Saint-Venant equation ($h_t+(hu{)}_x=0,u_t+u{u}_x=-g{\eta }_x-{\tau }_b/(\rho h)$)

The one-dimensional open-channel shallow water equations express conservation of water volume and depth-averaged momentum. Here $h$ is water depth, $u$ the mean velocity, $\eta$ the free-surface elevation and ${\tau }_b$ the resisting bed shear stress. The elevation includes the bed profile, so its gradient already includes gravitational driving by bed slope. A constant volume flux in space and time forces $h_t=0$ through the first equation; an unsteady reduction that imposes that flux while retaining momentum relaxation is consequently an approximation to this full system.

This scalar quasilinear partial differential equation results by imposing $h=Q/u$ in a frictional river momentum equation and measuring $X$ upstream. It retains a relaxation toward $u_0=Q^{1/3}$ and carries information at speed $c(u)=Q/(F^2{u}^2)-u$. Near equilibrium the relaxation time is $\epsilon F^2{u}_0/3$. A slowly moving downstream boundary can generate an attached boundary layer when $c(u)$ exceeds its speed throughout the relaxation. The model retains momentum dynamics only: exactly constant flux in both original Saint-Venant equations would instead force time-independent depth.