= Dry-bed dam break with cubic wetted area
{title2=$c^2=gh/3,\quad J_\pm=u\pm6c$}
A channel with width $b(z)=\alpha z^2$ has wetted area $A=\alpha h^3/3$. Its inviscid <shallow water equations> have characteristic speeds $u\pm c$ and the displayed <Riemann invariants>. Releasing a reservoir of depth $H$ at rest into a dry channel gives a left-family <rarefaction wave>: $c=(6c_0-x/t)/7$, $u=6(c_0+x/t)/7$, $h/H=[(6-x/(c_0t))/7]^2$ between $x=-c_0t$ and $x=6c_0t$, where $c_0=\sqrt{gH/3}$. This width law differs from a geometrically parabolic trough, whose width is proportional to $\sqrt z$.
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