= Hydraulic control with prescribed seepage
{title2=$Q(x)=Q_0-Sx,\quad E=h+Q(x)^2/(2gb(x)^2h^2)$}
For loss of fluid carrying its local axial velocity, the inviscid <Bernoulli equation> head stays constant while discharge decreases. On a flat bed a regular control requires a positive local maximum of $q=Q/b$. In the parabolic-width channel $b=b_0(1+x^2/L^2)$, this maximum occurs at $x_c=-SL^2/(Q_0+\sqrt{Q_0^2+S^2L^2})$, with $q_c=(Q_0+\sqrt{Q_0^2+S^2L^2})/(2b_0)$. Its critical depth is $(q_c^2/g)^{1/3}$. A control exists only when the prescribed total head equals the maximum critical-head envelope and the reach includes that point while retaining positive discharge.
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