670 Prof. 'T. H. Havelock on the Damping of the 
Suppose now that the heaving motion is given by C=asinot. The 
wave-length for a period of 7-42 sec. is about five times the beam of the 
ship. It is thus permissible to take a simple distribution, namely a 
uniform line source, of strength m cos ot per unit length, extending over 
the length L of the ship at some suitable mean depth f. We take the 
value of m to correspond to the rate of alteration in displaced volume 
of the ship, which is St or Sac cos ct. Hence we take 
m= SaG/2Aloe «sok we ah eis s,s aD 
We put this value of m in (6) and, using 
SNO202 = i rie ter i rae eon ete ta Le) 
we obtain NES(HOSHID ES, a 6 bo to 5 GS) 
For the corresponding logarithmic decrement we have 
3-20 = Hc Meu ath erat htt aca (1) 
Putting in the values already given for this case and taking f=20 ft. 
as a mean depth, (14) gives the value 5=1-4. The agreement with 
Horn’s estimate is, of course, merely a coincidence. Clearly, this large 
value of 6 goes beyond the assumption on which N has been calculated, 
namely that the damping is small enough to allow approximately simple 
harmonic waves to be established. Nevertheless the calculation is 
sufficient to show that wave motion is quite adequate to account for the 
large damping which has been observed in practice. 
6. There does not seem to have been any experimental work on cases 
of three-dimensional fluid motion. We shall examine the corresponding 
theory, as it will allow of more detailed calculation for other source dis- 
tributions and also of application to pitching. Consider a point source 
of alternating strength m cos pt at a depth f below the surface, that is, 
with the source at the point (0, 0, —f). In this case the surface elevation 
is given by 
Qipm . _ 2k k SiN Kf—Kg COS Kf «f , an! 
(aE el = I 7 cosh % Joy dic 
g Cage ER 
siege "H cat) Cee ceete Wer we LO) 
where r2=27+y?, p?=gky, Hy'=J y—7Yo, and the real part of the 
expression is to be taken. There is a corresponding expression for the 
velocity potential. The first two terms in (15) represent local standing 
oscillations of the surface, and the third term the symmetrical circular 
waves propagated outwards. For the present purpose we only require 
the wave motion at a great distance, and the first term in the asymptotic 
expansion of the Bessel functions is sufficient ; hence, for this part of 
the velocity potential and surface elevation, we obtain 
Y t 
b~2megm(— ) e trter sin (pet 7 — xg) Pai abet @ \)) 
TK 0”, 
496 
