= Solution
Assume the <shallow-water approximation>: the horizontal scale is long compared with the depth, the <hydrostatic approximation> applies, and the cross-section-averaged <velocity> is approximately uniform. Neglect viscosity and drag for this part. <Volume conservation> gives the constant discharge $Q=bhu$. The steady horizontal momentum equation is
$$
u u_x=-g(H+h)_x,
$$
so its integral is the <Bernoulli equation> head $\mathcal H=H+h+u^2/(2g)$. The <shallow-water specific energy> measured above the bed is therefore
$$
\boxed{E(h;Q,b)=h+\frac{Q^2}{2gb^2h^2},\qquad E+H=\mathcal H.}
$$
The <Froude number> compares the flow <velocity> with the long-wave speed $\sqrt{gh}$:
$$
\boxed{F=\frac{u}{\sqrt{gh}},\qquad E=h\left(1+\frac{F^2}{2}\right),\qquad
\left.\frac{\partial E}{\partial h}\right|_{Q,b}=1-F^2.}
$$
At fixed $Q,b$, the <shallow-water specific energy> tends to infinity as $h$ tends to zero or infinity. Its unique minimum occurs at the critical depth
$$
h_c=\left(\frac{Q^2}{gb^2}\right)^{1/3},\qquad F=1,\qquad E_c=\frac32h_c.
$$
The two depth branches above that minimum are <subcritical flow>, $F<1$ with $h>h_c$, and <supercritical flow>, $F>1$ with $h<h_c$. The long-wave <characteristic speeds> relative to the bed are $u\pm\sqrt{gh}$. Thus a <subcritical flow> can receive information from downstream, whereas both <characteristic curves> travel downstream in a positive <supercritical flow>.
A <hydraulic control> is a critical section where one <characteristic speed> vanishes and a smooth flow can pass between the two branches. Regularity there supplies a condition relating $Q$, the depth and the geometry; together with reservoir or inlet data, it can determine the discharge. Merely finding $F=1$ is not enough: a finite depth gradient also requires the geometric forcing to vanish in the critical momentum equation.
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