= Hydraulic control in a variable-width channel
{title2=$H_x=h_c b_x/b$}
For a smooth rectangular channel, conservation of discharge and <Bernoulli equation> head gives $(1-F^2)h_x=F^2h b_x/b-H_x$. A regular <hydraulic control> has $F=1$ and the displayed geometric compatibility condition. At fixed discharge the critical-head envelope is $T(x)=H(x)+3h_c(x)/2$, with $h_c=(Q^2/(gb^2))^{1/3}$. A simple smooth control requires a local maximum of $T$: $T_x=0$ and $T_{xx}<0$. A local minimum cannot be crossed at the critical head because the real depth branches disappear nearby.
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