Phillips 



^ a • V 

 b = y — sin X, + b , (8b) 



where x^. = k^ • x- n^t + e^., and G and b represent the small forced (nonreso- 

 nant) products of the interaction. The "amplitudes" a^ of the three primary 

 waves will be supposed to be slowly varying functions of time, in general, as 

 energy is exchanged in the interaction. The phase angle e^ may also vary 

 slowly as a result of the interaction. Equations that express the variations of 

 these quantities are obtained by the substitution of Eqs. (8) into Eq. (2). The 

 linear terms on the left are proportional to cos x^ and sin X^, and the quadratic 

 terms on the right generate quantities proportional to cos (X^. + X^) and 

 cos (X^-Xg), some of which, in view of the resonance conditions (6), have the 

 same spatial and temporal behavior as the primary waves themselves. Since 



X1 + X2 = X3+ e , e = £1+ ^2- ^3 ' 



then, in particular, the terms with wavenumber k^ and frequency n^ must bal- 

 ance. This condition leads, after some algebra, to 



kj^ (-2nj ttj •»/ + e'jaj-i/ + 2e j^a^^-v) sin Xj 

 -kj^(2ejnjaj-i/ +a.^-v-k^a^'v) cos Xj 



- Gjaa-jCij cos (Xj - X3) = ^23^20.3 cos (Xj - e) , (9) 



where 



Grs = Gsr = (n,-n3)(k^-k3) -i/k^k, COS 0^^ cos 6^^ 



+ I (n^-n3)(k^-k3)2 (k^ cos 0^3 cos 6^ - k, cos e^^ cos ^) 



+ |n |(k^-k3)2 - [(k^-k^) -vl^ . (k^ cos 0^3 + k^ cos 6?3^) (10) 



(no summation), with 6^, 0^^ being defined by 



k. "^s = k^a^ cos e„. (11a) 



ttj. • y = a^ cos 6^ . (lib) 



In general, in the weak interaction, the time scale of the variation of the 

 wave amplitudes is long compared with the wave period: 



laj/njaj « 1 , 



538 



