409 Prof. T. H. Havelock on the 
4. Suppose that the cylinder is wholly immersed in liquid with the 
axis of the cylinder horizontal and at a depth f below the free surface 
of the liquid, and let the cylinder be making small rotational oscillations 
about its axis. Let the angle 6 between the major axis of the section and 
the vertical be given by 
DSO MVC p54 a oe Mo ele e. CA) 
where 6, is small. 
For a first approximation we neglect the effect of the free surface. 
The angular velocity of the cylinder is of, cos ot, and the velocity potential 
is that due to a certain distribution of doublets along the instantaneous 
position of the major axis. We shall make a further approximation for 
small oscillations and assume that this distribution is along the mean 
position of the major axis, that is, the vertical through the centre of the 
ellipse. Thus we consider the motion to be due to a distribution of 
horizontal doublets of oscillating magnitude along the line between the 
foci of the ellipse in its mean position. From (6), the moment per unit 
length at a distance # from the centre of the ellipse is 
Fe Mate? —I?)!00s CET Te eee Ce) 
the limits for h being --ae. 
We replace M in (3) by this expression, write f+-/ for f, and integrate 
with respect to h; we obtain then the velocity potential for the given 
distribution when the condition at the free surface is satisfied. Similarly, 
from (4) we may obtain complete expressions for the corresponding 
surface elevation. This consists of a local oscillation whose amplitude 
diminishes rapidly with distance from the cylinder, together with regular 
waves travelling out on either side. We shall examine here only the 
amplitude of these regular waves ; from (4) and (8) the amplitude A of 
these waves, that is, the coefficient of sin (ct—« x) for positive values of x, 
is given by 
ae 
A=—k "8 Fi =j h(a2e2— h2)te—F + dh 
—ae 
=—K/?V (0? 0) (a+5)?0e-*° | ‘sin? Acos Ge "0% ©5 9q9,.. (9) 
0 
0 
This may be expressed in terms of the modified Bessel function I,(z), 
and we obtain 
same {gael o(Koae)—21,(kgae)}e““F. . . . (10) 
If «ae is small, that is, if the wave-length is large compared with the 
linear dimensions of the cylinder, the first term in the expansion of (10) 
gives, as an approximation, 
A=4n(a—b) (a+b) ew wee (2) 
464 
