In recent years, various groups of researchers have looked at the two-dimensional motions of an undamped infinite thin elastic plate lying under a uniformly moving incompressible inviscid fluid. The plate is driven, usually by a single frequency time-harmonic line-source switched on at a finite time. The system's behaviour is interesting as it can be shown to be absolutely unstable for flow velocities above a critical value, and below this the long-time solution is convectively unstable (downstream of the source) for a sufficiently low forcing frequency. These results do not appear particularly plausible from a physical point of view, and there is some question regarding the realisation of long-time steady behaviour, and so this article attempts to examine ways in which the model problem can be improved. In particular, the effects of introducing plate thickness and fluid compressibility to the model are studied. This is carried out by comparing the morphology of the original and modified solutions in the complex wavenumber space. It is found that, in the limit of small fluid-to-plate density ratio, the two problems exhibit qualitatively identical behaviour. However, the addition of structural damping is shown herein to lead to a very different solution -the initial boundary value problem is absolutely unstable at all flow velocities. Various other modifications to the original model, including finiteness of the plate, three-dimensional effects and nonlinearity, are discussed and their impact on the long-time response of the system is assessed.