CO ,00 



f /^ 



(8.11) lim 



N->oo ^-00 ^-00 ^ -^ ^-^ 



i[A*(a.p)]' 



co8[x'(a + c*) + y'(p + j3*)] 



+ cos [x'(a = a*) + y'{p- |3*)]j dx' dy' dc d(3 



►CO ^00 



/ 



= lim 



N-*0 '-c» '-OD 



[A*(a,(3)]' 



sin^(a+<z*) sin^(P+P*) sin-|(a-c») sin^ (p- p*) ] 

 (a + c*)(p + p*) (o - a*)(p - p*) J 



When Dirichlet's formula is applied, the result is finally that equation 

 (8. 12) is obtained. 



(8.12) [A*(#:^]^ + [A*{-c*,-p*)f =-^ 



it'' 



CO / 00 



1 



Q(x',y')cos(a*x' + P*y') dx' dy' 



00 '-CX) 



Note that substituting -a* and -p* for o* and p* leaves equation (8. 12) un- 

 changed. 



Equations (8. 9) and (8. 12) are the x, y plane analogues of the classical 

 time series equations which state that the covariance function is the Fourier 

 transform of the power spectrum and conversely that the Fourier transform 

 of the power spectrum is the covariance function. When applied to tl:.e problem 

 of finding the directional spectrum of a wind generated sea from discrete data, 

 the integrals have to be replaced by summations and formulas analogous to 

 the Tukey formulas for time series have to be derived. 



In deriving the equations used to determine the directional spectrum, the 

 analogy to the one-dimensional case given by Tukey [1949] will be shown. The 



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