Since 6 may be as large as 1/2, phase values computed from the 

 Finite Fourier Transform may be in error by as much as 90°. 



For a simplistic simulation of a wave train, it is sufficient to com- 

 bine three sinusoids with nearby periods propagating in the same direc- 

 tion. Letting A^ equal the amplitudes and k^, i - 1,2,3, the wave 

 numbers, the Fourier Transform of the combination is given by: 



_ ^ k-i sin ir6^ cos ($-z; - tt6-2^) 



i^j 2TrCm^ - m + 6^) [(m^ - m + 5^)'^ - 1] 



^ A7* sin tt6^ sinf<|i^ - ttS-/") 



hm= i: :-^ ^ ^ C^-IO) 



i=l 2tt(i?1£ - m + 6^)[Cm^ - m + 6^)^ - 1] 



Nearby periods are attained by setting 



A^ = (% - m) < 3 . (C-11) 



Let the coordinates of three nearby locations be (xj,yj)5 J = 1,2,3. 

 The only difference among the Fourier Transforms (eq. C-10) arising from 

 wave records at each location is the values of the ^^^'s. At each loca- 

 tion j, the $^ values are: 



*ij ~ ^i^'^n ^'^^ '^ "^ ^j ^^^ ^) ~ ^i > i = 1,2,3 , (C-12) 



H^ = k^f^j - (j)i 



where 



9.J = Xj cos a + y^-sina, j = 1,2,3. (C-13) 



Since the three sinusoids are assumed to have nearby periods, let 



ki = k2 = k3 = k . 

 Thus, 



Hj = ^^j - H- 



Then, for the wave record at location j : 



•5 Av sin it6^ cos (kf^^- - ^6^ - c))^) 



a^Ty = 2 *« :; ; — . 



i=l 2TTCAi + 6i)[(Ai + 6i)2 _ 1] 



3 A,' sin 7r6^- sinfkfi^- - Tr6,- - ^j} 



\) . - V ^ (C-14) 



"^ " i=l 27rCA^ -. 6^)[CA^ -H 5^)2 - 1] 



50 



