Inverting both sides of the equation, applying a little algebra and a trignometric 

 identity yields, 



_3k = 2w r 2 cosh 2 (kd) i (^ 



3u> g sinh(2kd)+2kd 



Substituting equation (1+5) back into equation (1*3) gives us, 



3to e = x r 2mUcos9 - a) 2 Ucos 8 . 2a> m 2 cosh 2 (kd) i , ^ n 

 3o> gtanh(kd) g Sinh 2 (kd) s sinh ( 2kd ) + 2kd J 



Finally, by substituting equation (l+l) into equation (U6) and manipulating (U6) 

 algebraically, we get the Jacobian, for any water depth as 



9a) e _ 1 2g)Ucos6 n _ 2kd i / ^\ 



3a) gtanh(kd) sinh( 2kd)+2kd 



For deep water, kd > tt, the term in the bracket approaches unity, as does the 

 hyperbolic tangent. The Jacobian of the transformation in deep water then becomes, 



3<t) e _ ± _ 2mUcqs9 ( ^8 ) 



3oj g 



The angular wave frequency, in the fixed coordinate system, can be determined 



from the encountered angular wave frequency. Rearranging equation (U2), we can 

 find a) in terms of u) e by 



bu 2 + a) - uu = (l+2b) 



where b = - 



Ucos e 



gtanh ( kd ) 

 The quadratic of to is 



- 1 ± /I + hht 

 0) = — 



1 (U9) 



2b 



As mentioned earlier, current effects may be included in the first analysis 

 section with mean wave directions, but has not yet been implemented in the second 



19 



