Reverting to the C^j , Qj[^ notation of Equation (4.13), we have, where 

 X. ■= -Xij and Yj. = -Y-lj , Cj^^ = C^j and Qj^ = -q±y The numerical 

 form of Equation (4.15) then becomes 



s(.,m.f) = b^. Cii(f) + a £ .IbJC (4) cos[^^((llx^.+ M.Y. )] 



Choice of b^^ values is arbitrary. A reasonable choice is b^. = 

 [N(N-l) + 1]~^ for all (i,j) (refer to following section). We now-'have 

 a basis for an approximation of S(J2,,m,f). Before exploiting this result, 

 we need a few side results. 



5. SPECIAL CROSS SPECTRAL MATRICES 



Assume that we have a real sea wave of frequency f > moving 

 from direction Qq where i^ = KgCoseQ and m = K^Sin Oq, Kq being the 

 wave number from Equation (3.1). We can write the wave as 



rt (X,H,1) — ACos(lTr(iTt + >n.'a -^Lt^af)) (5.1) 



Since the root-mean-square (rms) value of a cosine wave is A//27 we have 

 for the two-sided (-o° < f < «>) directional power spectrum of the wave in 

 Equation (5.1) 



or in polar form where K = / £ +in ; S = arctan(— ) 



+ i(e-le.-Tt))J(f*+,)]J(K-K.) 



18 



