S' (Jl,m,fo) = (zero where i? + m^ i' K^ 



a power density >_ for 9? + H = K^^. 



S'(£,m,fQ) thus defines power density at f = f^ for wave energy over 

 < 6 < Ztt. Figure 3 illustrates this case in wave number space. 



S(ji, m, fj 



S'(X,, m,, fj 



r + m^ = Kf 



FIGURE 3. DIRECTIONAL WAVE SPECTRUM AT A FIXED FREQUENCY, f 



We want to estimate the shape of S'(Jl,m,fQ) above the circle l^ + m'' 

 K in a directional wave train analysis. Remember, the S'(Jl,m,f ) 

 above is restricted to f >_ and is, in fact, equal to 



where 





if ri(x,y,t) is to be real. 



(3.2) 



Let us see how S' (£,m,f) might be found: we have said that 

 Ti(x,y,t) can be assumed to be a stationary Gaussian process. One char- 

 acteristic of such a process is that for fixed values (xq, y^, t^) qf_ 

 -the_pro£ess_ipdicas»_j5_Cx^i. y^.t^) is random ^y with a Gaussian 

 distribution; i.e.. 



?rek(>l(*..y*.0<>].)= jj:^^ e*^^^ i 



1 



