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 proposed by Rayleigh, Love, Bishop, Mindlin-Herrmann in which the lateral effects play an important role are also considered. The principles of construction of the multimode theories, corresponding equations and orthogonality conditions are considered. Dispersion curves, representing the eigenvalues versus real and imaginary values of the wave number, of these models are compared with the exact dispersion curves of an isotropic cylinder and conclusions on accuracy of the models are formulated.