Nonsinusoidal Osoitlations of a Cylinder in a Free Surface 



Here Ej is the exponential integral defined by- 

 poo .f 

 E,(z)=^ V "^^ ^^'^ |arg(z)|<iT. 



Case 2: h(x) = Ae^*^ ''^' +0(l/x2) as |x|-*oo (34) 



where K' (=^ K) Is a wave number and A is a complex constant 

 with A = Ac + jAg. 



Before trying to solve for W, we Introduce a harmonic 

 function W|(x,y) which is even in x and satisfies 



W, (x,0) - KW, = Ae 



W,y(x,-oo) = 



jK'lxl 



and 



jK'lxl , , 



W| ~ Be as | x | — oo 



where B is a complex constant with B = Bg + jBg. The solution 

 for W| is found in Appendix C as 



w/ ^ Ae'^'y jK'lxl . A ^„. f e"^'' E| (JK'z) e'"^'' E,(-iK'z) -| , 

 W,(x,y) =KrTK^ "JV^^'L K+K ' ^ W^ J '^^^^ 



Now we let 



Wg = W - W| in y < 0. 



It can then be shown that 



jKlxl 



where 



and that 



WgyCx.O) - KWg = h(x) - Ae = h2(x) 



h2(x) = 0(l/x^) as I X I — CO 



Wg^Cx.-oo) = and W2(x,y) = W2(-x,y). 

 917 



