Lee 



parts are needed, so whenever there appears an expression of the 

 product of two complex functions one of which is a time harmonic 

 involving with j = v- 1 , it should be understood that the real part of 

 the expression is to be taken. 



The convergence of these perturbation expansions will not be 

 discussed. As usual it is hoped that the first few terms of the ex- 

 pression would yield an adequate approximation to the exact solution 

 of the complex potential H(x,y,t)» 



The expansion given in Eqs,(17) and the Bernoulli equation (11) 

 suggest that we also assume the expansion 



Y(x,t) = e[Yj(x)e + Y^e ^] 



« -J2<T,t -jZo-gt 



+ €lY3(x)e +Y^e 



-J(o-, + 0-2)1 -j{o-,-<r2)t 



+ Yje ' ^ +Yge + Y^(x)] + O(e^) (18) 



where 



Yk=YKc + JY^3 for k=l,2..,.,6. 



Substituting these expansions into the Laplace equation and the boun- 

 dary conditions and equating the terms of the same order In e as 

 well as of the same harmonic time dependence, we obtain a set of 

 linear boundary-value problems. In this linearization process tfie 

 Instantaneous boundary of the fluid Is expanded In Taylor's series 

 about the undisturbed position of the fluid boundary. 



The linearized boundary conditions for the functions <pj 

 (j = 1,2,3,4,7) are shown In Appendix A where It is shown that In the 

 limiting case of a, = o-„ the relations ^, = (p^, <p^= <p^= <pJZ and 

 ^g = (p- can be established. These Identities mean that when o-| = 0-2 

 the perturbation expansion given in Eqs.(17) reduces to that for the 

 case of a simple harmonic oscillation which was Investigated by 

 Lee [ 1968] . 



The linearized boundary conditions for the functions <f, and 

 ^g are given next. 



3. 1 The Boundary- Value Problem for ^^(x.y) 



<p^ Is harmonic In y <0 except In the portion occupied by 

 the cylinder at Its mean position. On the free surface 



<p^y(x,0) - Kjj<p^=h^(x) (19) 



912 



