major part of the rapid growth on the forward face of the 

 spectrum can be attributed to the nonlinear transfer. 



The basic mechanism of the process is illustrated in 

 Fig. 6. According to the classical theory of lowest order 

 Bragg scattering, an incident wave i can be refracted 



by a sinusoidal disturbance c (e.g., one of the har- 



monics of a periodic lattice, or a second wave component) 

 into a scattered wave s , provided some form of non- 



linear coupling exists between c and the incident and 



scattered waves, and the three components satisfy the Bragg 

 relations for constructive interference (resonance) 



o . (jj . + a (jj - M 

 11 c c s 



-=^~ (2) 



a.k. + a k = k {a.o = ±) 



11 C C S 1 c 



It can be shown that these conditions cannot be satis- 

 fied by three gravity wave components — or in general by 

 any three waves having the same c IspL-icion relation for 

 which the .l; •" ture d^u/dk^ is negative throughout. 

 However, Bragg scattering can occur at second order. On 

 account of weak nonlinear interactions, the spectrum of 

 surface displacement continues not only free-wave components 

 ( ^r k ) but also all combinations of quadratic har- 

 monics ( ui ± (j02f 3si * IS2 ) generated by pairs of free 

 waves ( (^1, JSi ), ( 0)2/ k2 ) (plus cubic and 

 higher order harmonics of correspondingly smaller amplitude). 



25-17 



