Water surfaces profiles for open channel flow: non linear solution and method of singular point

The different possibilities of transition from a supercritical to a subcritical regime, and vice–versa, of open–channel flows with variable slope are usually analysed with the method of singular points of first–order ordinary differential equations. The method, introduced by H. Poincaré and developed by P. Massé, in the hydraulic field, enables to define – within the bounds of linear treatment – the pattern followed by flow profiles near the critical state condition. A different approach to the same problem is given by replacing the first–order ordinary differential equation which regulates the motion of linear flows with a third–order equation, referring to the motion of non–linear flows. This equation is achieved by adopting the hypothesis by J. V. Boussinesq, which allows to take into account the effect of vertical speed components for the same flow section. The equation of non linear motion for the study of the condition analysed by P. Massé with the singular point method has enabled to highlight details which otherwise could not be defined through the linear treatment.