the special conditions that follow from N = 2 can easily be applied here to 

 obtain simple equations. The case is not considered sufficiently interest- 

 ing to warrant writing out the equations here. 



The heaving motion can be obtained immediately, if desired, since 

 the ZQ-equation contains no coupling terms. In addition, the equation for 

 rotation about the z-axis is not coupled to the other equations. However, 

 the Yq and a motions are coupled; also the Xq and (3 motions. Similarly, 

 the couplings are simple enough for these equations to be solved directly. 



We should note that there are two resonance frequencies. In heave, 

 there is resonance when 



2 _ pgS(O) 



M + Mq/N 



[17a] 



In either of the coupled motions there is resonance when 



2MgrpS^ + ipaQS(O)- MqZq/N J m(z.')z.' dz/ 



r 2 r^o/N^ r 2 12 2 



2Ml|MaQ + pS^ +|p. /j^|+ J m(z.')(z/) dz-' - p S^ 



17b] 



Since the equations contain no damping forces, infinite response ampli- 

 tudes are predicted when resonance occurs. Of course, this is nneaning- 

 less in the linearized model and so the above equations ([I7a] and [I7b]) 

 can be valid only in frequency ranges, not including neighborhoods of the 

 two exceptional frequencies. When such neighborhoods are excluded from 

 consideration, the predictions should be fairly accurate if the snnall ampli- 

 tude and slenderness restrictions are observed, since damping forces are 

 of higher order than the forces considered. Near the resonance frequen- 

 cies, however, the damping forces are important, even if small. This 

 problem is considered in the next section. 



21 



