= Prismatic-channel shallow water equations
{title2=$A_t+(Au)_x=0,\quad(Au)_t+(Au^2+P)_x=0$}
In a channel whose transverse width $b(z)$ does not depend on downstream position, the wetted area is $A(h)=\int_0^h b(z)\,dz$ and the integrated <hydrostatic pressure> force divided by density is $P(h)=g\int_0^h(h-z)b(z)\,dz$. With cross-sectionally uniform velocity, <volume conservation> and <momentum conservation> give the displayed <shallow water equations>. Since $P'(h)=gA(h)$, the two <characteristic speeds> are $u\pm c$ with $c^2=gA/A'$. For $b(z)=Cz^p$, $c^2=gh/(p+1)$ and the <Riemann invariants> are $u\pm2(p+1)c$.
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