Exact Analysis of Wave Motions in Rods and Hollow Cylinders

Abstract

In a deformable isotropic infinitely long cylinder a discrete number of propagating guided modes regularly exists in a limited interval of frequency (f) and wavenumber (ξ). The calculation of the guided modes is best done via Helmholtz’s method, where the Bessel functions are used to scale the scalar and wave potentials. Solving the three-dimensional wave equations, leads to displacement and stress componenets in terms of potential to be found. By imposing the stress free boundary conditions for the inner and outer surface of the cylinder, the dispersion equation can be obtained. The dispersion equation shows how the phase velocity, c p = 2πf/ξ, change with the frequency. The group velocity, i.e. the speed of the propagating guided modes along the cylinder, can be obtained as c g = ∂(2πf)/∂ξ.