Longitudinal directions in media of arbitrary anisotropy

Abstract

A longitudinal direction is one in which three pure modes can propagate. It is known that every medium has at least three longitudinal directions. Every axis of symmetry is such a “bound” longitudinal direction, but in most media there are additional “free” longitudinal directions which do not coincide with symmetry directions. In particular, there are longitudinal directions even in triclinic media that possess no symmetry except the point symmetry. The maximum number of distinct longitudinal directions is thirteen. In a longitudinal direction the wave propagation is particularly simple: for the longitudinal mode, wave normal, displacement direction, and ray direction are identical, while for the two transverse modes the deviation of the ray direction from that of the wave normal depends on the type of longitudinal direction and the relative magnitude of a few stiffnesses expressed in a coordinate system aligned with the longitudinal direction. Similarly, the polarization pattern in the vicinity of a longitudinal direction depends strongly on the type of the longitudinal direction and on the relative magnitude of some off‐diagonal stiffnesses. Since the polarization pattern can be observed in suitable vertical seismic profiling (VSP) surveys, it might be inverted to obtain information on elastic parameters that is difficult to obtain by other means. For orthorhombic symmetry, there are always the three bound longitudinal directions coinciding with the axes. In each plane of symmetry a pair of free longitudinal directions can exist, and there can be four symmetrically disposed free longitudinal directions outside the planes of symmetry. Existence and direction of these free directions follows from simple expressions in the elastic stiffnesses.