ON 3-DIMENSIONAL LONG WATER-WAVES IN A CHANNEL WITH SLOPING SIDEWALLS

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 h0w«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, A=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<A≤1 the nature of the generated three-dimensional upstream disturbance depends on si 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)1/2. © 1990, Cambridge University Press. All rights reserved.