The Ultimate Approach to the Steady State in the Generation of Waves on a Running Stream

The initial-value problem is solved for the generation of two-dimensional waves by an oscillatory pressure acting at the surface of a running stream of finite depth. In place of the usual contour-integration techniques, the problem is treated with the aid of generalised functions. It is shown that in the ultimate steady state either two or four waves may exist, depending on the relative values of the speed of the fluid, its depth and the frequency of the applied pressure. At the values separating these two possible states the solution is found to be singular.