calculation. The result can well illustrate the effect of three- 

 dimensionality of the fluid motion which is remarkable at lower frequencies. 



In the first place, let us put § = e $ and express the two- 

 dimensional solution for a heaving cylinder in the form as 



i — °° 



(2D) 

 ' - - a. 



kz cos ky j, . Kz 

 e — J dk-7Tie cos Ky 



_ 



y a 2m 



^L/ (2m-l) ! 



m=l 



,2m-l 



v 2m-2 



+K 



„ 2m-l 2 2 J . 2m-2 I 2^ 2 



dz \ z +y / dz \ z +y 



(156) 



It is readily shown that this expression satisfies the free surface 

 condition 



_3$ 

 dz 



- K $ = atz=0 



(157) 



where K = V = co /g in the present case. Since the inner expansion of the 

 above function is 



,(2D) 



= a n [£nKr+Y+Kr cos (l-£nKr-Y)+Kr9 sin 6+TTi] 



Z 



m=l 



,2m 

 2m 



Kt 



cos 2m6+ -= — r- cos (2m-l)6 

 zm-1 



(158) 



where we have employed the cylindrical coordinates 



z = - r cos b, y = r sin 



(159) 



and Y is Euler's constant, 0.5772157. 



In order to make the inner expansion of the far field potential match the 



above, we employ the expression in the near field as 



59 



