On the Non–Linear Unsteady Water Flow in Open Channels

The procedures suggested by Boussinesq (1871, 1877), Lord Rayleigh (1876), Korteweg & de Vries (1895) for the study of the non–linear unsteady flow of water in open channels are recalled. It is emphasized that the hypotheses by which Boussinesq (1877) gained the well–known equation of second approximation do not contradict the d’Alembert principle provided that an average of the Eulerian equation at any cross section and at any time is performed. The conditions that allow the Cauchy relation to be associated to vectorial field used by Boussinesq (1877) are deduced. The mathematical equivalence between two procedures, the first adopted by Boussinesq (1871), Lord Rayleigh (1876) and Korteweg & de Vries (1895) for the study of the solitary wave, and the second proposed again by Boussinesq (1877) in order to recover the so called equation of second approximation is demonstrated.