In this paper a method for obtaining periodical solutions of the nonlinear equations of motion is proposed. The method is applied to the case of a falling liquid film i n wave motion. It consists i n a triple series expansion; the first is a Taylor expansion, with respect to the distance y1 from the free surface; a second one, representing the periodicity condition, is a Fourier expansion with respect to the variable z = k(x-ctl, and a third one is a Taylor expansion with respect to the amplitude Zi$l. The calculation of the different coeficients i s made easy by the fact that the algebraic equations obtained are linear and do not simultaneously contain a l l the unknowns. This allows the performance of the computation step by step i n increasing order of the powers of $1. The periodicity condition allows the determination of a l l physical quantities as function3 of one of them. The amplitude 12i@i/ was selected as the parameter.
The existence of a dimensionless quantity for the wave flow is outlined. Arguments are adduced in support of the fact that the amplitude 12i$ll depends only on + and a universal curve 12i@l] vs. $ is plotted on the basis of experimental data. Theoretical equations for the wave length, the wave velocity and the film thickness as a function of $ are established.
There is good agreement between the theoretical equations and experiment.
If a liquid film flows down a vertical plate it would be expected that the motion be laminar. Experiment shows, however, that the free surface of the film is not plane (Figure ) , but is disturbed by an unsteady periodical motion (1 to 7). The occurrence of this wave motion is due to the fact that the laminar steady motion is not stable to small perturbations and for this reason a new type of motion, stable to perturbations, is organized. Yih ( 8 ) , Benjamin ( 9 ) , Hanratty and Herschman (1 0) , and Whitaker ( ) have shown, by means of the linearized theory of instability, that the Nusselt velocity distribution is not stable to small perturbations; this theory however, is not able to provide information concerning the behavior of the stable state. Such information has been obtained by Kapitza (12, Wui!!