Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 71 / 3 / a / Solution

Treat the channel as prismatic, with horizontal bottom datum $z=0$, uniform cross-sectional velocity, and surface height $h(x,t)$. The area and depth-integrated hydrostatic pressure force divided by density areThe prismatic-channel shallow water equations areSince $P^{\prime }(h)=gA(h)$, they becomeTheir two characteristic speeds are $u\pm c$. Using $dc/dh=c/(2h)$ gives the Riemann invariantsThe factor $3$ replaces the familiar $2$ for a rectangular channel. It comes from the depth-dependent width, not from a different gravitational acceleration.