Dispersion relations for elastic waves in plates and rods

Wave propagation in homogeneous elastic structures is studied. Dispersion relations are obtained for elastic waves in plates and rods, for symmetric and antisymmetric modes using different displacement potentials. Some engineering beam theories are considered. Dispersion relations are obtained for phase velocity. The comparison of results based on the fundamental beam theories is presented for the lowest flexural mode. The Rayleigh-Lamb frequency equations are derived for elastic plate using the Helmholtz displacement decomposition. The Rayleigh-Lamb equations are considered in a new way. A new series expansion of frequency to any order of wave number, in principle, is obtained for symmetric and antisymmetric modes using an iteration method. Dispersion relations are shown in graphs for frequency, phase speed and group speed versus wave number. The obtained results are in good agreement with exact solutions. The cutoff frequencies for axial-shear, radial-shear and flexural modes are calculated and taken as starting points in dispersion relations for frequencies versus wave number. Different displacement potential representations are presented and compared. The Pochhammer-Chree frequency equations are derived for elastic rods using two displacement potentials, such as the Helmholtz decomposition for vector fields and Buchwald's vector potentials. Buchwald's representation enables us to find an efficient formulation of dispersion relations in an isotropic as well as anisotropic rods. Analysis of the numerical results on dispersion relations and cutoff frequencies for axial-shear, radial-shear and flexural modes is given.