468 Prof. T. H. Havelock on the 
are therefore subject to correction by a more complete analysis, bat 
they may serve to bring out a new point of view. 
2. It is interesting to recall the development of the similar problem 
in rolling. Some years ago Suyehiro (1924), experimenting with a small 
model, announced the discovery of a drifting force sideways upon a ship 
when rolling in waves. This interesting result does not seem to have 
been studied by other workers, and either confirmed or disproved. The 
effect is small and probably needs suitable conditions of forced rolling 
in resonance with the natural period of roll. Suyehiro himself ascribed 
the force to reflexion of the waves by the side of the ship and supported 
this view by observation of the motion of the fluid particles near the ship. 
No calculation was made of the reflexion or scattering of the waves by 
the ship, and this is a problem which still awaits solution. Here, again, 
no doubt this form of wave pressure contributes to the result, but there 
is no reason to suppose it adequate in itself ; moreover, the experiments 
showed a close association of the drifting force with the rolling of the 
ship. Recently an alternative theory has been put forward by Watanabé 
(1938). Starting from the Kriloff equations, Watanabé deduced an 
expression for the drifting force involving the angle of roll and the phase 
lag between the roll and the actuating moment; applied to Suyehiro’s 
model, this expression gave a force of rather more than half the observed 
value. 
In the following sections we derive similar expressions for the drifting 
force due to heaving and pitching when the ship is head-on to the waves ; 
we assume throughout the usual theory of irrotational waves of small 
height. 
3. Take the origin O in the undisturbed surface of the water, Ox 
horizontal and perpendicular to the wave crests and in the direction of 
the ship from stern to bow, Oy horizontal and Oz vertically upwards. 
We shall suppose first that the ship has no forward motion or, more 
precisely, we may suppose it constrained so that it is free to make small 
vertical oscillations and free also to make small rotational oscillations 
about a horizontal axis parallel to Oy through some point G. We consider 
plane waves of small amplitude h and of wave-length 27/« moving in the 
negative direction of Ox. To the first order the velocity potential is 
given by 
¢=(gh/c)e” sin (ot-+- Kx), oo Evo: Fee ee GD) 
with o?=gx, and the pressure by 
D=Po—992+ pL. oo 6 6 6 0 6 «6 6 o (4) 
=P)—9pz+gphe cos (ct+nxr), . . . . . (3) 
Po being the pressure at the free surface. 
484 
