  The vibration of a solid elastic cylinder is one of the classical applied
problems of elastodynamics. Many fundamental forced-vibration problems
involving solid elastic cylinders have not yet been studied or solved using the
full three-dimensional (3D) theory of linear elasticity. One such problem is
the steady-state forced-vibration response of a simply-supported isotropic
solid elastic circular cylinder subjected to two-dimensional harmonic
standing-wave excitations on its curved surface. In this paper, we exploit
certain previously-obtained particular solutions to the Navier-Lam\'{e}
equation of motion and exact matrix algebra to construct an exact closed-form
3D elastodynamic solution to the problem. The method of solution is direct and
demonstrates a general approach that can be applied to solve other similar
forced-vibration problems involving elastic cylinders. Two complete analytical
solutions are in fact constructed corresponding to two different but
closely-related families of harmonic standing-wave excitations. The second of
these analytical solutions is evaluated numerically in order to study the
steady-state frequency response in some example excitation cases. In each case,
the solution generates a series of resonances that are in correspondence with a
subset of the natural frequencies of the simply-supported cylinder. The
considered problem is of general interest both as an exactly-solvable 3D
elastodynamics problem and as a benchmark forced-vibration problem involving a
solid elastic cylinder.
