132 



where Un is the component of the Eulerian current in the direction of the nth wave. 

 C/„ = ^/u^+U~^ cos ("tan-i (^^ - Or^ (6.16) 



Amplitudes and Phases The amphtudes and phases the record are found by 

 separating the cosine and sine components so that the equations can be solved as a 

 linear least squares problem. 



Pd,c — O' cos [kxX + ky-y — ut -{■ a) 



= 6 cos [kxX -\- ky-y — mt) + csin {k^x + kyy — tot) 



(6.17) 



a = Vb'^ + (? 

 a = arctan {—c/b) 

 for the single wave method, and 



Pd.c — Ol COS (/Ca:,!^ + ky^iy — ut + Oi) + Gi COS {k:c,2X + ky^2y — tt)i + 03) 



= 61 COS {k:c,ix -\- ky^iy — ut) + Ci sin {k^^ix + ky^iy — ut) 

 +62 cos (A;a:,2^ + ky^2y — i^t) + C2 sin {k^,2X + ky^2y - ^t) 



«i= \/^? + c? (6.18) 



«2 = Y &2 + C^ 



q;i = arctan (—ci/6i) 

 02= arctan (—C2/62) 



Refining the Linear Estimates These rough estimates are refined by optimizing 

 for a best linear wave theory fit to the record: 



;v M 

 minimize(9(X) = J^ J^ {Pd,c,m,n - F^ (X; x„, y„, tm)f (6.19) 



n=l TO=1 



where: 



/^"^ (X; Xn, yn, tm) = CI COS (k^Xn + kyy^ - Utm + Oc) 



X = {kx,ky,a,a) 



K = Jkl-^kl 



