Analytical integration of the kinematic equation for runoff on a plane under constant rainfall rate and Smith and Parlange infiltration

An analytical solution to the kinematic wave approximation of the Saint-Venant equations describing surface water flow with infiltration is presented. Adopting the Smith and Parlange formulation for the pending time and the postponding infiltration rate, a change of variable among time and infiltration rate allow the analytical integration of the characteristic equations when the flow rate is expressed as the second power of the water depth. In the case of any other real exponent a simple numerical integration method may be used for the computation of the characteristic curves. An illustration is made to compare the solutions for different values of the exponent, as well as to assess the accuracy of some numerical schemes.