Supercritical surface gravity waves generated by a positive forcing

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+λϵ 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 λ0 of λ. For λ⩾λ0 there are steady solutions, while for 0⩽λ<λ0 no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for λ>λ0, the solution of FKdV with zero initial condition tends to a stable steady solution, whilst for 0<λ<λ0 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.