1/2 

 have assumed that the Froude number is of the order of £ , we expand 



the far field potential which has been given in the preceding section by U 



and discard terms of higher order. The result is 



§1 (x) = - 2 J dm ^| ) sgn(x-x') £n(2|x-x'|) dx' 



+ ^v|m(x , ) [H (v|x-x'|)-Y (v|x-x'|)] dx' 



+ 2TTiflf^l H (v|x-x'|)-Y (v|x-x'|) 



+v|x-x'| I -H-j^Cvlx-x' |)+Y 1 (v|x-x' |) dx' 

 + 2 TT i v(W)fm(x') e 2iV " (x - X,) H< 2) (v|x-x' | ) dx' 



- 4 tt V nfmCx') e 2iVfi(x_x,) H< 2) (v |x-x' | ) dx' (133) 



2 

 where V = to /g 



fl = to U/g 



H n = Struve function 



(2) 

 Y ,Y ,H = Bessel functions of the second and third kinds 



Because of the term zg (x) , the boundary value problem becomes a little 

 different from the pure two-dimensional solution for (j) . The most 

 remarkable feature of this case is that the two-dimensional solution is 

 related to the free surface condition 



(2D) (2D) 



- V cfA Z ; = (134) 



dz 



instead of that for the double body flow given in the preceding section. 

 This is the consequence of taking the second term of the perturbation 



51 



