2 2 2 

 + oj„ A^ sin (k2X - oj t + 6„) 



2 2 2 

 + 00. A, cos (k,Xj^ - uj^t + 6^) 



+ 2a).cj2A,A„ cos(k X - u) t + 6^)cos(k2X - ^>^^ + 62) 



2 2 2 

 + 0)^ A„ cos (k_x - OJ t + 62) . 



Using the trigonometric relationships: 



2 2 2 2 2 2 2 



gn (x,t) = 0), A, sin (k,x - co^t + 6^^) + ^2 A2 sin (k2X - ^s^^ + ^^ 



1 2 

 + J 1^1 ^1^2 ^'^°s[(k - k2)x - (wj^ - t02)t + <5jl - <52] 



- cos[(k + k2)x - (toj^ + a)2)t + "5^ + 52]"} 



1 2 



+ y w A^A2 {cos[(k^ - k2)x - (oj^ - ^^t + 5^ - 62] 



- cos[Ck + k^)x - (co + 0)2^^ '*" "^1 "^ '^2-''^ 



1 2 2 2 2 



- 2" {t^i A + 0)2 A2 } - 0)^^0)2 A.,^A2 cos[(k^ - k2) x 



• - (ooj^ - a)2)t+ "S^ - (S2]- 



Combining further: 



gn'^^-'(x,t) = - \^^^^ cos[2{k^x - cj^t + 6^}] 



- \ ^^k^ cos[2{k2X - ao2t + 62}] (E-IS) 



12 2 



+ J ((jj - 2u) 0)2 + (^2 )A^A2 cos[(k^ - k2)x - (o)^ - ^^X "^ "^1 " ^2-'' 



which is the final form for the second-order term for free-surface ele- 

 vation. 



153 



