De-Saint Venant equation

The equation of motion (conservation of momentum) for one-dimensional, unsteady flow in open channels. The system of equations representing the conservation of momentum and mass is given by

where Q is the discharge, A is the cross section area, g is the acceleration of gravity, y is the height of the water surface above the lowest point in the cross section, x is the distance along the channel, t is time, q is the lateral inflow per unit length of the channel, S_f is the friction slope, and S₀ is the bottom slope of the channel (positive with decline downstream).