End Effects and Time-Harmonic Longitudinal Wave Propagation in a Semi-Infinite Solid Cylinder

Axisymmetric end problems of longitudinal wave propagation are studied in a semi-infinite isotropic solid circular cylinder which is free of traction on its cylindrical surface. An accurate and computationally efficient method of solution is presented which can exploit the asymptotic behavior for high harmonics in the radial direction. The stresses and displacements are expanded in terms of the eigenfunctions of the case of a lubricated-rigid cylindrical surface condition. The expansions are used to construct a stiffness matrix relating the harmonics of stress and displacement for the traction-free case, which is shown to approach asymptotically that of the case of the mixed condition. Unlike other approaches such as finite element or boundary integral methods, which typically require the solution of large systems of equations for rapidly varying end conditions, the present formulation can lead to a coupled system of equations for lower spatial harmonics and a weakly coupled system for higher spatial harmonics. Due to the small number of equations in the coupled system, the present approach is very effective in handling general boundary conditions, and is particularly efficient for end conditions with rapid spatial variation.