First-order hyperbolic form of velocity-stress equations for waves in elastic solids with hexagonal symmetry

This paper reports a theoretical framework to analyze wave propagation in elastic solids of hexagonal symmetry. The governing equations include the equations of motions and partial differentiation of elastic constitutive relations with respect to time. The result is a set of nine, first-order, fully-coupled, hyperbolic partial differential equations with velocities and stress components as the unknowns. The equation set is then cast into a vector form with three 9 × 9 coefficient (or Jacobian) matrices. Physics of wave propagation are fully described by the eigen structure of these matrices. In particular, the eigenvalues of the Jacobian matrices are the wave speeds and a part of the left eigenvectors represents the wave polarization. Without invoking the plane wave solution and the Christoffel equation, two- and three-dimensional slowness profiles can be calculated. As an example, slowness profiles of a cadmium sulfide crystal are presented.