Prismatic triangular-channel shallow water equations

For fixed channel width $b(z)=\beta z$, cross-sectional area is $A=\beta h^2/2$. With reduced gravity $G\varphi$, uniform cross-sectional fields and downward settling velocity magnitude $w$, volume conservation, momentum and particle deposition flux giveThe hydrostatic pressure force contains ${\int }_0^h(h-z)\beta zdz=\beta h^3/6$. No streamwise widening term occurs in a prismatic channel. Deposition uses the projected boundary width $\beta h$ under the vertical-settling model.

For the prismatic triangular-channel shallow water equations, $c^2=G\varphi h/2$. The characteristic speeds are $u,u\pm c$. Along $dx/dt=u$, $d\varphi /dt=-2w\varphi /h$. Along $dx/dt=u\pm c$, the hyperbolic system has compatibility relationThe concentration differential cannot generally be discarded to obtain Riemann invariants $u\pm 4c$. Positive depth and concentration are required for this nondegenerate form.