Heaving and Pitching Motion of a Ship. 673 
with 2n/o the natural period of pitching when damping is neglected. 
With this relation, the logarithmic decrement is given by 
_7N 7a N 33 
Y ol YF gpVm (23) 
Collecting these results, and expressing the integrand in (30) in terms 
of Bessel functions, we find 
cs 
SiikcgUslmee a d6 
=a 6 | alice! 0058). Sse eae (are) 
To obtain a numerical result we take a case from an early paper by 
Kent (1922). The relevant data for the ship are: length 400 ft., beam 
52 ft., draught 22 ft., displacement 10,000 tons, longitudinal metacentric 
height 458 ft., natural pitching period 6-11 sec. 
In (34) we take the distribution of the same length as the ship, that is 
J=200 ft.; we assume a mean beam of 40 ft., and we take f=20 ft. 
These values suffice for a rough approximation. The integral in (34) 
was computed from tables of Bessel functions ; and we obtain finally 
the result 6=1-6. The same general remarks apply to the limitations 
of this calculation as in the case of heaving motion; however, it is 
interesting that the decrement 6 comes out at about the same value in 
the two cases. 
9. Summary.—Using expressions for the wave motion due to alternating 
sources in a liquid, application is made to the heaving and pitching 
motions of a ship, and, in particular, to estimating the damping from the 
rate of propagation of energy outwards in the wave motion. This 
method of approximation assumes the damping to besmall, and the results 
obtained are too large for much importance to be attributed to the 
actual numerical values. Nevertheless, it may be concluded that the 
wave motion gives rise to large damping for both heaving and pitching, 
and that the decrements are probably comparable with those obtained 
experimentally. 
References. 
T. H. Havelock, Phil. Mag. xxix. p. 407 (1940). 
F. M. Lewis, Trans. Soc. Nav. Arch. New York, xxxvii. p. 1 (1929). 
Browne, Moullin and Perkins, Proc. Camb. Phil. Soc. xxvi. p. 258 (1930). 
A. Schuler, Zeit. f. Ang. Math. u. Mech. xvi. p. 65 (1936). 
F. Horn, Jahrb. Schiffbau. Ges. xxxvii. p. 153 (1936). 
Kent and Cutland, Trans. Inst. Engrs. and Shipbrs., Scotland, p. 212 (1941). 
J. L, Kent, Trans. Inst. Nav. Arch. Ixiv. p. 63 (1922). 
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