On three-dimensional long water waves in a channel with sloping sidewalls

Abstract

A theoretical model is presented for the propagation of long, weakly nonlinear water waves along a channel bounded by sloping sidewalls, on the assumption that h0/w [Lt ] 1, where 2w is the channel width and h0 is the uniform water depth away from the sidewalls. Owing to the non-rectangular channel cross-section, waves are three-dimensional in general, and the Kadomtsev–Petviashvili (KP) equation applies. When the sidewall slope is O(1), an asymptotic wall boundary condition is derived, which involves a single parameter, [Ascr ] = A/h02, where A is the area under the depth profile. This model is used to discuss the development of an undular bore in a channel with trapezoidal cross-section. The theoretical predictions are in quantitative agreement with experiments and confirm the presence of significant three-dimensional effects, not accounted for by previous theories. Furthermore, the response due to transcritical forcing is investigated for 0 < [Ascr ] [les ] 1; the nature of the generated three-dimensional upstream disturbance depends on [Ascr ] crucially, and is related to the three-dimensional structure of periodic nonlinear waves of permanent form. Finally, in an Appendix, the appropriate asymptotic wall boundary condition is derived for the case when the sidewall slope is O(h0/w)½.