Direct asymptotic integration of the equations of transversely isotropic elasticity for a plate near cut-off frequencies

Direct asymptotic integration of the equations of transversely isotropic elasticity, for a layer with in-plane axis of transverse isotropy and zero surface traction, is carried out in the vicinity of the cut-off frequencies. In direct contrast to the corresponding isotropic case, in which there is only one family of thickness shear resonance frequencies, two such families are observed to exist. Consequently, the two-dimensional equations for the associated long-wave amplitudes are each scalar equations, rather than the single vector equation which arises in the isotropic case. These equations, together with that corresponding to thickness stretch resonance, are obtained. The exact dispersion relation is also derived and asymptotic expansions, giving frequency as a function of scaled wave number, are obtained in the neighbourhood of the cut-off frequencies. This both reveals the appropriate asymptotic orders of stress and displacement components to help facilitate the direct asymptotic integration and retrospectively acts as a check on the coefficients of the two-dimensional equations. The mathematical equivalence of approximations derived from the exact solutions and exact solutions derived from asymptotically approximate equations is therefore verified.