= Hydraulic control over a smooth hump
{title2=$D+h+Q^2/(2gh^2)=\text{constant}$}
A slowly varying, steady, inviscid channel flow has constant discharge $Q=uh$ per unit width and the displayed <Bernoulli equation>. The bed-height function of depth has its maximum at $h_c=(Q^2/g)^{1/3}$, where the <Froude number> is one. At a nondegenerate critical crest, a differentiable profile changes between the <subcritical flow> and <supercritical flow> branches. With subcritical upstream squared Froude number $F<1$ and upstream depth $h_1$, the shock-free supercritical downstream depth is $h_1(F+\sqrt{F^2+8F})/4$.
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