1.Introduction. In this note we consider possible wave motions on the steady laminar flow of a thin liquid film down an inclined plane. A systematic expansion procedure is developed for the propagation of long waves of arbitrary amplitude on the film. Certain limiting situations, dependent on the relative importance of the nonlinearity, are capable of an analytical treatment. The results of a long wave linearized stability theory [1] [2] are recovered as special cases of the nonlinear analysis. Possible finite amplitude permanent wave solutions are found, and the weakly nonlinear stability problem is discussed for two dimensional waves.

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A uniform train of periodic waves may be unstable to large scale variations so that some incoherence can develop. The analysis is presented for the case of gravity waves, and for a class of interaction problems typical of many physical systems.

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