Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 90 / 1 / b / Solution

Differentiate the Bernoulli equation at fixed discharge, using $u_x/u=-b_x/b-h_x/h$. This givesAt a smooth hydraulic control, $F=1$ and the right-hand side must vanish. For the specified geometry,For the intended positive parameters $\beta ,\eta ,a>0$, positive finite depth requiresThe control lies between the bed summit and the width minimum, rather than necessarily at either one. For $a<0$ the corresponding interval is $a<x_c<0$. If $a=0$, the coincident summit and throat admit $x_c=0$ without imposing a depth through this first-derivative relation. The limiting cases $\beta =0$ or $\eta =0$ must be handled before division: their control is at the bed summit or throat, respectively.

At $x_c=a/2$, the critical-depth formula simplifies toThis establishes the requested depth without assuming that the bed summit and throat coincide.

There is a further regularity qualification if a critical candidate is to be a simple, smooth transcritical hydraulic control. At fixed $Q$, define the critical-head envelopeReal depths require $\mathcal{H}\ge T(x)$. At a control $\mathcal{H}=T(x_c)$, so $T$ must have a local maximum, not a local minimum. The first condition $T_x=0$ is precisely the compatibility equation above, andFor the positive-parameter case, a nondegenerate control therefore additionally satisfiesIndeed $T_{xx}=-2\eta [3a+\beta (a-x_c{)}^2(3a-10x_c)]/[3(a-x_c)b(x_c)]$. The interval $0<x_c<a$ gives all positive-depth candidates, while this inequality excludes candidates with no smooth critical crossing. At $a/2$ it becomes $\beta a^2<6$ for a simple control; the requested depth relation remains necessary whenever such a control exists. Equality is a degenerate critical point requiring higher-order analysis. For example, $a=1$, $\beta =16$ makes the midpoint a local minimum of $T$, so positivity of the depth alone would incorrectly identify it as a regular control.