Nonsinuo sidal Oscillations of a Cylinder in a Free Surface 



equation such as Eq. (19) or (24). The terna on the right-hand side 

 of the equation of the first-order free-surface condition represents 

 the pressure distribution on the free surface. Thus the problems 

 for the second-order potentials <p. and (p^ presented in Sections 3. 1 

 and 3. 2 are the same type of boundary- value problems as those for 

 the first-order potentials except for the "non-constant pressure" on 

 the free surface. If we assume there is no body in the fluid, then 

 these problems can be treated as problems for surface waves arising 

 from variable free-surface pressure distributions. Solutions for 

 these problems are given in Wehausen and Laitone [ I960] . If we 

 denote the velocity potential associated with the problem of variable 

 pressure distribution on the free surface by W and if we assume 

 that it is known, we can use it to find the potentials (pg and ^g. 

 This is done by introducing a new function G = <p - W, where <p 

 could be either <p^ or ^g,so that the free-surface condition for G 

 is given by a homogeneous equation such as Gy(x,0) - KG = where 

 K is either K5 or Kg. The boundary-value problem for G is 

 then identical to those for the first-order potentials and the solutions 

 to these are well known. Once G is known the solution for <p is 

 readily obtained from ^ = G + W. This scheme was used by Lee 

 [1968] to find the second-order potentials ^3 and <p^. However 

 there are certain requirements on the "free-surface pressure 

 functions,' h_ and hg given by Eqs. (20) and (25) respectively, 

 to be satisfied before the known methods can be used to 

 find the potential W. These requirements are that the functions 

 h5(x) and hg(x) should be absolutely integrable in (-00,00) and 

 should satisfy the HcJlder condition. Although the proof of these 

 statements may not seem obvious from Eqs. (20) and (25), it can be 

 shown that both hg and hg satisfy the Holder condition and as 

 |x| — ^ CO 



h5=0(l/x2) (29) 



and 



jl(K|-K2)lxl-i8} , - 



hg= aoe'^ '+0(l/x2) (30) 



where aj, and P are given by 



^o=^'Q,Q2K,K2(cr, -(T^) (30a) 



and 



P = q, - q2- 7r/2. (30b) 



Here the quantities Qj^ and q|^ for k = 1 ,2 are associated with the 



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