High‐Wave‐Number/Frequency Attenuation of Wind‐Wave Spectra

Properties of surface‐elevation spectra representative of nonlinear deep‐water wind waves are examined theoretically. For a wave field characterized by second‐order nonlinearities, expressions describing the spatial covariance and the directional wave‐number spectrum of the surface geometry are derived. The nature of these quantities are then examined with emphasis on the high‐wave‐number attenuation of spectral amplitudes. It is found that, if the spectrum of the firstorder linear wave field decays as $k^{-p}$ toward the high‐wave‐number extreme, then the spectrum of the nonlinear wave field must decay as $K^{-p+2}\text{.}$ This condition coupled with the saturation/equilibrium range concepts is shown to necessitate the existence of certain upper‐limit asymptotes to the high‐frequency attenuation of linear‐wave spectra. Practical implications of this result are explored with reference to low‐pass filtering of wave records, and the representation of Gaussian sea waves based on various empirical and/or theoretical forms of wind‐wave spectra.