal 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 canno t 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 s heet has a concave region embracing the four-fold axis. The points of vanis hing transverse curvature on the slowness surface in symmetry planes of tet ragonal and cubic media are identified by concise relations, facilitating t heir explicit analysis. (C) 2000 Acoustical Society of America.