Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 76 / 3 / d / Solution

Bed and sidewall stress, turbulent mixing at the interface, wave radiation and hydraulic jumps convert or remove mechanical energy. In the specified local head-loss model, differentiate $B=u^2/2+g^{\prime }h$ while keeping $Q=bhu$ constant:At a regular hydraulic control the coefficient of $h^{\prime }$ vanishes. Therefore $B^{\prime }=-\alpha$ requiresThus the control moves to the widening downstream side, $x_c>0$. This is the exact local compatibility condition; it is the head-loss correction to hydraulic control with loss expressed as Bernoulli potential per unit length.

For a small loss and a control remaining near the throat, use the lossless depth $h_c=2H/3+O(\alpha )$ and $b^{\prime }/b=2x_c/L^2+O(x_c^3/L^4)$. The leading displacement isFor a finite loss, the local derivative alone does not specify the accumulated loss from the far-upstream reservoir, so it does not fix a unique revised discharge or numerical control depth. Given $Q$, the exact condition can instead be solved implicitly for $x_c$; if the inviscid discharge is imposed, it becomes $\alpha =(4g^{\prime }H/3L)(x_c/L)[1+(x_c/L{)}^2{]}^{-5/3}$. The small-displacement root is the one continuous from the throat. This distinction avoids treating the original upstream energy as unchanged after an unspecified finite upstream loss.