= Piston-driven shallow-water bore
{title2=$F^2v^2=1+\frac{3}{2(v-1)}+\frac1{2(v-1)^2}$}
A wall moving at constant <speed> $U>0$ into stationary water of depth $h_0$ produces a <hydraulic bore> of speed $V>U$ in the uniform-state <shallow water equations> model. Behind the bore the fluid <velocity> equals $U$ and its depth $H$ obeys $H/h_0=V/(V-U)$. Put $F=U/\sqrt{gh_0}$ and $v=V/U$. The moving <Rankine-Hugoniot conditions> give $F^2v=[(H/h_0)^2-1]/2$, hence the displayed equation for $v>1$. Its left side is a <strictly increasing function> from $F^2$ to infinity, while its right side is a <strictly decreasing function> from infinity to one. The <intermediate value theorem> therefore gives exactly one admissible $V$ for every $F>0$.
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