Drifting Force on @ Ship among Waves. 469 
A complete solution would include an addition to (1) necessary to 
satisfy the boundary condition at the surface of the ship in its actual 
motion and also the condition of constant pressure over the free surface 
of the water. We are, meantime, neglecting this additional term, and 
assuming the conditions such that for a first approximation we may 
calculate the resultant forces from the pressure given by (3). The 
resultant horizontal force backwards is given by 
Hs | [GS oe ape 8 ©) 
taken over the immersed surface of the ship in any position, (J, m, ) 
being the direction-cosines of the outward drawn normal at any point. 
This may be transformed into a volume integral taken throughout the 
immersed volume V of the ship, and using (3) we have 
rf 
=~gprh| | { e sin (GiseKx)adVig 5) (8) 
Let S,, V, be the immersed surface and volume, respectively, when the 
ship is in its equilibrium position in smooth water. If the ship is held in 
this position in the waves, the corresponding force F, calculated from (5) 
is a purely periodic force with mean value zero (1937). Suppose now the 
ship to be in a slightly displaced position S due to heaving and pitching. 
The additional horizontal force is given by (5) integrated throughout the 
volume between S, and S. If dv is the distance from any point of So 
normally outwards to S, we have dV=SvdS,. Let the pitch be measured 
by a small angle @ of rotation round an axis through a point G on Oz 
at a height c above O, taking 0 to be positive with the bow up ; and let 
the heave be given by a small vertical displacement ¢ upwards. Then, 
to the first order in ¢ and 0, we have 
ou—net ne —1 (=e ue ee) (8) 
Hence the horizontal force backwards in the new position is given by 
F=F,—gpkh¢ ff e” sin (ot-+Kx)ndSo 
—gpkhé ff e sin (ot+xx){na—I(z—c)}dS, . (7) 
where the integrals are taken over the equilibrium position of the ship’s 
surface. 
Calculations may be made directly from this expression, but we put 
it into another form to show that it leads to an average steady force 
backwards. 
Let B be the extra buoyancy for the ship in its equilibrium position 
due to the wave motion, that is, the additional force upwards arising from 
