69 



At third order: 



(79 = (25i + 52] 



ail = {Si + 2S2] 



as = (352) 

 <7io = (25i - ^2) 

 CT12 = (5i - 252) 



And at fourth order: 



^i3 = (45i) 

 <Ti5 = (35i + 52) 

 (7iT = (25i+252) 



cri9 = (5i +352) 



^14 = (452) 



<Ti6 = (35i — 52) 

 (7i8 = (25i-252) 



<''20 = (5i — 352) 



(Ti,(T2,cr3,cr4,<77,(j8, (Ji3, and <Ti4 are self-wave interactions. The balance of the cTj- are 

 wave-wave interactions. 



Because the phases are arguments of the sine in the potential function, a sim- 

 ple expansion of Eq. 4.11 results in redundant terms that have been removed. For 

 example, if there are two components: 



Bi sin(5i — 52) and B2 sin(52 — 5i) 

 Because sine is an odd function: 



B2 sin(52 — 5i) = —B2 sin(5i - 52) 

 the two components can be combined into a single component: 

 (5i - B2)sin(5i - 52) = A,sin(5i - 52) 



(4.19) 



(4.20) 



(4.21) 



If both components were included, the effects of Bi and B2 would be indistinguishable 

 in the optimization; they must be combined into the single coefficient, Ai. 



This form for the potential function results in 28 unknowns at fourth order {2QAi 

 plus kxi ky, u, and a for each of the two intersecting waves) that must be found 



