From the PaitosopuicaL Maaazine, Ser. 7, vol. xxxiii. p. 666, 
September 1942. 
The Damping of the Heaving and Pitching Motion of a Ship. 
By T. H. Havetock, F.R.S. 
1. In a recent paper (1940) I discussed the damping of the rolling of a 
ship in still water due to the radiation of energy in the wave motion set 
up by the rolling. The following note is a similar examination of heaving 
and pitching oscillations ; an attempt is made to estimate the dissipation 
of energy in wave motion and comparison is made with such experimental 
results as are available. 
The problem may be stated first in relation to heaving motion. Con- 
sider a body of mass M floating freely in water, and suppose it is acted 
on by a periodic force E cos pt and is making small vertical oscillations ; 
let ¢ be the vertical displacement upwards from the equilibrium position. 
The equation of motion, for a frictionless liquid, is 
Mt=X—Mg-+E cos pt, Soe Po eran vege, CL) 
where X is the vertical resultant of the fluid pressures on the immersed 
surface. For an exact solution we should have to determine the velocity 
potential of the fluid motion so as to satisfy the boundary condition at 
the moving surface of the solid and the condition of constant pressure 
over the free surface of the liquid. Failing such a solution we proceed 
