The definitive sketch of the problem is given in Figure 16, where an 

 arbitrarily shaped floating object is moored with elastic lines. The spring 

 constants, the initial tension, and the number of attachment points of the 

 mooring lines are also arbitrary. The center of gravity, G, and the center 

 of buoyancy, B, are shown at hydrostatic conditions. Because of the inci- 

 dent wave, the object is assumed to undergo a small amplitude oscillation 

 resulting in an alteration of the center of buoyancy and the metacentric 

 height. The fluid motion is assumed to be inviscid and irrotational, so there 

 is a velocity potential which satisfies the Laplace equation. The floating 

 object reflects and transmits the incident wave and creates a local standing- 

 wave system near the object, usually called scattered waves. The potential 

 function for the scattered waves was expanded into an infinite series of 

 scattered wave terms which decays exponentially with distance from the object. 

 If imaginary boundaries are taken sufficiently away from the object to elimi- 

 nate the effects of the local standing wave, it can be assumed that only the 

 incident and reflected waves exist in the region (0), and that only the 

 transmitted waves exist in the region (0') (see Fig. 16). The potential 

 functions for these regions are straightforward. The potential function for 

 region (I) (Fig. 16) was determined from Green's identity formula and the 

 equations of motion for the floating structure. 



-~ X 



W'l-J.-h ) 



E (c ojl O 



W( *,-h) 



Figure 16. Definitive sketch of moored floating object (after 

 Yamamoto, Yoshida, and Ijima, 1980). 



Yamamoto, Yoshida, and Ijima (1980) considered only the two-dimensional 

 problem; the values of forces, moments, volumes, and spring constants have the 

 dimensions per unit length of the floating body. At a given instant, the 

 location of the center of gravity, G, and the rotation, 9, of the body are 

 given by 



,iu)t 



x„ - x„ = Xe J 



(23) 



Ze- 



6 = 0e J 



(24) 

 (25) 



where oj is the incident wave frequency, and X, Z, and are the complex 

 amplitudes of the three modes of motion to be determined. From Newton's sec- 

 ond law of motion, the equations of the floating body for sway, heave, and 

 roll modes are given as 



45 



