= Solution
Assume a steady, incompressible, <inviscid flow> with a nearly uniform <velocity> $u(x)$ across each water section. The channel varies slowly enough for vertical accelerations to be negligible, <pressure> to be hydrostatic and the water surface to be nearly horizontal across the width. Atmospheric <pressure> is uniform; neglect <surface tension>, wall friction and energy losses. The hydraulic equations apply away from jumps and short nonhydrostatic regions.
The <Euler equation> for horizontal momentum, with <hydrostatic pressure>, gives
$$
u\frac{du}{dx}=-g\frac{d(H+h)}{dx}.
$$
Integrating gives a constant <Bernoulli equation> head $\mathcal B=H+h+u^2/(2g)$. <Conservation of mass> gives the constant volume discharge $Q=bhu$. Thus the <shallow-water specific energy>, measured above the local bed, is
$$
\boxed{E(h;x)=h+\frac{Q^2}{2gb(x)^2h^2},\qquad H(x)+E(h;x)=\mathcal B.}
$$
Define the <Froude number> by $F=u/\sqrt{gh}$ for the positive flow direction. Then
$$
\boxed{E=h(1+F^2/2),\qquad \left.\frac{\partial E}{\partial h}\right|_{Q,b}=1-F^2.}
$$
At fixed discharge and width, energy tends to infinity at both zero and infinite depth. It has a unique minimum at
$$
\boxed{h_c=\left(\frac{Q^2}{gb^2}\right)^{1/3},\quad F=1,\quad E_c=\frac32h_c.}
$$
Indeed $E_{hh}=3Q^2/(gb^2h^4)>0$. Above the minimum there are two depth branches: the shallower one is <supercritical flow> and the deeper one is <subcritical flow>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-83-specific-energy.png]
{title=Specific-energy minimum and the subcritical and supercritical depth branches}
The local long-wave propagation speeds from the <shallow water equations> are $u\pm\sqrt{gh}$. In subcritical positive flow, one wave family travels upstream and one downstream. In <supercritical flow> both travel downstream. At critical flow the upstream-directed speed is zero, so the flow can provide <hydraulic control> separating upstream adjustment from downstream conditions. This statement concerns long waves in the slowly varying hydraulic approximation, rather than all short-wave disturbances.
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