The demonstrated importance of nonlinear energy transfer spurred further 

 investigation. Numerical study of the nonlinear interactions for a Pierson- 

 Moskowitz spectrum by Webb (1978) provided new insight. An imaginative 

 approach to computing the strength of nonlinear energy transfer within the 

 peak of a narrow spectrum was given in a paper by Longuet-Higgins (1976) and 

 a companion paper by Fox ( 1976) . Longuet-Higgins made use of the nonlinear 

 Schrodinger equation that describes the envelope of a weekly nonlinear wave 

 train. Both Longuet-Higgins and Fox concluded that nonlinear transfer within 

 the peak is of dominant importance; however, contrary to observations, it 

 tends to broaden the spectral peak and reduce spectral asymmetry. 



Although the derivations of Hasselmann (1962, 1963) include finite as well 

 as infinite depth, the finite-depth application had not been explored further 

 until recently (Shemdin, et al., 1978, 1980; Herterich and Hasselmann, 1980). 

 This application appears promising for future investigation. 



The stability of a wave spectrum to small oblique perturbations has been 

 considered by Alber (1978), Alber and Saffman (1978), and Crawford, Saffman, 

 and Yuen (1980). A range of conditions giving rise to instability in a 

 Gaussian random surface wave train is identified. Instability is found to 

 exist for a sufficiently narrow spectrum and for sufficiently small perturba- 

 tion wave angles. The effect of randomness is to reduce the importance of 

 instability. These studies are a counterpart to the deterministic approach of 

 Benjamin and Feir (1967) discussed below. 



Most of the above work pertains to random-phase Gaussian sea states. With 

 few exceptions, the descriptions of nonlinear energy transfer represent phase- 

 averaged exchanges. 



b. Evolution of Nonlinear Wave Train with Significant Steepness . A 

 deterministic approach to the evolution of a sea state was taken by Benjamin 

 and Feir (1967) and Benjamin (1967). The studies centered on the unexpected 

 discovery of Benjamin and Feir that finite-amplitude, progressive waves in 

 deep water (Stokes-type waves) are unstable to small perturbations at certain 

 frequencies . The instability was found in the laboratory and also shown 

 theoretically . 



The BF instability accounts for the unbounded growth of initially small 

 perturbations, and it provides insight on the ultimate disintegration of a 

 coherent wave train. 



Another study at about the same time dealt with large, but extremely grad- 

 ual perturbations in a deepwater wave train (Lighthill, 1965). The study also 

 revealed evidence of instability for a wave packet. However, the solution 

 became singular in finite time and could not predict the ultimate evolution. 



The BF analysis shows that the frequency and amplitude dispersion terms 

 required to maintain a steep wave of permanent form give rise to unbounded 

 growth of perturbations with frequency f ± "^^o' where f is the carrier 

 frequency, when 6 is in the range 



< 6 < ^2 ak (1) 



15 



