Nonlinear Waves in a Channel

Abstract

INTRODUCTION In this paper, we consider some approximate equations for the study of nonlinear water waves in a channel of variable cross section. For finite amplitude waves, a system of shallow water equations are given; for small amplitude waves, we present a K–dV equation with variable coefficients. Some of their applications are discussed. Some problems deserving more study are mentioned at the ends of the following two paragraphs and in the conclusions to Sections 3 and 4. One of the interesting problems of water waves in a sloping channel concerns the breaking of a wave moving toward a shoreline, the development of a bore, and the movement of the shoreline after the bore reaches it. For the two dimensional case corresponding to a rectangular channel of variable depth, the bore run-up problem was studied by Keller et al. (1960), Ho and Meyer (1962), and Shen and Meyer (1963a,b) on the basis of shallow water equations (Stoker, 1957). Later Gurtin (1975) derived a criterion for the breaking of an acceleration wave in a two-dimensional channel, and his result was extended by Jeffrey and Mvungi (1980) to the case of a rectangular channel of variable width and depth. We generalize Gurtin's result to predict the breaking point of an acceleration wave in a channel of variable cross section and review some existent results regarding the bore run-up problem for a rectangular channel with a uniformly sloping bottom.