The Resistance of a Ship among Waves 
By T. H. Havetock, F.R.S. 
(Received 25 March 1937) 
1—The wave resistance of a ship advancing in still water may be calculated 
under certain assumptions, which amount to supposing the forced wave 
motion to be small so that squares of the fluid velocity may be neglected; 
moreover, the ship is supposed to advance with constant velocity in a 
horizontal line. It does not appear to have been noticed that we may super- 
pose on the solution so obtained free surface waves of small amplitude, and 
that the addition to the resistance may be calculated, to a similar degree of 
approximation, as the horizontal resultant of the additional fluid pressures 
due to the free surface waves; this additional resistance, which may be 
negative, depends upon the position of the ship among the free waves. 
Various calculations are now made from this point of view. We consider 
first transverse following waves moving at the same speed as the ship, and 
then a ship moving in the waves left by another ship in advance moving at 
the same speed; finally, we examine the more general case of a ship moving 
through free transverse waves of any wave-length. All the cases are discussed 
with reference to such experimental results as are available. 
2—We treat the problem first as one of steady motion with the ship at 
rest in a uniform stream of velocity c in the negative direction of Ox; we 
take the origin O in the undisturbed water surface, and Oz vertically 
upwards. The velocity potential is given by 
b= cx+¢y, (1) 
where ¢, represents the disturbance due to the ship. This, on the usual 
approximations, may be regarded as due to a source distribution over the 
longitudinal section of the ship; the source strength per unit area is 
(c/27) dy/dx, with y = f(x, z) as the equation of the surface of the ship, and 
it is to be noted that dy/dx is assumed to be small. 
We now take $ =cx+¢,+¢’, 
p' = heetv cos(kyx — f), (2) 
where ky = g/c?. The additional term represents standing surface waves of 
elevation hsin(kyx—f). We should, of course, require further terms in 
order to satisfy exactly the condition at the surface of the ship; but such 
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