The dispersion relation for a nonlinear random gravity wave field

The dispersion relation for a random gravity wave field is derived using the complete system of nonlinear equations. It is found that the generally accepted dispersion relation is only a first-order approximation to the mean value. The correction to this approximation is expressed in terms of the energy spectral function of the wave field. The non-zero mean deviation is proportional to the ratio of the mean Eulerian velocity at the surface and the local phase velocity. In addition to the mean deviation, there is a random scatter. The root-mean-square value of this scatter is proportional to the ratio of the root-mean-square surface velocity and the local phase velocity. As for the phase velocity, the nonzero mean deviation is equal to the mean Eulerian velocity while the root-mean-square scatter is equal to the root-mean-square surface velocity. Special cases are considered and a comparison with experimental data is also discussed.