The forces on a submerged spheroid moving in a circular path 304 
The radial force can be obtained by the same method from the remaining terms in 
the velocity potential; but the expressions are lengthy and not suitable for numerical 
computation. 
If we take limiting values as h > 00, Q>0, hQ—c, the couple G becomes zero and 
R reduces to the wave resistance given previously (Havelock 1931) for the linear 
motion of a spheroid. On the other hand, if we take h = 0, we find that L, = 0, 
M,,,,, = 9, and we obtain the couple for pure rotation as 
25766 co 
G = ET OME 0482 Sn Ph, exp [— 8n2Q%flg], 
1 
1 
with P;,, =| (1—w?) Jy, (4n?O7%aeu/g) du. (43) 
0 
8. For numerical computation the integrals for L, M can be expanded in various 
forms; for instance, one which proved useful can be derived from the expansion 
JA p( + u*)*} sin (n tan wv) = (Jp4+Jn41) (2p) 
(3pu)? 
) 3! 
the Bessel functions having the argument p. For some values of the parameters it 
was found more convenient to evaluate the integrals by direct quadrature. 
La (Jn—s ar 3I 4 a 3S 44 ar In+3 Hee (44) 
hQ// (Gf) 
FIGURE 2 
As a particular case we take a spheroid for which 2a = 5b, so that e = 0-9165; and 
for the depth we take f = 2b. This was one of the cases for which calculations were 
made previously for rectilinear motion. To bring out the effect of curvature we take 
for the radius of the path h = 5b. The results for the wave resistance are shown in 
figure 2. The ordinates are values of R/7gpb?, and the abscissae are hQ/,/(gf). The 
curve A is for linear motion and is taken from the paper already quoted (Havelock 
561 
