Serre Equations in Channels and Rivers of Arbitrary Cross Section

Abstract

A variational approach was used to derive a set of Serre equations for fully nonlinear, dispersive waves in channels of arbitrary cross section. A family of travelling waves was found, as well as the relation between amplitude and celerity of solitary waves. An upper bound is proposed for the solitary wave amplitude as a function of the Froude number in trapezoidal cross-sectional canals, and it showed good agreement with existing theory. For waves of moderate amplitude, cnoidal waves result with a soliton limit; these waves and their properties (celerity and wave number) are written as functions of the channel bank slope and channel bank curvature. The theoretical findings are in agreement with well-established results in the literature, in particular with more-recent Boussinesq-type theories. A validation is proposed against existing experimental data.