= Saint-Venant equation
{c}
{title2=$h_t+(hu)_x=0,\quad u_t+uu_x=-g\eta_x-\tau_b/(\rho h)$}
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.
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