Specific directions of longitudinal wave propagation in anisotropic media

Abstract

Plane elastic waves can be propagated in every direction of an unbounded elastic medium. It is known that associated with each direction there are three independent waves, the displacements of which form a mutually orthogonal set. In general none of the three displacement vectors coincides with the vector of the normal to the wave front, that is, in general that waves are neigher longitudinal nor transverse. The purpose of this paper is to find specific directions in a medium of given anisotropy, along which the displacement of one of the three possible waves is exactly parallel to the direction of wave propagation. A method is developed which leads to the complete set of such "longitudinal" directions, if the matrix of the elastic coefficients is known. The method is applied to several groups of crystal symmetry, namely to the trigonal, hexagonal, tetragonal, and cubic systems, and the general conditions are established under which pure longitudinal waves exist. For α quartz, for example, the numerical calculation shows that there are five such distinct directions, not counting the ones which are equivalent by symmetry properties. As a by-product of the results, special conditions between the elastic constants are obtained under which longitudinal waves could be propagated in any direction in a hypothetical anisotropic medium fulfilling these conditions. Owing to their relation to an early investigation by G. Green in 1839, the latter are called specialized Green's conditions. © 1955 The American Physical Society.