411 Prof. T. H. Havelock on the 
used for the doublet distribution loses accuracy with diminishing depth 
of submergence of the body ; moreover, when the surface of the body 
cuts the free surface some of the expressions for the surface elevation may 
take infinite values. It may be noted, however, that such infinities 
generally occur in the local part of the disturbance, the expressions for 
the amplitude of the regular waves at a distance from the body remaining 
finite. 
For the rolling ship the period is such that the wave-length of the 
corresponding waves is large compared with the draught of the ship. Thus 
if we consider the analogous problem of the oscillating plate with its upper 
edge in the surface, the quantity x,)a of the previous section is small, 
in most cases about 0-1. In these circumstances, treating the motion 
as two-dimensional, we propose to regard the ship as a single oscillating 
doublet at a depth which is small compared with the wave-length ; 
thus, from (4), we take 277 oM/g as a first approximation for the amplitude 
of the waves at a distance from the ship. Further, as we cannot expect 
more than an estimate of the order of magnitude from this assumption, 
we shall regard the ship as a plank, of length L and draft D, oscillating 
about the water-line through an angle 0, on either side of the vertical ; 
using the result given at the end of § 4 and writing T for the complete 
period of rolling, this gives for the height of the regular waves 
4D? 
tat g?T4 0° 
It should be noted that the wave-height, as the term is commonly used, 
is measured from trough to crest and is twice the amplitude. 
h (16) 
6. Before applying this result, we may review briefly calculations which 
have been made from a different point of view. 
The part played by wave propagation in causing resistance to rolling 
was first recognized by W. Froude (1872) and was advocated by him in 
a series of papers. Froude showed that the energy propagated outwards 
in the wave motion corresponds to a resisting couple proportional to the 
angular velocity of rolling, and also that the energy actually dissipated 
in rolling, or a large part of it, could be accounted for by waves ot 
extremely small height ; in one case, for example, his calculation gave 
a height of 1} inches for waves 320 ft. long. The same method has been 
applied by other writers subsequently, and it may be worth while repeating 
the argument in a somewhat different form from that in which it is usually 
given. 
Suppose the ship to be rolling about a horizontal axis through its 
centre of gravity, and take the equation of motion in its simplest form as 
NOL ie=O, 2 oe ee 8 aa) 
466 
