Sedimenting rectangular-channel characteristic compatibility ($dx/dt=u,u\pm \sqrt{bh}$)

With reduced gravity $b$ and settling velocity $V_s$, a mixed rectangular-channel current obeys $Dh=-h{u}_x$, $Du=-b{h}_x-h{b}_x/2$, $Db=-V_{s}b/h$, where $D={\partial }_t+u{\partial }_x$. The characteristic speeds are $u,u\pm c$, with $c^2=bh$. Along the latter curves,The density differential matters: $u\pm 2c$ are Riemann invariants only when the reduced gravity is constant without deposition.