Waves produced by the Rolling of a Ship. 410 
Consider a cylinder of given vertical dimension 2a, with varying breadth 
2b. Naturally, for the circular cylinder (b>=a) the disturbance is zero. 
It is of interest to note that for the approximation (11) the maximum 
wave amplitude occurs for b=}a, its value being then nearly twice the 
value for the limiting case. of the flat plate (b5=0). 
Suppose that the cylinder has its major axis vertical, and is making 
linear horizontal oscillations in which the displacement is d sin of. ‘Then 
from (4) and (5) the amplitude of the regular waves is 
Ne aS i (a2e2—h2)te— "+ dh, 
a— —ea 
t 
= med (7) TA (ee) ema Maen nei) Pena (2) 
a—b 
If xoae is small, this gives, approximately, 
Meola, 9 5 oo ot oo (lS) 
Finally, combining (10) and (12) with d=a6,, we obtain the amplitude 
for an elliptic cylinder with its major axis vertical in its mean position, 
and making small angular oscillations given by 8=@) sin of about the 
upper end of its major axis; in this case we obtain 
a+b\} i 6) 2 —Kof 
Ant (=) {kyae(a-+ b)1p(K gue) —2(a+- b4- ya") 1, (x ae) }e~ “7. (14) 
For « ve small, the first term in the expansion is the same as (13) with 
d=abp. 
In all these cases the expressions take simpler forms in the limiting 
case of the flat plate, for which we made 6 zero; but it should be noted 
that the ideal solution then implies infinite fluid velocity at the edges 
of the plate. In particular, consider a plate of height 2a, making small 
oscillations about its upper edge, the centre of the plate being at a depth f. 
Tf «oa is small, the first term in the expansion of (14) gives 
bene MOO KIT ee 3 a8 oo 0 (LS) 
This, naturally, is equivalent to replacing the oscillating plate by a single 
doublet at its centre. If, in addition, «,f is small, we may take 7K97a*6, 
as a first approximation for the amplitude of the regular waves. A 
similar approximation could be made for a cylinder of any cross-section, 
using the corresponding inertia coefficient for lmear motion and the mean 
horizontal velocity of the cylinder. 
Rolling Ship. 
5. The expressions given in the previous section are approximations 
suitable for wholly submerged bodies ; it is not permissible, in general, 
to apply them to the oscillations of floating bodies. The approximation 
465 
