ipA (l qr) e- ""+'"'+ "^'j^ ' (1 + cos u) 



.exp{-iKR(l - cos u)]Pi{u + ff)du (22) 



where in the last integral we have replaced the variable 

 by M = X + 9 — /3. Since the radius of the control 

 surface is large, KR » 1, and we may evaluate the 

 integral over u by the method of stationary phase: 



+ e«74-2.ARy(,)| +o(l/ft) 



Thus, 



X, = -Aipg - e-«*+'"'P,(/3) 



(23) 



Substitution of the appropriate expressions for P,(/3) 

 gives the six exciting forces, as functions of the angle of 

 incidence /S. 



In the special case of a spheroid at zero speed, the fore- 

 going results are in agreement with Havelock's expres- 

 sions, obtained directly by finding the diffraction po- 

 tential and integrating the pressure over the surface of 

 the spheroid. 



We now restrict our attention to the roll moment X4: 



X4 = —%itpgAK'^aia-!fi3{a2'^ 



where 



2)D4sin/3e-**+-'-'4-^ (24) 

 5 



q = /C[(a,' - a^) cos^ + (a,= - 03^) sin^ j3]'« 



In the sf)ecial case of beam waves, /3 = 5r/2, and thus 



X, = -%TpgAa,a,a^ie-''''+'"\h[K{ai' - 03')"'] (25) 



We note that this moment depends on the length 2ai 

 through the factor aiD4. The spherical Bessel function 

 ji oscillates about zero for real values of its argument, 

 and thus for 02^* — az^ > 0, or a beam-depth ratio greater 

 than one, the roll moment coefficient will oscillate about 

 zero.' This holds for all values of /3. On the other hand, 

 for 02^* — 03^ < 0, the parameter q will be either real or 

 imaginary, according as 



^ ai^ - 03= 

 tan= /3 ^ — -, 



For the angles of incidence between head (or following) 

 waves and the critical angle, where 



tan2 /3 = T — 



5 will be real and the roll-momept coefficient will oscillate. 

 For angles between the critical angle and beam waves, q 

 will be imaginary and J2{q)/q'' wll be a (real) monotonic 

 increasing function of q. Thus for these angles the co- 



o.oor 



- 











oooa 



- 



/ 



A 







aoos 



- 



f 



\ 



\ 





Y'o.oo4 



- 



/ 



\ 



\ 





0003 



- 



h 



\ 



V 



\90* 



0.002 



- 



-TV 



/ Vts- 



\ 



\ 



\ 



0.001 





/V'\ 



\ 



\ 



y \ 







-oooos. 



1 



19' \ ' 



V 



\ 



\ V^____^ 



1 1 r 







1 





2 3 4 



.1/. I.. • 



Fig. 1 Coefficient of roll-exciting moment for ellipsoid ai/ai ^ 

 l/7,a3/a, = 1/14, b/ai = 2/7, for various angles of incidence 



efficient of the moment .X4 wll be positive and non- 

 oscillatory, rising from zero at zero frequency to a maxi- 

 mum, and then decreasing to zero at large frequencies, 

 due to the exponential factor e"'^* . However for fairly 

 slender ellipsoids, with oi ^ » 03' > 02', this sector of 

 angles will be quite narrow. In all cases the roll moment 

 is 90 deg out of phase with the wave height at the 

 centroid. 



Computations of the roll moment X, are easily per- 

 formed from equation (24). The inertia coefficient D4 can 

 be computed directly using tabulated values of the inte- 

 grals a J in Zahm [6], or in terms of the entrained inertia 

 coefficient 



(a^^ — 03') ("2 — 03) 



' It is assumed in this discussion that ai > Oj. 



"" 2(02" - 03^) -I- {a, - a,)ia2' + 03')' 



which is plotted in Kochin, Kibel', and Rose, [7], and 

 tabulated (with the notation im = A„') by Zahm [6]. In 

 both references, the semi-axes are denoted (a, b, c) 

 rather than (ai, 02, as), with the restriction a > b > c. 

 Thus (a, b, c) should be replaced by (oi, 02, 03) if ai > 02 > 

 03 or by (oi, 03, 02) if Oi ^ ds ^ a?. The spherical Bessel 

 function can be evaluated from various tables, or from 

 the relation 



