Stationary spectra of surface gravity waves in a finite-depth sea

The known analytic studies of stationary solutions to the four-wave kinetic equation for the spectra of surface gravity waves are restricted to the ca se of deep water. This study extends these results to the case of a finite water depth h. Our analysis is based on a simplified version of the kinetic equation, namely, on the directional approximation, which is valid for nar row angular spectra of waves. A fundamental difference is found in stationa ry spectra between gravity waves in deep and finite-depth water. In the dee p-water approximation, these spectra have a nonintegrable singularity as k --> 0, and the divergence of all integral dynamic characteristics of gravit y waves (in particular, of their energy) correspond to these spectra. The l ong-wavelength cutoff of deep-water spectra is achieved only in the context of a nonstationary model. Allowance for a finite depth h leads to the long -wavelength cutoff of the stationary spectra of gravity waves and to their finite integral energy.