A general analysis of transonic states in an anisotropic elastic body

Abstract

The basic characteristics of surface-wave propagation in an anisotropic elastic body depend crucially on the transonic states, defined by the sets of parallel tangents to a centred section of the slowness surface. The number of tangents parallel to a given direction in the plane of the section is at least 3 and at most 15 and, according to the number of points of contact and the number of branches of the slowness section to which they belong, there are 6 types of transonic states. The homogeneous plane wave represented by a point of contact is called a limiting wave and a transonic state is said to be exceptional when each of its limiting waves leaves free of traction the planes orthogonal to the tangent. For an elastic material of general anisotropy with positive definite linear elasticity tensor it is shown that at most 2 transonic states can be exceptional and that precisely 4 combinations of exceptional transonic states can occur, involving only 3 of the 6 types. Necessary and sufficient conditions for the existence of each of the possible combinations are given. As an application of the general results a complete characterization of exceptional transonic states in elastic materials with hexagonal symmetry is obtained.