pitch oscillations. 



In deriving the hydrodynamic forces and moments acting on the body, 

 we shall assume that the incident waves and the oscillations of the body are 

 small, and thus we shall retain only terms of first order in these ampli- 

 tudes. We shall also assume that the body is slender. The analysis w^ith 

 only first-order terms in the body's diameter leads to undamiped resonance 

 oscillations of infinite amplitude. To analyze the motions near resonance, 

 it is necessary to introduce damping forces which are of second order with 

 respect to the diameter-length ratio. 



THE FIRST-ORDER VELOCITY POTENTIAL 



We shall consider the hydrodynamic problem of a floating slender 

 body of revolution with a vertical axis in the presence of snnall incident 

 surface waves. Let (x,y,z) be a fixed Cartesian coordinate system with 

 the z-axis positive upwards and the plane z = situated at the undisturbed 

 level of the free surface. The x-axis is taken to be the direction of propa- 

 gation of the incident wave system, and the motion of the body is assumed 

 to be confined to the plane y = 0. We shall also employ a coordinate sys- 

 tem (x',y',z') fixed in the body, with z' the axis of the body, so that with 

 the body at rest, (x, y , z) = (x' , y ' , z' ); and a circular cylindrical system 

 (r , 9, z), where x = r cos 6 and y=rsin6. If£,t,, and 4^ a-re the instan- 

 taneous amplitudes of surge, heave, and pitch, respectively, relative to 

 the body's center of gravity, it follows that 



X = ^ + x' cos ijj + (z' - Z/-) sin \\i 



y=y' [1] 



z = t, - x' sin \\) + (z' - Zp) cos ^ + zL 



where zL is the vertical coordinate of the center of gravity in the body- 

 fixed system; see Figure 1. The displacements ^, t,, and ijj are assumed 

 to be small oscillatory functions of time; we shall consistently linearize 

 by neglecting terms of second order in these functions or their products 

 with the incident wave amplitude A. Thus Equation [l] may be replaced by 



