I 





















B«b«am 

 T. draft 

 H-B/2T 





1 



V^H.O 













\\\ 



V^H"! 













"\\ 



\\^H'Zli 













-\ 



'pT H»4 













- 



1 1 



1 



1 1 



1 



H- 



\ 



~1 





1 Z 



s 



4 



S 





6 





7 I 



Fig. 3 Coefficient of heave-exciting force for a floating ellipse Fig. 4 Coefficient of heave-exciting force for a floating ellipse 



or the amplitude of the heave exciting force, per unit 

 ampUtude of the incident wave, is 



A 



^A 



(42) 



This relation is of practical importance since the wave- 

 height ratio A has been computed [8-12] for various 

 floating cylindrical forms, including the circular cylinder, 

 ellipse, flat plate, and the so-called Lewis-forms and their 

 generalizations. 



Figs. 3 and 4 show the nondimensional heave-excit- 

 ing force coefficient 



C3 = 



^3'°' 



pgAB 



A. 

 KB 



where B = beam, for various elliptic cylindrical sections. 

 In all cases the coefficient A was obtained from the calcu- 

 lations of Porter fll]. In Fig. 3 the abscissa is the con- 

 ventional parameter, <iy'B/2g, while in Fig. 4 the abscissa 

 is u^T/g, with T = draft. Also shown in the latter 

 figure is the thin-ship theory cun'e for an ellipse, which 

 may be regarded as the limiting curve for an ellipse of 

 small beam-draft ratio, or as the thin-ship approximation 

 to an arbitrary ellipse. This thin-ship result corresponds 

 to a source distribution on the centerline of the ellipse, of 

 strength proportional to the normal velocity on the sur- 

 face of the ellipse. One obtains in this manner the ex- 

 pression 



Cz= 2,rlL-i(/C7') - h{KT)\ 



where L-i and /i are the modified Struve and Bessel 

 functions, respectively, defined by the series 



L-i(x) = E 



{x/2f 



r(n -I- -A) r(n -I-V2 ) 

 " ( V2)^"+' 



^'(^^ = „?onT(;my! 



It is evident in Fig. 4 that the dependence upon beam is 

 relatively small, and thus that the thin-ship result is a 

 fairly good approximation, at least for small or moderate 

 frequencies and moderate values of the beam-draft 

 ratio. 



The coefficient C3 is nondimensionalized with the force 

 pgAB, or the hydrostatic buoyancy force due to the wave 

 amplitude A . Thus in the limit of low frequency, or long 

 waves, Cz = 1.0. However Figs. 3 and 4 show that in 

 practice, this limit is a poor approximation since the 

 values of d fall off very rapidly for finite frequencies. 

 In the limit of high frequencies, it can be shown that 



m 



KT 



(1 + H), 



KT» 1 



for the ellipse of beam B and draft T, and H = B/2T. 

 Thus for large frequencies, 



