Internal Wave Interactions 



SO that the terms on the left side of Eq, (9) of the second order in the time de- 

 rivative can be neglected. This then reduces to the pair of first-order equations 



2n J k j^ cosfi'jO.j = "Gjjttja^ sin e, (12a) 



2n J k j^ cos t'jttjej = ~ G23 cij a^ cos e. (12b) 



Similar pairs of equations specifying the rate of change of amplitude and phase 

 of the other two components can be found in a similar way. They are 



(13a) 



2n2k2^ cos (^2°^ 2'^ 2 ~ ~^3i'^3°'^i ^'^^ ^ (13b) 



and 



2n3 kj^ cos Q -^0-2, ~ ^i2°^i^2 ^i" ^ • (14a.) 



2n3 k3^ cos d ^a^'e ^ - ~ ^X2°'\'^2 ^^^ ^ ' (l^b) 



where 



«rs = -("r+"s)('^r + ''s) ' ^ •< r ^ s ^OS 6^^ COS 9^^ 



+ 9 (nr + "s)('*r+'^s)^ (^r '^"S 6^^ COS 6^ + k^ COS 0^^ COS 6 J 



.In 



(k^+kj2 _ [(k^+k^) -u] 



■ (k^ COS 0^^ + k^ COS 9^^ . (15) 



The individual phase angles e^.ej.ej can be eliminated from the second of each 

 of these sets, giving 



■2 °-i °-i °-i '^l ^^2 



-G.3 ,/ , ^°3. ,/ ■ -".. ,.' ' . (16) 



\ kj COS -^ (9 J kj COS ■'6' 2 ^{' cos'' 6^/ 



since n^. = N cos 6^. This, together with 



2Nkj^ cos^i^j o-j = -G23 sin e a^ a3 , (17a) 



2Nk2^ cos^i92 a^ - -G31 sin e aj aj , (l^b) 



2Nk3^ cos^b'j a^ - \{^^ sin e a^a.^, (17c) 



provide four equations for the four unknowns, e the relative phase and the three 

 amplitudes aj,a2,a3 of the wave components involved. They are characteristic 



539 



