Denoting hereafter the FFT coefficients for the water surface displacement 

 as a(n) and b(n), it is noted that the coefficients have the following prop- 

 erties: 



-^ N-1 „ „ 

 n2 = Z [a2(n) + b2(n)] 

 n=l 



(17) 





''N + k = "^^ - k 

 2 + k 2 '^ 



(18) 



and thus 



N/2 



n^ = 2 E [a2(n) + b^ (n) ] 

 n=l 



(19) 



Therefore, the total (kinetic plus potential) energy, E(n) , associated with 

 a particular frequency component, n, is 



(20) 



E(n) = 2Y[a^(n) = b^(n)] 



For wave direction, consider the definition sketch in Figure 16 and the 

 following representation for ri(x,y,tj): 



N — [kjjCn) cos a(n) cos a (n) + ky(n) sin a(n) - e (n) ] 



Ti(x,y,t4) = Z F(n)e 

 n=l 



i2Trnj 



= Z [a(n) - ib(n)]e 

 n=l 



Gofc 1 ^ 



^- 1 (shoreword ) 



(21) 



Figure 16. The two-gage array notation and directional ambiguity. 



31 



