Evolution of weakly nonlinear short waves riding on long gravity waves

A nonlinear Schrödinger equation, describing the evolution of a weakly nonlinear short gravity wavetrain riding on a longer finite-amplitude gravity wavetrain, is derived. This equation is then used to predict the steady envelope of the short wavetrain relative to the long wavetrain. It is found that approximate analytical solutions agree very well with numerical solutions over a realistic range of wave steepness. The solutions are compared with corresponding studies of the modulation of linear short waves by Longuet-Higgins & Stewart (1960) and Longuet-Higgins (1987). We find that the effect of the nonlinearity of the short waves is to increase the modulation of their wavenumber, significantly reduce the modulation of their amplitude, and reduce the modulation of their slope when compared with the predictions of Longuet-Higgins (1987) for linear short waves on finite-amplitude long waves. The question of the stability of these steady solutions remains open but may be addressed by solutions of this nonlinear Schrödinger equation.