the spar radii are small enough compared to wavelength and spar sepa- 

 rations that slender body theory may be used. Equations of nnotion are 

 derived on these bases. These equations predict motions which are un- 

 damped; thus they are valid only for frequencies which are not near the 

 resonance frequencies. 



Near the resonance frequencies, it is necessary to consider the 

 damping due to wave generation, and this report shows that forces of high- 

 er order in terms of spar radii must be included. The leading damping 

 forces are found, thus providing equations of motion valid near resonance. 



In addition, this report indicates that viscous forces depend linearly 

 on the velocities for axial motions, and these forces are found explicitly. 



GEOMETRY AND COORDINATES 



It is convenient to define several coordinate systems. With the 

 structure floating at rest, we place the origin of a space-fixed reference 

 frame at the undisturbed free surface over the center of gravity of the 

 structure. Let the Cartesian coordinates of a point in this system be 

 (x,y,z), with the z-axis directed upwards. In this sanne system we de- 

 fine cylindrical coordinates (R, 9, z): 



r2 = x^ + y^ ; e = tan"^ - 

 ^ X 



X = R cos 9 ; y = R sin 9 



z is here the sanne as the Cartesian coordinate z. 



Let the undisturbed axis of the i spar be located at R = aQ, 9 = 9^. 

 We define another set of space-fixed coordinates, (x^, y^, z^), with origin 

 at R = aQ, 9 = 9., z = 9. Let the cylindrical coordinates of a point in this 

 system be (R^, \-^, z^), with the latter having the same orientation as the 

 previous cylindrical system. 



In the undisturbed condition, the surface of the i spar will be 

 specified by the equation: 



