Kaplan 



where S Is the spar cross section area at the waterline intersection. 

 The mooring force will depend upon the mooring arrangement and 

 geometry, and is deleted temporarily from consideration. The wave 

 exciting force can be evaluated and the wave generation damping is 

 determined from work in [ 11] . 



The velocity potential and the pressure on a slender cixisym- 

 metric body in waves are found using the results of [ 12] for the case 

 of a vertically rising body in waves, at zero forward speed and 

 evaluated for the condition where only the submerged portion from 

 the waterline down is of interest. The fluid pressure on the body in 

 regular sinusoidal waves is (from [ 12] ) 



2-rrC/\, 4'irR -., * i a.\ /co\ 



p = pga e ( — r — cos 9 cos oot - sin cot) (58) 



where | and 9* are the longitudinal and angular coordinates of the 

 spar hull and R is the local hull radius. The local vertical force 

 on a section of the spar buoy is then 



^ = - 2R tahorl p d9» (59) 



dZ 



with tan a = dR/d^ (the slope of the body contour) , the local vertical 

 force is 



dZu. , ^ Z'tCA dS' 



-^ = pga sin cot • e -^ (60) 



where S' is the local hull cross section area, leading to 



.0 



z 



^ = pga sin cot \ e -^ d| (61) 



where k = Zir/X - ^ /%, which can be simplified further by integration 

 oy parts (with S'(L) = 0). 



The surge equation for the spar hull form is represented as 



mx = - p ^ S'(e)[x + (e - |g)e] de + Xd + X« + X^ (62) 



where ^g is the C. G. location along the ^-axis, and the wave force 

 is obtained as 



1058 



