Directional spectra of wind-generated waves

TL;DR: In this paper, the directional spectrum of wind-generated waves on deep water is determined by using a modification of Barber's (1963) method, and the results reveal that the frequency spectrum in the rear face is inversely proportional to the fourth power of the frequency.

Abstract: From observations of wind and of water surface elevation at 14 wave staffs in an array in Lake Ontario and in a large laboratory tank, the directional spectrum of wind-generated waves on deep water is determined by using a modification of Barber's (1963) method. Systematic investigations reveal the following: (a) the frequency spectrum in the rear face is inversely proportional to the fourth power of the frequency $\omega $, with the equilibrium range parameter and the peak enhancement factor clearly dependent on the ratio of wind speed to peak wave speed; (b) the angular spreading $\theta $ of the wave energy is of the form sech$^{2}$ ($\beta \theta $), where $\beta $ is a function of frequency relative to the peak; (c) depending on the gradient of the fetch, the direction of the waves at the spectral peak may differ from the mean wind direction by up to 50 degrees, but this observed difference is predictable by a similarity analysis; (d) under conditions of strong wind forcing, significant effects on the phase velocity caused by amplitude dispersion and the presence of bound harmonics are clearly observed and are in accordance with the Stokes theory, whereas (e) the waves under natural wind conditions show amplitude dispersion, but bound harmonics are too weak to be detected among the background of free waves.