 Solution

Differentiate the specific energy at fixed discharge:In the lossless flat channel $E$ is constant. A smooth hydraulic control has $F=1$, so regularity requires $b^{\prime }=0$. For the prescribed width this occurs at the minimum width $x=0$. The required minimum specific energy is largest at the narrowest section; a controlled flow supplies exactly that head.

Far upstream $b\to \infty$, so the velocity of a finite-discharge flow with depth tending to $H$ tends to zero. Thus the Bernoulli equation gives $E=H$. At the control $u_c^2=g^{\prime }{h}_c$ and $E=3h_c/2$, henceThe printed width normalization has $b(0)=1$. Downstream on the accelerating controlled branch the flow becomes shallower and supercritical.