472 Prof. T. H. Havelock on the 
The model is symmetrical fore and aft, and for simplicity we assume 
G, the position of the axis of rotation, to be at O ; thus we take c=0 in 
(9). The integrations in (8) and (9) over the sides and keel of this model 
are readily carried out, and we obtain eventually 
8g pbh } ; 
B— a e“{sin (p+p1)—p cos (p+p;)—sin p,} cos ot. . . (20) 
8gpbh_. , : 
We a [(p°-+P1— 3) sin (p+-p,)—(p1+ 3p) cos (p+p,) 
+3 sin p;—P, Cos p,}e- 2+ (l—e4—ge 4) {sin (p+-p,) 
—p cos (p-++-p,)—sin p,}]sin of, ~ (21) 
where p=kl, p,=ka, q=kd. 
Tn (21) under usual conditions, the second part is small compared with 
the part which is factored by e~“; if this latter factor be also neglected, 
the expression is simply the conventional pitching moment obtained 
from the hydrostatic pressure due to the wave elevation integrated over 
the water-plahe section of the ship. This form of model is not quite 
suitable for the approximations on which the calculations have been 
made, but there are no experimental results available for simple 
symmetrical models of small beam. 
From experiments, carried out at the National Physical Laboratory, 
on a model of a single-screw cargo ship, Kent and Cutland (1941) have 
obtained some very interesting results. We give the relevant data for 
the ship: length 400 ft., beam 55 ft., draught 24 ft., and displacement 
11,332 tons. The natural pitching and heaving periods are given as 
6:2 and 7:42 sec. respectively. Measurements were made of pitch and 
heave, of resistance and of other quantities, under various conditions in 
waves of 175 ft., 350 ft., and 490 ft. In the shortest waves it is probable 
that a considerable part of the increased resistance arises from the 
reflexion of the waves by the ship. Applying an expression which I 
gave recently (1940) for this resistance, Kent and Cutland show that it 
accounts for rather more than half the measured resistance for a ship 
moored in the waves. That expression gave a limiting value for a ship 
of great draught held at rest, assuming total reflexion by the front 
portion of the ship. It seems reasonable, therefore, to suppose that the 
force arising from reflexion would be much smaller for a ship of usual 
form floating on the water, and especially so for the large wave-lengths. 
Moreover, in the short waves the pitch and heave are slight and are 
irregular in period ; in the medium waves the periods are approximately 
equal to the period of encounter between ship and wave, but the amplitude 
changes from a minimum to a maximum in regular cycles: in the long 
waves the pitch and heave are approximately uniform. In the present 
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