Reprinted without change of pagination from the 
Proceedings of the Royal Society, A, volume 201, 1950 
The forces on a submerged spheroid moving 
in a circular path 
By T. H. Havetock, F.R.S. 
(Received 28 December 1949) 
Expressions are obtained for the tangential and radial forces on a sphere moving in a circular 
path at constant depth; similar calculations are made for a prolate spheroid, including in this 
case the couple acting on the spheroid. Numerical computations have been made, and curves 
are given to show the effect of curvature of the path upon the wave resistance. 
1. The forces on a ship moving in a curved path are, no doubt, affected to some 
extent by the wave motion produced, but it is not easy to estimate the magnitude 
or nature of this influence. In the following paper an approach is made to some 
aspects of this problem by considering some cases of a submerged body moving in 
a circular path, namely, a sphere and a prolate spheroid. The motion of a sphere has 
been examined recently by Sretensky (1946), but the results given by him are in- 
correct. In the present work a different method is adopted; it is one which can be 
used for bodies of other forms, and also for non-uniform motion. 
2. We may derive first expressions for the ideal case of a simple source moving in 
any manner at constant depth f below the free surface of the water. We take fixed 
axes with O in the free surface, Oz vertically upwards, and we use cylindrical co- 
ordinates (o, 6,z). If at time 7 the strength of the source is m and its horizontal 
distance from O is wp, the velocity potential due to an infinitesimal step in the motion 
is given, as in equation (27) of a previous paper (Havelock 1949), by 
gb = 2mgtdr ie Jy(k@) e*F sin {g?k?(t —7)} Kt dk. (1) 
We may regard the effect due to a point source, varying in strength and moving in 
any manner, as made up of the superposition of small steps of this nature. In 
particular, for the present problem, we suppose the source of constant strength and 
to be moving in a circle of radius h; further, we take the motion to start at ¢ = 0 and 
the angular velocity to have a constant value ©. Hence we obtain the velocity 
potential at any time ¢ as 
p= am a + 2ng* | ar | * o(KWy) e *F sin {g?x?(t—7)} Kt dk, (2) 
1 
where 7? = wo? +h?— 20h cos (8— Mt) + (2+f)?, 
r= w2+h?— 20h cos (8— Qt) + (2—f)?, (3) 
we = w+ h? — 2h cos (9— Q7). 
For the relative steady state which is ultimately established we require the limiting 
form of (2) as too. We substitute in (2) 
Ig Dp) = Jol) Jp ch) + 25 (kt) Jy(kh) cos n(0 — Or). (4) 
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