On the theory of water waves

Abstract

Abstract The aim of this paper is to formulate a two-dimensional theory for the propagation of fairly long water waves. The approach differs from the usual in that the theory is set up via two-dimensional postulates. Subsequently, it is shown how a simple three-dimensional approxi­mation enables us to relate the two-dimensional theory to the three-dimensional theory. The resulting equations are used to discuss the unidirectional propagation of waves. lt is shown how the results obtained from the theory proposed here are related to the results of Korteweg & de Vries (1895) and to those of Benjamin, Bona & Mahony (1972).