A hydroelasticity theory  and a hydroelastic analysis are presented to describe the dynamic characteristics of a circular non-uniform beam in a bounded, viscous fluid. In vacuo, the principal mode and natural frequency characteristics are uniquely defined but when the combined influences of fluid-structure-external boundary interactions are accounted for in the mathematical model, significant variations in the prinipal coordinates, resonance frequencies, responses, etc., occur. That is, resonance frequencies, etc., are not uniquely defined and reasons for this are found in the changes in the magnitudes of the generalized fluid actions. These are shown to be functions of the principal mode shapes, frequency of oscillation, the annular clearance between the cylinder and the rigid tube boundary and the viscosity of the fluid. The latter is mathematically modelled as an incompressible Stokes' fluid [2,3] and the principal modes of the dry free-free non-uniform cylinder are determined by using a Prohl-Myklestad approach [1,4].