Using a separable Buchwald representation in cylindrical coordinates, we
show how under certain conditions the coupled equations of motion governing the
Buchwald potentials can be decoupled and then solved using well-known techniques
from the theory of PDEs. Under these conditions, we then construct three parametrized
families of particular solutions to the Navier-Lamé equation in cylindrical coordinates.
In this paper, we specifically construct solutions having 2π-periodic angular parts.
These particular solutions can be directly applied to a fundamental set of linear elastic
boundary value problems in cylindrical coordinates and are especially suited to problems
involving one or more physical parameters. As an illustrative example, we consider
the problem of determining the response of a solid elastic cylinder subjected to a
time-harmonic surface pressure that varies sinusoidally along its axis and we demonstrate
how the obtained parametric solutions can be used to efficiently construct an
exact solution to this problem. We also briefly consider applications to some related
forced-relaxation type problems.