0.007 



- 



/ 



C\ 









OOM 



- 



/ 



\ 







1 



aooe 



- 



/ 



\ 









O00« 



- 





\ 





V \*°* 





aoo3 





A 



V 



L. 



V** 







0.001 



"1 



A«- 



\ 







. 



0.001 



1 



'A 



^ 



\ 





5^ 





1 1 1 



e 







1 





z 



3 4 



KviTv 



Fig. 2 Coefficient of roll-exciting moment for ellipsoid a2/ai = 



1/14, ai/ai = l/7,b/ai = 2/7, for various angles of incidence 



Mz) 



U "J 



For illustration we shall compute the roll-moment 

 coefficient 



C, = 



X, 



Sir pgAaia^ciiDt cos oit 

 = -K'iai^ - ai') sin /Se"^* 



q' 



(26) 



in the case of the two elUpsoids for which damping com- 

 putations were given in [2] ; the first of these has a beam- 

 length ratio oz/oi = 1/7, a beam-depth, ratio a^/az = 2, 

 and a depth of submergence h = 2 a^, or equal to the 

 beam. The second ellipsoid has the beam and the depth 

 interchanged, or ai/ai = 1/14, aa/oi = 1/7, h = 4a2. 

 Curves of the coefficient C4 for various angles of inci- 

 dence /3 are shown in Figs. 1 and 2, as functions of 



In view of the definition of C4, the curves for ,8 = 90 deg 

 or beam waves can be considered as independent of the 

 length 2a,, for any ellipsoid with the given beam-depth 



ratios and depth of submergence. Fig. 2 also shows the 

 critical angle /3 = 83 deg where the coefficient C* ceases 

 to oscillate about zero. 



Relation Between Damping and Exciting Forces 



The fact that the exciting forces can be determined 

 from the far-field asymptotic behavior of the radiation 

 potentials <pj implies a relation between tlie damping 

 forces and the exciting forces, since it is well known that 

 the damping forces can be found from energy radiation at 

 infinity. In fact, for an arbitrary three-dimensional body 

 at zero speed, the s'ix principal damping coefficients Bjj 

 are given by the integrals [2 ] 



TTOl 



Jo 



■du 



(27) 



where for the particular body considered, the functions 

 Pj characterize the far-field potential, in accordance with 

 equation (17). 



The functions Pj can be replaced by the exciting forces 

 Xj, from equation (23). It is convenient for this purpose 

 to define the exciting-force amplitudes X/°', where 



Xj = X /">€'' 



Then, from (23), 



Pj(^) 



^PqA 



gfA^^W (fi) 



(28) 



(29) 



where X/"'(|8) denotes the exciting force amplitude for 

 waves at an angle of incidence |8. With this notation, it 

 follows from (27) that 



B,, = 



oK 



r'^'" 



m\'d^ 



(30) 



'i-npg^A^ , 



Thus the damping coefficients Bj, are proportional to 

 the integrals of the squares of the corresponding exciting- 

 force amplitudes, integrated over all angles of incidence. 

 This relation is valid for an arbitrary three-dimensional 

 floating or submerged body, since its derivation only 

 requires that the far-field potentials ipj be of the form (17), 

 with the functions Pj corresponding to the particular 

 body under consideration. 



Equation (30) allows us to compute the damping 

 coefficients, if wie know the exciting forces for waves of 

 all angles of incidence. However in practice it is more 

 likely that one may desire the inverse, i.e., given the 

 damping coefficients, can we find the exciting forces? In 

 general this is not possible, for the damping coefficients 

 are constants while the exciting forces depend on the 

 angle of incidence. One exception is for bodies of revolu- 

 tion with a vertical axis of symmetry, such as a sphere or 

 spar buoy. Then clearly the heave exciting force is in- 

 dependent of the angle of incidence, /3, while the remain- 

 ing nonzero exciting forces will depend linearly on cos j3 

 or sin /3. Thus, for example, for surge, 



Xi<»>(/S) = X,<»'(0) cos /3 



and from (30) it follows thpt 



