Shallow-water shock into stationary unit-depth water ($s=\sqrt{h(h+1)/2},u=(h-1)\sqrt{(h+1)/(2h)}$)

For the dimensionless shallow water equations with gravity coefficient one, let the left state be $(h,u)$, $h>0$, and the right state $(1,0)$. The Rankine-Hugoniot conditions are $s(h-1)=hu$ and $shu=h{u}^2+(h^2-1)/2$. For $h\ne 1$ and $s>0$, eliminating $u$ gives the displayed formulas. The characteristic speeds are $u\pm \sqrt{h}$ and $\pm 1$. Because $u-\sqrt{h}<s$ always, the positive branch can only be an admissible family-plus shock. Its remaining Lax shock admissibility conditions, $1<s<u+\sqrt{h}$, both reduce to $h>1$. At $h=1$ there is no jump and the labelled interface has no determined shock speed.