Passage through the critical Froude number for shallow-water waves over a variable bottom

Abstract

We study the behaviour of shallow (of order δ [Lt ] 1) water waves excited by a small (of order ε [Lt ] 1) amplitude bottom disturbance in the presence of a uniform oncoming flow with either constant or slowly varying Froude number F . When F * ≡ | F − 1|ε −½ [Gt ] 1, the speed and free surface perturbations are of order ν = O (ε); these grow to become of order ε ½ if F * = O (1). Therefore, the asymptotic expansions of the solution for ε → 0 depend on the order of F *. These expansions are constructed in a form which remains valid for times of order ν −1 ; they are then matched to provide results which are also valid for all F . The analytic results exhibit the interesting effects of weak nonlinearities including the steepening of waves and eventual formation of bores if δν −½ [Lt ] 1, the surface rippling due to dispersion if δν −½ = O (1), the strong interaction of waves and the periodic generation of upstreampropagating solitary waves if F * = O (1), etc. All these results are confirmed by numerical integration of the governing equations.