Resonant transcritical flow over a wavy bed

Near-resonant free-surface flows over a wavy bed of either finite or semi-infinite extent are considered. For a bed of finite extent wavy profile of a small amplitude, it is shown that the linear model is valid and a steady state exists even within the resonant regime, with an ''upstream influence''. If the bed extends semi-infinitely to far downstream, the linear model breaks down in the near field, where the motion is governed by a series of forced KdV equations, which are matched to the far-field linear solutions.