Piston-driven shallow-water bore ($F^2{v}^2=1+\frac{3}{2(v-1)}+\frac{1}{2(v-1{)}^2}$)

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 $F=U/\sqrt{g{h}_0}$ and $v=V/U$. The moving Rankine-Hugoniot conditions give $F^{2}v=[(H/h_0{)}^2-1]/2$, 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 $F>0$.