where a and k are the amplitude and wave number of the wave train. The 

 fastest growing instability corresponds to 



6 = ak (2) 



Thus a wave train which initially has significant energy only at frequencies 

 f and its higher harmonics (2fQ, 3f , etc.) can be expected to develop 

 strong concentrations of energy at sideband frequencies fgd + ak) and £„(! - 

 ak) . The energy in both sidebands is approximately equal due to their coupled 

 growth. 



With well-developed sidebands, the wave train can be considered as a mod- 

 ulated carrier wave. It was noted by Benjamin and Feir (1967) that pertur- 

 bations corresponding to values of 6 near zero in equation (1) represent 

 mainly phase modulation which gradually gives way to pure amplitude modulation 

 as 6 is increased to /2 ak. For the most unstable mode, given by equation 

 (2), phase and amplitude modulation are equal. The modulation time scale. 

 I.e., the time between wave groups in a wave train with well-developed side- 

 bands, is 



"~^ ■ ^ "> 



Both Benjamin (1967) and Whitham (1967) showed that the instability can 

 occur in finite depth, d, as well, on the condition that 



kd > 1.363 (4) 



Contrary to the phase-averaged nonlinear transfer discussed earlier, the 

 BF sideband interactions depend crucially on phase. 



The concept of a sea state as a perturbed carrier wave train has stimu- 

 lated theoretical study of the evolution of the envelope of such a train. 

 Early studies of evolution of the envelope of a train of weakly nonlinear 

 dispersive waves were done by Benney and Newell (1967) and Zakharov (1968). 

 Subsequently, Chu and Mei (1970, 1971) derived envelope equations which over- 

 came the singularity found by Lighthill ( 1965) , but which for other reasons 

 could not be extended to infinite time. 



Hasimoto and Ono (1972) derived the nonlinear Schrodinger equation in the 

 context of the envelope of water waves. The equation has been solved exactly 

 by Zakharov and Shabat (1972) for pulselike initial conditions which approach 

 zero sufficiently rapidly. The solution predicts that any initial pulse of 

 waves will eventually disintegrate into a series of wave packets, or solitons, 

 and a dispersive oscillatory tail. Each soliton is a permanent progressive 

 wave solution to the nonlinear Schrodinger equation. The solitons are stable 

 features which can pass through each other with no change of form except pos- 

 sibly a phase shift. Thus the nonlinear Schrodinger equation may be a tool 

 for predicting the ultimate evolution of a sea state. 



Laboratory data were given by Yuen and Lake (1975) to show that the non- 

 linear Schrodinger equation provides a useful quantitative description of the 

 long-term evolution of wave packets. 



16 



