The deformation of steep surface waves on water ll. Growth of normal-mode instabilities

Studies of the normal-mode perturbations of steep gravity waves (Longuet-Higgins 1978 b, c) have suggested two distinct types of instability: at low wave steepnesses we find subharmonic instabilities with fairly low rates of growth, and at higher wave steepnesses there are apparently local (‘superharmonic’) instabilities leading directly to wave breaking. Between these two types of instability is an intermediate range of wave steepnesses where the unperturbed wave train is neutrally stable. In the present paper we employ the time-stepping method of an earlier paper (Longuet-Higgins & Cokelet 1976) to test the rate of growth of each type of instability. For the initial linear stages of each instability, the computed rates of growth are accurately confirmed, and it is verified that the local instability does indeed lead to breaking. The later nonlinear stages of the subharmonic instabilities are further investigated. In the two examples so far computed it is found that the gradual rates of growth of the subharmonic instabilities are maintained, and that ultimately every alternate crest develops a fast-growing local instability which quickly leads to breaking.