Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate

Summary The dispersion relation associated with harmonic wave propagation in an incompressible, transversely isotropic elastic plate is derived. Such a material is characterized by only three material constants, contrasting with five in the corresponding compressible case. Motivated by a numerical investigation, asymptotic expansions, giving phase speed and frequency as functions of wave number, are derived in both the long and short wave regimes. These approximations, which owing to the constitutive simplifications are readily available, are shown to provide excellent agreement with the corresponding numerical solution. It is envisaged that the detailed investigation carried out in this paper will aid numerical inversion of the transform solutions often used in impact problems. Additionally, the asymptotic investigation provides the necessary basis for future studies to derive asymptotically approximate models to describe long and short wave motion.