Bounds on elastic wave speeds in crystals: theory and applications

The propagation of plane waves is considered in the context of the linearized theory of elastic crystals (or anisotropic materials) which exhibit orthorhombic, tetragonal, hexagonal, or cubic symmetry (‘RTHC crystals’). Although not explicitly considered, the analysis is valid also for waves of infinitesimal amplitude superposed on finite static homogeneous deformations of isotropic elastic bodies.An extremely simple procedure is presented for finding a lower bound and an upper bound on the speeds of propagation of all possible plane waves which may propagate in a given crystal of these classes. There is a drawback, however, in that the procedure may fail to give a meaningful lower bound (negative lower bound for the squared wave speeds). Also, the bounds are not always attained, but the procedure immediately shows in which cases they are attained. Even so, because of its simplicity the procedure may be of value particularly when searching for crystals with desirable properties.The procedure is then applied in turn to each of the RTHC crystal systems. Numerical values of the bounds are presented for several specific crystals, illustrating the various possibilities arising in the theory.