The momentum equation for open-channel flow may be found by starting from the incompressible Navier-Stokes equations :where  is the pressure,  is the kinematic viscosity,  is the Laplace operator, and  is the gravitational potential. By invoking the high Reynolds number and 1D flow assumptions, we have the equations:<math display="block">\begin{aligned}
{\partial u\over{\partial t}} + u{\partial u\over{\partial x}} &= -{1\over{\rho}}{\partial p\over{\partial x}} + F_{x} \\
-{1\over{\rho}}{\partial p\over{\partial z}} - g &= 0
\end{aligned}, where the channel depth  is the difference between the free surface elevation  and the channel bottom . Substitution into the first equation gives:where the channel bed slope . To account for shear stress along the channel banks, we may define the force term to be:where  is the shear stress and  is the hydraulic radius. Defining the friction slope , a way of quantifying friction losses, leads to the final form of the momentum equation: