Polynomial mode approximations for longitudinal wave dispersion in solid and hollow circular cylinders

The characteristic equation for wave dispersion in solid or hollow circular cylinders is a transcendental equation: no closed-form solution exists and it can only be solved numerically. The characteristic equation for a hollow circular cylinder is substantially longer and more complex to solve than the equation for a solid cylinder. This paper examines, at different orders of approximation, two polynomial approximations (Jacobi mode approximation and an approximation by Widehammar, Gradin and Lundberg) which simplify the computation of dispersion curves. We focus on the approximation error for the first four modes of axially symmetric longitudinal waves, for both solid and hollow circular cylinders. The two polynomial approximations yield accurate dispersion curves for a solid cylinder, provided a sufficient order is used. For a hollow circular cylinder, the numerical limits of thinner cylinders prevent the use of the highest orders of approximation, limiting the accuracy of the dispersion curves. Classic theories for thin and thick cylindrical shells (by Naghdi and Cooper, and Herrmann and Mirsky), remain useful alternatives when computing dispersion curves for the first propagation mode for very thin hollow circular cylinders. We find that Widehammar's approximation yields more accurate dispersion curves than Jacobi mode approximation for hollow cylinders, for a given order of approximation, as well as being more straightforward to program.