Reprinted without change of pagination from the 
Proceedings of the Royal Society, A, volume 231, pp. 1-7, 1955 
Waves due to a floating sphere making periodic 
heaving oscillations 
By Str THomas Have tock, F.R.S. 
(Received 19 February 1955) 
The paper gives a discussion of the fluid motion due to a sphere, floating half-immersed in 
water, which is made to describe small heaving oscillations. The velocity potential is obtained 
as a series for which the unknown coefficients are given by an infinite set of equations. These 
are solved approximately so as to obtain curves showing the variation with frequency of the 
virtual inertia coefficient and of the equivalent damping parameter. 
1. When a floating solid is made to describe periodic oscillations wave motion is 
produced, and it is required to determine the resultant pressure on the solid and the 
energy radiated outwards in the wave motion. The problem has been studied in 
general form by John (1950), especially as regards the necessary conditions for the 
uniqueness of the solution of the potential problem. The only cases which, to my 
knowledge, have been worked out in any detail are two-dimensional problems. In 
particular, Ursell (1949, 1953) has examined fully the heaving motion of a circular 
cylinder half-immersed in water. Similar work has been carried out by Grim (1953) 
for cylinders with various forms of cross-section, more especially with a view to 
application to ship problems in estimating virtual inertia and damping coefficients 
for heaving motion. In all these cases the virtual inertia coefficient approaches an 
infinite value as the frequency becomes small; this is no doubt connected with the 
fact that the condition at the free-water surface then approximates to that for 
a rigid boundary, and the two-dimensional potential problem with that boundary 
condition is indeterminate. This does not arise for three-dimensional motion; the 
general case approximates to determinate potential problems in the two limits as 
the frequency approaches zero or infinity. The point of special interest is the 
variation of the virtual inertia coefficient with frequency between these limiting 
values. The general character of the variation has been surmised, but there do not 
seem to have been any actual calculations. In this paper we consider the simplest 
case, a sphere half-immersed and making small vertical oscillations. The calcula- 
tions show that the virtual inertia coefficient rises to a maximum with increasing 
frequency, falls to a minimum and then presumably rises gradually to its final 
limiting value. The variation of the equivalent damping coefficient is also obtained. 
A solid of ship form would come between the two extremes of an infinite cylinder 
and a sphere, and could be represented better by, say, a spheroid. The limiting 
values of the virtual inertia coefficient for a spheroid can readily be calculated, 
but the general solution for any frequency leads to expressions too complicated for 
computation. 
2. We take the origin O in the undisturbed water surface, with Oz vertically 
downwards. The water is assumed incompressible and frictionless and the motion is 
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