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Wind waves are known to have frequency spectra proportional to ω^4 for frequencies w higher than a few times the spectral peak frequency. This frequency spectrum leads to the wavenumber spectrum propo...rtional to k^-<5/2> via the linear dispersion relation k ex ω2, where k denotes the wavenumber. The field measurement by Banner et al., however, shows wavenumber spectra proportional to k^-3 in the saturated or equilibrium range, suggesting some dispersion relation other than that of linear waves. In order to examine this possibility, nonlinear dispersion relation is calculated for typical wind-wave spectra, on the basis of the third-order weakly nonlinear theory developed for irrotational gravity waves subject to the quasi-Gaussian, stationary, and homogeneous stochastic process. The theory predicts that nonlinear interaction increases the phase speed of waves propagating in the main direction. Waves propagating in the opposite direction, however, have reduced phase speeds on account of nonlinear interaction. That is, the nonlinear dispersion relation depends on (1) the location of the wave component in concern in the whole spectrum of the wavenumber-frequency space, (2) the degree of nonlinearity of wind-waves, and (3) the form of the twodimensional spectrum. Therefore no simple algebraic form of dispersion relation is expected for the entire domain of frequencies and propagation directions. It is shown nevertheless that, for waves propagating in the main direction, there is a significantly wide range of frequencies where the resulting nonlinear dispersion relation is approximated by k ∝ ω^<3/2> , which dispersion relation transforms ω^4 frequency spectra into k^-3 wavenumber spectra. Thus the nonlinear dispersion relation studied here makes the wavenumber spectrum measured by Banner et al. consistent with the well-established frequency spectrum, though within a certain wavenumber-frequency range that depends on the propagation direction, nonlinearity, and the spectral form.続きを見る
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