From the PutLosopHicaL MaGazine, Ser. 7, vol. xxxiii. p. 467, 
June 1942. 
The Drifting Force on a Ship among Waves. 
By T. H. Havetocg, F.R.S. 
1. WHEN a ship is advancing through a train of waves it experiences 
an average steady resistance greater than that at the same speed in 
smooth water. ‘There are no doubt several factors operative in producing 
this result ; one, for instance, may be described as wave pressure due 
to the reflexion or scattering of the ocean waves by the surface of the 
ship (Kreitner, 1939). This must certainly be taken into account in a 
complete theory, but investigation of it involves second-order terms in 
the hydrodynamical equations and a satisfactory solution of the problem 
would be difficult. Certain calculations which I have given recently 
(1940) seem to show that this cause is not likely to account for more 
than a small’ fraction of the observed results. Experiment shows that 
the effect is most prominent when the period of encounter of the ship 
with the waves is near the natural periods of the ship’s oscillations ; 
whether directly or indirectly, the phenomenon is clearly associated with 
the heaving and pitching motions of the ship. In the paper already 
quoted (1940) it was suggested that it may well be that interactions 
between first-order effects which in themselves are purely periodic may, 
through phase differences, give rise to steady additional resistances. 
The object of the present note is to give some tentative calculations 
amplifying and illustrating this suggestion. For this purpose we fall 
back on the approximate theory which neglects the disturbing effect of 
the ship’s surface upon the wave motion. In suitable cases we may 
perhaps regard the necessary additions for the reflected waves to be 
small corrections, as, for instance, for a long narrow ship (1937). This 
assumption was the basis of the theory developed by W. Froude in his 
work on the rolling of a ship among waves, in which case the wave- 
length is assumed large compared with the beam of the ship. It was 
also used explicitly by Kriloff in his well-known analysis of the heaving 
and pitching of a ship among waves. This latter work dealt only with 
the oscillations of the ship, and not with the extra resistance to motion 
which is now under consideration. It is true that the problem involves, 
in some form at least, second-order terms, and any partial separate 
examination of such terms is unsatisfactory ; the following calculations 
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