and substituting this into (37)? we get, 



♦((ife,8)(— )du = <frU,e)do» (39) 



9u> 



where <|>(<o e ,9) is the wave spectrum in the encountered frequency demain, <J>( cd, 6) 

 is the wave spectrum in the fixed frequency domain, u) is the encountered 

 frequency expressed in radians per second, 6 is the angle between the buoy drift 

 and wave propagating directions, and to is the angular frequency in the fixed coor- 

 dinate system. 



9to e 

 The Jacobian of the transformation, , can be determined from the dispersion 



9(0 

 relation 



(o e = (o-kU cos 9 ( 1+0 ) 



and the angular wave frequency, 



u 2 = kg tanh(kd) (Hi) 



where U is the relative velocity between the moving and fixed coordinate systems, 

 and d is the water depth. Now, if we substitute equation (kl) into (1+0) and dif- 

 ferentiate with respect to co, we get the following 



p 

 a) Ucos 9 , , , 



(0 p = (0 ; (kp) 



e gtanh(kd) lH ^ ; 



and 



8( "e _ 1 r 2(pUcos9 _ (Q 2 d cos 9 9k i (^) 



9(o gtanh(kd) g sinh 2 (kd) 9u 



The inverse of the last term on the right hand side of equation (1+3) can be found by 

 implicitly differentiating equation (l+l) with respect to k, 



2a)— = gtanh(kd) + kgd 1 {hk] 



9k cosh 2 (kd) 



18 



