Denoting the water surface displacements at the two gage locations as 

 n(xi,yi,t) and n(x2,y2,t) and calculating the cross spectrum, Si2(n), 



Si2(n) = [ai(n) - ibi(n)] [azM - ib2(n)] ^22) 



where the overbar denotes the complex conjugate, 



Si2(n) = [ai(n)a2(n) +bi(n)b2(n)] - i[a2(n)bi(n) - ai(n)b2(n)] 



= ci2(n) - iqi2(n) (23) 



and ci2(n) and qi2(n) are the "cospectrum" and "quadspectrum," respectively of 

 ni and ri2- Si 2(^1) '^^'^ also be expressed as 



-i[k^(x2 - X2) cos a(n) + k^(y2 - Vi) sin aCn)] (24) 

 Si2(n) = F2(n)e 



and noting from Figure 16 that the separation distance and orientation of a 

 line joining the two gages are denoted by I and 3, respectively, S|2(n) 

 can be written as 



-ikjjJl cos [a(n) - B] 

 S,2(n) = F2(n)e 



(25) 



= F2(n) cos|kn£ cos [a(n) - B] | - iF^ (n) sinlkn^ cos [a(n) - 6] J 



Comparing equations (23) and (25), it is apparent that the wave direction, 

 a(n), relative to the x-axis can be expressed as 



r \ o ^ -1/1 -1 

 a(n) = 6 ± cos { tan 



>12 



(n)' 



12 



(n) 



(26) 



where the ± is the result of a directional ambiguity associated with a two- 

 gage array (Fig. 16) and is avoided by choosing the minus sign which selects 

 the wave arriving from the seaward half -plane adjacent to a line joining the 

 two gages. 



There are two conditions for which it was not possible to calculate the 

 wave direction, a(n): (1) poorly conditioned wave data, presumably due to 

 spectral leakage, and (2) spatial aliasing due to the fairly large separation 

 distance («;23 meters) between the two gages. If the first condition exists, 

 the absolute value of the quantity within the brackets { } in equation (26) may 

 exceed unity, clearly a physically impossible condition since the extreme values 

 of the cosine function are ±1. This tended to occur for very long waves for 

 which both the energy and the value of k^ were small. The percentage of 

 energy within an individual record when this condition occurred was relatively 

 small, averaging 2 to 3 percent with a maximum of approximately 10 percent. 

 The second condition requires that the wavelength be equal to or greater than 

 twice the projection of the wave gage separation distance in the direction of 

 wave propagation. Referring to Figure 16 



32 



