EQUATIONS OF DAMPED MOTION 



In the previous section, we considered only buoyancy and accelera- 

 tion forces on the spars because, generally, these were the forces of 

 Icwest order in the small parameter a(z). At resonance, these forces 

 cancel each other, thus they are no longer the lowest order forces. We 

 must re-examine the previous analysis and include terms of a higher 

 order in a-(z) to obtain equations of motion which have meaning at and 

 near resonance. 



The boundary condition, Equation [V] , was valid only to first order 

 in a-(z). If we now^ include second-order quantities (in a.(z)) in the bound- 

 ary condition. Equation [7], and add the necessary corresponding terms in 

 the potential function. Equation [8], we simply obtain more terms in the 

 acceleration and buoyancy forces. Since these terms are much smaller 

 than those already considered, they can alter the response only slightly, 

 principally by changing the resonance frequencies somew^hat. They still 

 contribute no damping forces. 



Nevertheless, the desired damping forces can be obtained from the 

 second summation in the potential function. Equation [8] . These terms 

 were discarded earlier because they contributed forces of higher order 

 than those being considered. It is easily seen, ho-wever, that these terms 

 do lead to damping forces, which will be the lowest order forces at reso- 

 nance. We also see that the terms in this second sunn which involve the in- 

 cident wave amplitude A do not contribute damping effects. They simply 

 affect the driving force, again by an amount of higher order. 



So now we consider the potential 



N Q 

 $ = TTGok y < a- [zQa.'+ aQ a.' ( o- sin 9. - p cos 9.)] 



i=l 



+ af [xo+ (3^ - \ao sin 9-] ^ 



8x. 



+ af[yQ- C.C+ YaQCOs 9.] J_\ [ e'^^^i^^^ Jq (k R. )] d^ 



22 



