where x is the space coordinate along the channel axis, t denotes time, A (x, t) is the cross-sectional area of the flow at location x, u (x, t) is the flow velocity, ζ (x, t) is the free surface elevation and τ(x, t) is the wall shear stress along the wetted perimeter P (x, t) of the cross section at x. Further ρ is the (constant) fluid density and g is the gravitational acceleration.