Lee 



order in a perturbation series in the ratio of motion amplitude to 

 half-beam. In the present work this cylinder-oscillation problem is 

 extended to the case where the cylinder is oscillated at the sum of 

 two monochromatic frequencies. In this case the second-order 

 forces acting upon the cylinder include the effects of interactions 

 between two frequencies, in particular , the sum and difference fre- 

 quencies of the basic spectrum. The magnitudes of these second- 

 order hydrodynamic quantities provide a measure of the non- 

 linearity of the frequency response of an inviscid incompressible 

 fluid to a periodic disturbance generated by an oscillating body in a 

 free surface. 



Hydrodynamic quantities such as added mass and damping 

 obtained from the theory of oscillating cylinders in a free surface 

 with a monochromatic frequency are extensively used in the studies 

 of ship motions. Most of these studies are based on the assumption 

 of linear frequency response of ships to waves. Recently Tasai 

 [ 1969] and Grim [ 1969] emphasized the necessity of further investi- 

 gation on nonlinear ship responses to waves. The present investiga- 

 tion is an attempt to provide information on the nonlinear relation 

 between the motions of a body and surrounding fluid. This informa- 

 tion might lead to the study of nonlinear ship motions in waves, 

 perhaps by using the scheme suggested by Hasselmann [ 1966] . 



The problem to be investigated in this work is the following. 

 An infinitely long horizontal cylinder which is symmetric about its 

 verticeil axis is semi-submerged and forced to oscillate vertically 

 at the sum of two monochromatic frequencies. The maximum dis- 

 placement of the cylinder from its mean position is assumed to be 

 small compared to the hgilf-beam of the cylinder. The fluid in which 

 the cylinder oscillates is assumed inviscid, incompressible, and 

 infinitely deep. The motion is assumed to have existed for a period 

 significantly long that the initial transient phenomenon of the response 

 of the fluid has completely decayed. This problem can be formulated 

 as a boundary-veilue problem for a velocity potential. The kinematic 

 and dynamic conditions to be satisfied on the free surface are non- 

 linear and the position of the free surface is a priori unknown. An 

 exact solution of this problem in a closed form cannot be attained, 

 so an approximate solution based on a linearization of the problem 

 is pursued in this work. The linearization of the problem is carried 

 out by a perturbation expansion of the velocity potential in terms of 

 a small parameter formed by the ratio of the half- be am to a typical 

 displacement amplitude of the cylinder motion. The first-order 

 perturbation potential consists of two potentials , 4>i(x,y,t) and ^^, 

 each of which involves only one of the two fundamental frequencies. 

 The second-order perturbation potential consists of five potentials. 

 Two of them are ^, and <|>^ which are associated respectively with 

 frequencies of twice the fundamental frequencies. Two more are 

 (}>- and <|>g which are associated respectively with the sum and the 

 difference of the fundamental frequencies, and the last one, <j)^(x,y), 

 Is Independent of the frequencies and Is a steady potential. The 



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