Waves produced by the Rolling of a Ship. 412 
where I is the moment of inertia, W the weight, m the metacentric height, 
and N@ the resisting couple. 
The exact solution of (17) gives damped oscillations with a damping 
coefficient h=N/2I, and the rate of dissipation of energy is N6?. Suppose 
now that the dissipation is small and assume an undamped motion 
6=6) sin ot, holding approximately for a sufficient time, with 
o?'=Wm/1=47?/T?. 
With this assumption, the average rate of dissipation of energy 
=4No"6.?=Wmhé,?. 
In the usual notation for the rolling of ships, a0) is the decrement of rolling 
angle for one swing ; hencea=3hT. Thus the average rate of dissipation 
of energy is 2Wmaé,?/T. Assume, with Froude, that when the ship 
is rolling, regular straight-crested waves are sent out on either side, the 
breadth of each train being approximately equal to the length L of the 
ship ; further, let A be the amplitude of the waves, the wave-length, 
T the period, with A=gT?/27. In each train energy is propagated out- 
wards at half the wave velocity V, that is, at an average rate }gpA?VL 
on each side. Hence, equating the average rate of dissipation of energy 
to the average rate of propagation of energy outwards in the waves 
on both sides, we have 
2Wmab,?/T=t9pA2VA, 
or 
Wi PpSeokeln 5 5 2 a 0 a o (U8) 
This is, in effect, the equation given by Froude and used by later writers, 
the left-hand side of (18) being the loss of energy in one swing ; the other 
side of Froude’s equation was, however, twice that given in (18), owing 
apparently to neglect of the difference between group velocity and 
wave velocity. The statement given here, besides including this correc- 
tion, shows the various assumptions and brings the argument into line 
with the usual method of approximating to the damping coefficient in 
isochronous damped oscillations when the damping is sufficiently small. 
Froude recognized that his solution was not in any sense rigorous and 
hoped that it would be supplemented by some direct estimate, even if with 
no greater exactness, of the wave-making property of a ship when rolling ; 
it is also of interest that he proposed to attempt direct observation of the 
waves produced by the rolling of models of simple form. However, 
nothing further seems to have been done on this particular aspect of the 
problem since that time. Other writers have used Froude’s expressions 
to estimate the wave height, and it appears to be accepted that wave 
motion accounts for a large part of the dissipation of energy in rolling, 
that due to fluid friction or eddy-making being relatively small apart 
467 
