Drifting Force on a Ship among Waves. 471 
except for general descriptive purposes. Among other reasons, there is 
a lack of precise information about the damping of natural heaving and 
pitching. A common statement is that the damping in both cases is 
large, the natural oscillations in uniform waves diminishing rapidly and 
the motion reducing to the forced oscillations. On the other hand, this 
is difficult to reconcile with certain experimental results when the period 
of encounter with the waves is near a natural period ; in such cases the 
amplitude of the resultant oscillation has a slow periodic variation from 
a minimum to a maximum in a manner suggesting the superposition of 
natural and forced oscillations of nearly equal period. The only published 
estimate from experimental results appears to be that given by Horn 
(1936), who states that the damping of heaving and pitching is of the 
same order of magnitude ; his estimate gives a logarithmic decrement of 
about 1-4, an extremely large value compared, for instance, with the 
damping of rolling. 
We have assumed the ship to have no forward motion, but, so far as 
the present approximation goes, we may suppose it moving with uniform 
speed; the only difference is that the quantity o in (13) is such that 
27/o is the period of encounter of the ship with the waves. 
We may make a rough estimate of the order of magnitude of the extra 
resistance given by (16). For a cargo boat of 400 ft. in waves of 500 ft. 
in wave-length and of height 6 ft., the amplitude P, of the pitching 
moment may be about 80,000 ft.-tons while the amplitude B, of the 
buoyancy might be, say, 300 tons. Hence from (16) we should have 
Teo, gira GY Gh Sa 85 og oo 5 oo 0 (ile) 
If the period of encounter is not near a natural period we might assume 
a total heave of 4 ft. and a pitch of 3° ; whence 
R= 3-ssin)P-- U3isingBy | ss ae 2 (8) 
A value of 15° for the phase lags 6, 8’ would not be unreasonable. This 
would give R=4-4 tons, of which three-quarters would be associated 
with pitching and one-quarter with heaving. 
5. For a more detailed analysis, we consider a simple form of wall- 
sided model of uniform draft d, with a rectangular middle body of length 
2a and beam 2b, and with entrance and run each of length J and of 
parabolic form. Thus with O at the centre of the parallel middle body, 
the equation of the contour from «=a to x=I-+a is 
YSHI—@=GHP se goo 0 6 o o (UD) 
at all depths, from z=—d to z=0; and there is a similar equation for 
the run at the stern from x=—(l/--a) to x=—a. 
487 
