A nonlinear Schrodinger equation, a high-order wave-action conservation, and a nonlinear dispersion relation are derived through perturbation and variational principle methods, for describing the evolution of short waves riding on long waves. The formulations of the nonlinear Schrodinger equation derived through these two different approaches are found identical.
The nonlinear Schrodinger equation, the high-order wave-action conservation and the nonlinear dispersion relation are then used to predict the steady solution of a short-wave train riding on a long wave. When the wavelength ration (of the short wave to the long wave) is small (say 0.01), the predictions obtained either through the nonlinear Schrodinger equation or through the high-order wave-action conservation and the non-linear dispersion relation are almost identical. However, when the wavelength ratio is relatively large (say 0.1), only the solution through the high-order wave-action conservation and the nonlinear dispersion relation is accurate. The reason can be explained by the relationship among a Schrodinger equation, a wave-action conservation and a dispersion relation.
The steady solution of a nonlinear short-wave train riding on a long wave is then compared with that of a corresponding linear short-wave train (Longuet-Higgins 1987). We find that with the increase of the short-wave steepness, the modulation of the short wavelength increase slightly in the case of a small wavelength ratio, but significantly in the case of a relatively large wavelength ratio. However, with the increase of the short-wave steepness, the modulation of the short-wave (first-harmonic) amplitude ad(1) (accurate up to 0(e32)) decreases in the case of a small wavelength ratio, but increases in the case of a relatively large wavelength ratio. Therefore, it is expected that a stronger wave modulation occurs when a short-wave train rides on a relatively long wave.
Using the nonlinear Schrodinger equation, we study the side-band instability of a short-wave train riding on a long wave. It is found that with the increase of the long-wave steepness, both the resonant bandwidth of the normalized perturbation wave number and the normalized resonant growth increase; while with the increase of the short-wave steepness and the decrease of the wavelength ratio, both of them decrease. The numerical results of the instability are compared with the approximate analytic solution, and the agreement between them is satisfactory. The effects of the side- band instability of the short-wave train on the assumption of a steady short-wave envelope are investigated and their implication to the remote-sensing technique is explored.