[Reprinted from the PROCEEDINGS OF THE Royat Soctgty, A, Vol. 144.] 
The Calculation of Wave Resistance. 
By T. H. Havetocr, F.R.S8. 
(Received January 25, 1934.) 
1. The wave resistance of a body moving in a frictionless liquid has been 
calculated by various methods. In a few cases it has been found directly as 
the resultant of the fluid pressures on the surface of the body. Another 
method, which has been more generally useful, involves the introduction of 
a certain type of fluid friction into the equations of motion. The wave 
resistance is then found by calculating the rate of dissipation of energy and 
taking the limiting value when the frictional coefficient is made vanishingly 
small. This method has certain important analytical advantages, nevertheless 
it is highly artificial, A third method, dealing directly with a frictionless 
liquid, consists in examining the flow of energy in the wave motion ; this has 
hitherto been used only for two-dimensional problems when the wave motion 
consists of simple waves with straight parallel crests, the usual theory of 
group velocity being directly applicable. 
In the followmg note this method is extended to three-dimensional fluid 
motion. Although no new special results are obtained so far as expressions 
for wave resistance are concerned, it seemed of sufficient interest to obtain 
them by this direct method, namely, by considering the flow of energy and the 
rate of work across planes far in advance and far in the rear of the moving 
body. 
These quantities are examined first for a free wave pattern of simple type. 
Then a general expression is given for wave resistance in terms of the velocity 
potential of the free wave pattern to which the disturbance approximates at 
a great distance in the rear, and this is applied to a general form of wave pattern 
and to some special cases. Finally, a similar examination is made of a certain 
problem when the water is of finite depth. 
2. With the origin O in the free surface of deep water, and Oz vertically 
upwards, the surface condition is 
OD 6 = 2=0, (1) 
where ¢ is the velocity potential and ¢ the surface elevation, For a wave 
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