668 Prof. T. H. Havelock on the Damping of the 
motion would then be very small. However, the experiments show that 
damping by fluid friction and eddies was also negligible. Reference 
may be made specially to work by Schuler (1936) with a vibrating prism 
of rectangular section, in which direct measurement was made of the 
amplitude of the waves. The logarithmic decrement was also measured, 
and it was concluded from the dimensional form of the results that the 
damping was due to wave motion, viscous and other damping being 
negligible in comparison. Schuler gives no theoretical calculation of the 
damping, and unfortunately the data necessary for making a comparison 
with theory are not recorded, such as the effective mass and restoring 
force and the free or forced nature of the vibrations. 
Coming to the ship problem, as far as published work is concerned 
there is practically no accurate information about the damping of 
natural heaving. It is usually stated to be very large, any natural 
vibrations dying out very quickly. The only numerical estimate appears 
to be that given by Horn (1936) and said to be an average result derived 
from a large number of models ; his estimate gives a logarithmic decre- 
ment for natural heaving of about 1:45. This is very large, and would 
mean that the amplitude is reduced by about one-half in each swing. 
It is also stated that the decrement for natural pitching oscillations is 
of the same order of magnitude. 
3. We now examine the waves produced by an oscillating body, and 
we adopt the method of replacing it by some suitable distribution of 
alternating sources. 
We consider first two-dimensional fluid motion, and we take the origin 
O in the free surface, Oa horizontal, and Oz vertically upwards. If 
there is a source of strength m cos pt per unit length at a depth f, that 
is at the point (0, —f), the velocity potential is given by 
, on (OO 2 COS LEE 
=—me'P log -2me™ | i Seana 
) me” Jog pias me ea ea K, (4) 
where p®=gky, 7,2=2x?+(z+f)?, ro2=a+(z—f)?; and we take the 
limiting value of the real part of the expression as y is made zero. 
This leads to a surface elevation given by 
2amp 2mp.. r® ¢ COS Kf—Ky Sin Kf 
C= —— e—f cos (pt x ,x)— —— sin ot | - : et dk, (5) 
g (p ae 0 ) g Je K+ Ko? 
where the upper or lower signs are taken according as 2 is positive or 
negative. 
The first term in (5) gives the regular waves propagated outwards on 
either side ; if A is the amplitude of these waves and E the mean rate 
of propagation of energy outwards per unit length, we have, taking 
account of both sides of the origin, 
E=gppA2/2x)=20?m2ppe 7, nries oa asd lg ae (6) 
By summation, or integration, we can obtain the corresponding expression 
for any given distribution of periodic sources in the liquid. 
494 
