A note on the growth of the spectrum of wind-generated gravity waves as determined by non-linear considerations

Abstract

Non-linear wave properties lead to the conclusion, according to Phillips, that “where non-linear effects are important” the spectrum of wind-generated gravity waves must have the form S(μ) = αg2ω−5. To second order, it has been shown by Tick that non-linear effects are important in producing sharp crests above a frequency corresponding to twice the frequency of the dominant peak in the wave spectrum and that the second-order contribution is due to contributions from the linear part obtained from both sum and difference frequencies. In the representation due to Tick it is possible to represent the wave spectrum as a first-order part, S1 (μ, υ), and a second-order correction, S2 (μ, υ). If these results are accepted, and if several additional assumptions are made, it is shown that the first-order family of wave spectra, S1 (μ, υ), for fully developed seas, cannot form a nested family of curves. The results suggest that the various theoretical forms for the first-order spectrum of a fully developed sea are each correct over a limited range of wind speeds. However, since they are nested curves, they cannot be correct over the whole range of sea states.