Supercritical surface gravity waves generated by a positive forcing

Abstract

Forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small bump on a horizontal rigid flat bottom are studied. The wave motion on the free surface is determined by a nondimensional wave speed F, called Froude number, and F = 1 is a critical value of F. If F = 1 + λ ϵ with ϵ > 0 a small parameter, then a time-dependent forced Korteweg–de Vries (FKdV) equation can be derived to model the wave motion on the free surface. Here, the case λ ⩾ 0 (or F ⩾ 1 , called supercritical case) is considered. The steady FKdV equation is first studied both theoretically and numerically. It is shown that there exists a cut-off value λ 0 of λ. For λ ⩾ λ 0 there are steady solutions, while for 0 ⩽ λ < λ 0 no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for λ > λ 0 , the solution of FKdV with zero initial condition tends to a stable steady solution, whilst for 0 < λ < λ 0 a succession of solitary waves are periodically generated and continuously propagating upstream as time evolves. Moreover, the solutions of FKdV equation with nonzero initial conditions are studied.