The effect of a fixed barrier on an incident progressive wave in shallow water

Abstract

The effect of a fixed barrier on the propagation of an incident wave inside an ideal fluid is investigated within the frame of the shallow-water theory in two dimensions. The fluid is supposed to occupy an infinite channel of constant depth. Further, the horizontal extent of the barrier is assumed to be small. An asymptotic double series expansion for the solution is used. This procedure enables us to obtain analytic expressions for the local perturbations. The results of the first-order approximation indicate that no reflections exist. The second-order approximation of the solution is found to be the superposition of a progressive wave and local perturbations. For approximations of order higher than two, a secular term which increases monotonically with time appears in the expressions for the progressive wave. This unacceptable result is due to certain aspects in the mathematical procedure used. For this reason, the procedure is modified by using a suitable transformation of variables which reduces the determination of the transmitted wave to the solution of the K.d.V. equation. As an illustration, the special case of the incident uniform flow is considered and the stream lines of the resulting flow are drawn.