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. Needless to say, the use of shallow water equations for the study of bore propagation may be open to criticism. The issue would be settled if we knew the precise conditions for the validity of shallow water equations. Up to date, the shallow water equations for a two-dimensional channel with analytical initial data have been justified by Kano and Nishida (1979), and for the three-dimensional case with a priori assumptions on the free surface by Berger (1976). At present we may accept shallow water equations as model equations, and the bore run-up problem for a general channel certainly deserves further investigation.