Prof. T. H. Havelock 667 
by approximations. One part of the force X-Mg is the additional 
hydrostatic buoyancy, gpS¢ upwards, assuming the solid to have vertical 
sides near the water-line and § to be the area of the water plane section. 
Suppose now that the motion of the body is a forced vibration of frequency 
p and that the energy radiated is relatively small; then, as in similar 
problems, for instance the scattering of sound waves by a movable 
obstacle, it is assumed that the rest of the resultant fluid pressure may 
be expressed as the sum of two terms, one proportional to ¢ and the other 
to ¢. The factor of the first term represents the so-called added mass, 
while the second corresponds to the loss of energy by propagation of 
waves outwards. In these circumstances, equation (1) is reduced to the 
form 
M’C-+N¢+gpSf=Ecospt, . . . . . . (2) 
where M’ is the total effective mass. 
2. Various empirical formule have been devised for the effective mass 
of a ship for heaving motion, and for flexural vibrations. Reference may 
be made in particular to Lewis (1929) for ship forms, and to Browne, 
Moullin and Perkins (1930) for the added mass of prisms floating in 
water. The basic assumption in these studies is to neglect the wave 
disturbance and to suppose the fluid motion to be that due to a certain 
solid moving in an infinite liquid, the solid being made up of the immersed 
part of the floating body and its reflexion in the free surface of the 
water. The experiments of Browne, Moullin and Perkins showed that 
this leads to a reasonable value of the added mass, the calculated values 
being rather higher than those deduced from the experiments. 
It is the second term of equation (2), namely NZ, with which the present 
paper is specially concerned. Instead of calculating the fluid pressures, 
an alternative method is to work out the mean rate of propagation of 
energy outwards in the wave motion, and equating this to the mean 
value of NZ? we obtain a value for N for the given frequency. This 
procedure is permissible under the assumed conditions under which the 
motion is a forced simple harmonic vibration and the radiated energy 
is small. To obtain the corresponding logarithmic decrement for the 
damped natural vibrations, these may be taken as approximately of 
period 2z7/c, with 
Geis GS etm tei bil het ante! af ssh) 
Then the logarithmic decrement is given by 7N/oM’, with N having its 
value for the frequency co. There is very little work, theoretical or 
experimental, to which reference can be made. Browne, Moullin and 
Perkins (1930) measured the damping for prisms vibrating in air and when 
immersed in water; they conclude ‘The damping added by the water 
is negligible compared with the damping due to the supports, a result 
which might not have been expected.” But in those experiments the 
prisms were not floating freely and the frequency was of the order of 
13 per second ; it can readily be estimated that the energy in the wave 
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