Phillips 



of those describing second-order resonant interactions, other examples having 

 been given by McGoldrick (7) and Thorpe (6). From arbitrary initial conditions, 

 the development in time can be traced and the full solution given in terms of 

 elliptic functions (as in the manner described by Bretherton (10), 



Some simple properties of the interaction can, however, be demonstrated 

 without recourse to the general solution. It is evident that there are two sta- 

 tionary solutions for the phase angle, namely, ±77/2, the value e = 77/2 being 

 stable if the coefficient in the large parentheses of Eq. (16) is of the opposite 

 sign to aja2a3, and the value e - -77/2 being stable if the coefficient is the same. 

 When any one of aj, a^, or a^ passes through zero, the relative phase can 

 change by 77. 



Two partition integrals follow immediately from Eqs. (17). From Eqs. (17a) 

 and (17c), 



(Hjjki^ cos2ei)aj2 + (G^sk^ 005253) a32 = const., (18) 



and similarly, from Eqs. (17b) and (17c), 



(Hijk/ cos252)'^2^ + (Gsika^ cos26>3)a32 = const. (19) 



Since a^^ represents the mean energy in the component with wavenumber k^. , 

 these evidently describe the partition of energy among the three component 

 modes. Not all of G23, G31, and Hjj can have the same sign. If, say, Hjj and 

 G23 do have the same sign, then as the energy of the k, component increases as 

 a result of the interaction, that of the kj and k^ components decreases; this 

 continues until (from Eqs. (17)) either a^ or a^ vanishes. At this point ij = 0; 

 the energy in the 1 component is a maximum and decreases at later times as 

 energy is transferred back to the 2 and 3 components. 



The conservation of total energy in the three wave components can be dem- 

 onstrated, either from these partition integrals, or directly from Eqs. (17): 



N •jT (a.j2 + a^ + a^) = - sin 



kj^ cos^e^ kj^ cos2 02 ^3^ cos^^j 



It is perhaps not obvious that the term containing the interaction coefficients on 

 the right vanishes identically. The proof of this involves some elementary but 

 tedious algebra and requires the use of the auxiliary conditions (6), a v - 0, 

 [a,k,i/] = 0, and the set 



kj cos ^12 = kj cos 6'3 2 , (20a) 



kj cos ^23 - ~kj COS i9j3, (20b) 



k3 COS 6^^ = kj COS £'21 ■ (20c) 



These last conditions are readily derived from 



540 



