= Solution
Here $s$ is downstream distance, $A$ the wetted cross-sectional area, $u$ the section-averaged downstream <velocity>, and $M$ the lateral water input per channel length and time. Thus $A_t$ is local water storage and $(Au)_s$ the divergence of discharge. Their balance is <mass conservation>.
The momentum terms $u_t$ and $uu_s$ are local and advective acceleration. The bed slope $S$ gives the downstream gravitational acceleration $gS$. <Wall shear stress>[Bed shear stress] $\tau$ acting over wetted perimeter $l$ produces resisting force per unit mass $\tau l/(\rho A)$; the <hydraulic radius> is $R=A/l$. Finally $-g\overline h_s$ is the hydrostatic free-surface <pressure>-gradient term in the chosen bed-following description. The Saint-Venant approximation depth-averages the flow, assumes a long shallow geometry and <hydrostatic pressure>, and closes the boundary shear through a <surface roughness>[roughness] law.
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