= Shallow-water shock into stationary unit-depth water
{title2=$s=\sqrt{h(h+1)/2},\quad u=(h-1)\sqrt{(h+1)/(2h)}$}
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 $shu=hu^2+(h^2-1)/2$. For $h\ne1$ and $s>0$, eliminating $u$ gives the displayed formulas. The <characteristic speeds> are $u\pm\sqrt h$ and $\pm1$. 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.
Back to article page
