Asymptotic Theory of Unsteady Three-Dimensional Waves in a Channel of Arbitrary Cross Section

Abstract

A mathematically tractable theory is developed for the study of unsteady three-dimensional gravitating waves in a channel of arbitrary cross section. Three-dimensional effects due to bottom topography on the lateral change of free surface elevation are fully accounted for, and a unified approach to the derivation of various expansions is achieved. This paper deals mainly with finite-amplitude waves in a homogeneous incompressible inviscid fluid. Three cases are studied, which correspond to different orders of magnitude of a parameter representing the product of the square of the axial length scale of the motion with the amplitude of the derivation of surface elevation from that of conventional shallow-water theory. The bottom topography has a pronounced effect on the surface elevation when that parameter is of the same order as the cube of the hydraulic radius. Asymptotic equations governing finite-amplitude waves are then derived, and a Neumann problem defined on a variable domain is posed for determinin...