= Sedimenting rectangular-channel characteristic compatibility
{title2=$dx/dt=u,\ u\pm\sqrt{bh}$}
With <reduced gravity> $b$ and <settling velocity> $V_s$, a mixed rectangular-channel current obeys $D h=-hu_x$, $D u=-bh_x-hb_x/2$, $D b=-V_sb/h$, where $D=\partial_t+u\partial_x$. The <characteristic speeds> are $u,u\pm c$, with $c^2=bh$. Along the latter curves,
$$
du\pm\frac ch\,dh\pm\frac h{2c}\,db=\mp\frac{V_sc}{2h}\,dt.
$$
The density differential matters: $u\pm2c$ are <Riemann invariants> only when the <reduced gravity> is constant without deposition.
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