Nonsinusoidal Osailtations of a Cylinder in a Free Surface 



solutions for the first-order potentials were given by Ursell [ 1949] , 

 Tassai [ 1959] , Porter [ I960] , and Frank [ 1967] among others. 

 The solutions for three of the second-order potentials, (^y <|)^, and 

 <^_, were given by Paras sis [ 1966] and Lee [ 1966] . In the present 

 investigation, the solutions for the remaining second-order potentials, 

 <^5 and <j)6, will be given. 



These solutions are based on the method of multipole expan- 

 sions similar to that employed by Lee [ 1966] . An interesting prob- 

 lem arising from the present work is a surface-wave problem con- 

 cerning a non-decaying pressure distribution on the free surface. 

 The solution of this pressure-distribution problem is shown in detail 

 in Appendix C. The potential <|>g, associated with the difference fre- 

 quency, is of particular significance in practical problems. <J>6 is 

 a potential which is slowly-varying in time if two fundamental fre- 

 quencies are close. In the case of bodies with insignificant restoring 

 forces, such as submersibles and floating platforms, any hydro dy- 

 namic force, which is constant in time or varies slowly with tinme , 

 could cause large excursions from the mean positions of such bodies 

 if it acts for a long time. 4>6 ^nust be calculated in order to deter- 

 mine this slowly-varying hydrodynamic force. 



In the present work numerical results obtained from the solu- 

 tion of <|>6 are shown. These include the pressure-distribution about 

 a semi-submierged circular cylinder, the hydrodynamic force acting 

 on it, and the outgoing waves. These results are shown with other 

 first- and second-order quantities for conaparison purposes . 



II. FORMULATION OF THE PROBLEM 



A Cartesian coordinate system is used with origin at the 

 interaction of the undisturbed free surface and the vertical line of 

 sym^metry of the cylinder. The x-axls Is in the undisturbed free 

 surface and the y-axis is directed upward. 



Any point in the space is described in complex notation by 



z = X + iy = re . (1) 



The region outside the cylinder and the cylinder boundary Is mapped 

 from the region outside a circle In the t,-plane and Its circumference 

 by the conformal transformation 



n=0 

 ; = e + i^ = Xe'*" , \ > 1 , (3) 



907 



