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 + lambda epsilon with epsilon > 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 lambda >= 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 lambda(0) of lambda. For lambda >= lambda(0) there are steady solutions, while for 0 <= lambda < lambda(0) no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for lambda > lambda(0) the solution of FKdV with zero initial condition tends to a stable steady, upstream solution, whilst for 0 < lambda < lambda(0) a succession of solitary waves are periodically generated and continuously propagating as time evolves. Moreover. the solutions of FKdV equation with nonzero initial conditions are studied. (C) 2008 Elsevier Masson SAS. All rights reserved.