Nonsinusoidal Osailtations of a Cylinder in a Free Surface 



where 



h5(x) = - jo-,/(2g){^,(x,0)(.p2yy - Kg^gy) - 2(<P„'?'2x'^^ly*P2y)> 



- JO-2/(2g){^2(x,0)(<p,yy - K,<jP,y) - 2(<p,^^2x "^ ^ty'Pzy^^ <20) 



in which 



K,=^, 1 = 1,2. 



On the cylinder surface, 



«> (x ,y )f'{x ) - 0^ = Tnnlx ,y ) (21) 



where 



or, In terms of streann functions, 



^5(^0' Yo) = - J I (^lyK'Yo) + *2y)- (23) 



In the far field (p^y"" as y — *• - cx) and ^. should represent out- 

 going plane waves as |x| — ^ 00. Furthermore there is a symmetric- 

 flow condition which is expressed by ^ (x,y) = ^g(-x,y). 



3. 2 The Boundary- Value Problem for yg(x,y) 



^g is harmonic in y < except In the portion occupied by the 

 cylinder at its mean position. On the free surface 



where 



K« = (cr, - o-J /g. 



913 



