At a given section, Q = VA; thus equation 1 becomes
Because the hydraulic depth D = A/T and _A = T _Y, the above equation may be written
The above equations are all forms of the continuity equation for unsteady flow in open channels. For a rectangular channel or a channel of infinite width, equation 1 may be written
where q is the discharge per unit width.
Equation of Motion
In a steady, uniform flow, the gradient, , of the total energy line is equal in magnitude to the "friction slope" =/(CR) where c is the chezy coefficient and r is the hydraulic radius. Indeed this statement was in a sense taken as the definition of ; however in the present context we have to consider the more general case in which the flow is nonuniform and the velocity may be changing in the downstream direction. The net force, shear force and pressure force, is no longer zero, since the flow is accelerating. Therefore, the equation of motion becomes