Comparison of classical and modern theories of longitudinal wave propagation in elastic rods

Abstract

A unified approach to derivation of different families of differential equations describing the longitudinal vibration of elastic rods and based on the Hamilton variational principle is outlined. The simplest model of longitudinal vibration of the rods does not take into consideration its lateral motion and is described in terms of the wave equation. The more elaborated models were proposed by Rayleigh, Love, Bishop, Mindlin-Herrmann, and multimode models in which the lateral effect plays an important role. Dispersion curves, representing the eigenvalues versus wave numbers, of these models are compared with the exact dispersion curves of isotropic cylinder and conclusions on accuracy of the models are deduced. The Green functions are constructed for the classical, Rayleigh, Bishop, and Mindlin-Herrmann models in which the general solutions of the problem are obtained. The principles of construction of the multimode theories, corresponding equations and orthogonality conditions are considered.