= Characteristic compatibility for an entraining triangular-channel current
{title2=$c^2=g'h/2$}
For positive depth and <reduced gravity>, the <characteristic speeds> are $u,u\pm c$. Along the material characteristic, $Dg'=-2g'w_e/h$. Along $D_\pm=\partial_t+(u\pm c)\partial_x$, the left-eigenvector compatibility equations are
$$
D_\pm u\pm\frac{2c}{h}D_\pm h\pm\frac{h}{3c}D_\pm g'
=-\frac{2w_e}{h}\left(u\mp\frac c3\right).
$$
If $w_e=0$ and $g'$ is spatially constant, these reduce to the <Riemann invariants> $u\pm4c$. Variable buoyancy or entrainment prevents that simplification.
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