Propagation of Longitudinal Waves in Circular Cylindrically Orthotropic Bars

Recent years have seen a revival of interest in the propagation of waves in anisotropic materials such as crystals, earth crust, reinforced plastics, and others. This paper investigates a rigorous theory of propagation of small disturbances in infinite cylindrically orthotropic bars of circular cross section. The field equations represent two coupled second-order differential equations for the radial and longitudinal displacements and involve seven elastic constants. They are solved by means of the Frobenius method using developments in power series in the radial coordinate, and assuming sinusoidal variation in the longitudinal coordinate and time. The boundary conditions of vanishing tractions on the curved surfaces of the bar supply two equations, whose nontrivial solution leads to the frequency equation. The first and the second-order approximations to the wave frequency are found which, for transverse isotropic and isotropic case, reduce to the classical results of Chree and Chree-Pochhammer. A simple formula for long waves is suggested.