Derivation of Approximate Long Wave Equations in a Nearly Uniform Channel of Approximately Rectangular Cross Section

Abstract

Estimates are derived on the time and space derivatives of a shallow fluid flow in a channel of approximately rectangular cross section. These estimates depend upon the velocity and pressure being bounded and the free surface having a long wave length compared to the breadth and depth of the channel. As a consequence of these results, the horizontal velocity along the channel is shown to be independent of depth, and the pressure hydrostatic to the first approximation. This enables one to derive the classical equations of shallow water flow in nearly rectangular channels by a rigorous process which yields estimates for the error of the neglected terms which are related to the geometry of the channel. In particular we are able to give a rigorous derivation of the Boussinesq and Korteweg-deVries equations for channel flows as true approximations to the three-dimensional theory.