Anomalous solutions to simple wave equations

Standard fourth and sixth-order differential wave equations for beams and for pairs of fluid-coupled plates are shown to give rise to apparently anomalous solutions describing waves that grow in amplitude in the direction of propagation and thus violate conservation of energy when the propagating medium is semi-infinite, despite the fact that the original equations conform to this principle. The second-order wave equation for a string does not have these problems, nor do they occur in the other cases if the propagating medium is finite in extent and has simple boundary conditions at both ends. This apparent paradox can be resolved by consideration of the group velocity, which is shown to be negative in the case of the anomalous waves, thus preventing them from propagating into the half-space of the medium.