Stationary, transcritical channel flow

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Abstract
A compact (streamwise scale small compared with characteristic length) pressure distribution, which models a ship and is equivalent to a compact bottom deformation of cross-sectional area A,exerts a net vertical force ρgA on, and advances with speed U over, the free surface of a shallow canal of upstream depth H. The hypotheses of weak dispersion, weak nonlinearity and steady, two-dimensional flow in the reference frame of the force yield, through a generalization of Rayleigh's (1876) formulation of the (free) solitary-wave problem, a cnoidal wave downstream of the force matched to a null solution on the upstream side if , at least under the restrictions of weak dispersion and weak nonlinearity. Comparisons with experiment and numerical solutions of the nonlinear initial-value problem provide some confirmation of the cusped solitary wave but suggest that the cnoidal wave may be unstable in the absence of dissipation.
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