= Hydraulic flow in an inverted channel
{title2=$E=Q^2/(2W^2h^2)+g'h-g'H$}
A light shallow layer flows below a roof of height $H(x)$ and width $W(x)$ through a deeper approximately motionless ambient. <Hydrostatic equilibrium> and inviscid steady <Bernoulli equation> give the displayed specific <energy> for constant <volume flux> $Q$ and <reduced gravity> $g'$. At fixed position, $E_h=g'(1-\mathrm{Fr}^2)$ and $h_c=[Q^2/(g'W^2)]^{1/3}$. A roof depression raises the minimum admissible energy, just as a bed elevation does for a heavy lower layer.
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