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

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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 ε½ 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.