there is resonance in pitch and surge. To determine the oscillation am- 

 plitudes in the vicinity of these resonance frequencies, it is necessary to 

 consider the damping mechanisnn due to energy dissipation in outgoing 

 ■waves. Thus, for these frequencies, ■we must consider the free-surface 

 effects on the restoring forces. For this purpose ■we must retain some 

 terms ■which are of second order in the radius of the body. 



THE DAMPING FORCES 



The damping forces ■will follo^w by considering the last terms in 

 Equations [13] to [l6] and ■will consequently be of higher order in R than 

 those terms ■which ■we retained in the previous analysis. This procedure 

 is nevertheless consistent, since at resonance the lo^wer order restoring 

 forces vanish. In other ■words, we are retaining the lo^west order force 

 or moment of each phase separately. For a further discussion of this 

 point, see Reference 4. 



We proceed, therefore, to study the danriping forces, or the forces 

 in phase ■with each velocity. The only contribution from Equations [13] 

 to [l6] is the potential 



4.* = TTuKe^^ j /t,R|^ + [e +4^{zi - Zc)]R^ g^je^^l jQ(Kr)dzi 



[39] 



Since J^(Kr) = 1 - ;f (Kr)^ + .... it follows that on the surface r = R{z), 

 the leading terms are 



Jo(Kr) S 1 



and 



— J„ (Kr) S _ i K^x = - 1 K^R cos 

 9x ^ 



Thus, to second order in R the damping potential on the body is 



14 



