= Finite-reservoir-head draining free streamline
{title2=$a=\pi/(\pi+2)$}
For a planar reservoir draining below a vertical lip of height one into a unit-speed stream, the potential strip $0<\operatorname{Im}w<a$ and first-quadrant unit velocity disk satisfy
$$
\frac1{1+e^{\pi w/a}}=\left(\frac{q^2-1}{q^2+1}\right)^2.
$$
On the <fluid free boundary> put $q=e^{-i\theta}$, $-\pi/2\leq\theta<0$. Then $dz/d\theta=-2ae^{i\theta}\cot\theta/\pi$, so the downstream height is $1-2a/\pi$. Equating it to $a$ gives $a=\pi/(\pi+2)$.
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