Salve sen 



The first-order complex potential for the given singularity distribution 

 (Eqs. (14)) and satisfying the condition given by Eq. (11) can be obtained from 

 Wehausen and Laitone (6a): 



10 10 



'^•^(z)= E^-tn (z-^). L^^n (z-^- 2ib) 



j=0 j=0 



10 

 j = 



L2^ 1 z - — - 2ib 



where the exponential integral l(0 is defined as 



!(„.. P.V.J _ 



du - i77 e" 



(17) 



(18a) 



and has the expansion 



I(0 - -e-^ [y + Ui+ i77 + ]2 — 



(18b) 



where Euler's constant is 7 = 0.5772 .... 



In accordance with the assumption that the cylinder -wall condition will be 

 satisfied only to the first-order approximation, it follows from Eq. (12) that the 

 second-order complex potential vi'-^^z), must satisfy the conditions 



where 



1. V^ w( 2) = , 



2. Re 



7'(''> 2 



- 1 I.., (1)1 



;(2) 



- Im 



pu 



f (x) at z = X + ib , 



.(1) 



Re 



z z z 



(19a) 

 (19b) 



3. lim wl^^ = 



as Im z -» -00 , 



(19c) 



4. 1 im w 



( 2) 



as Re z -» -co . 



(19d) 



This is exactly the same differential system as for the linear problem of a 

 fixed pressure distribution f(x) on the free surface of a uniform stream of ve- 

 locity U. Thus f(x) may be interpreted as a pressure distribution due to the 

 first-order wave system. 



602 



