Note on the undular jump

Abstract

A study is made of the asymptotic equations governing gravity waves on water which cause a transition from one surface level to a slightly higher one and approach a steady wave-form as time increases, at least at the head of the wave. A two-parameter family of limit processes is surveyed in each of which the time scale and the horizontal length scale tend to infinity in a definite relation to the amplitude, as that tends to zero. Small-amplitude linearization is shown to be possible at most during a transitory stage of the wave development. Arbitrarily close approach to steadiness at the head of the wave is found to imply that a substantial part of the transition wave must be ultimately governed by a nearsteady variant of the non-linear equation of Korteweg & de Vries (1895) and must take the form of a train of cnoidal waves characterized by a parameter which changes slightly from crest to crest.