[Reprinted from the PROCEEDINGS OF THE Roya. Society, A. Vol. 95] 
Wave Resistance: Some Cases of Three-dimensional Fluid 
Motion. 
By T. H. Havetock, F.R.S. 
(Received November 27, 1918.) 
1. Calculations of wave resistance, corresponding to a pressure system 
travelling over the surface, have hitherto been limited to two-dimensional 
fluid motion ; in those cases, the distribution of pressure on the surface is 
one-dimensional, and the regular waves produced have straight, parallel 
crests. The object of the following paper is to work out some cases when 
the surface pressure is two-dimensional and the wave pattern is like that 
produced by a ship. A certain pressure system symmetrical about a point is 
first examined, and more general distributions are obtained by superposition. 
By combining two simple systems of equal magnitude, one in rear of the 
other, we obtain results which show interesting interference effects. In 
similar calculations with line pressure systems, at certain speeds the waves 
due to one system cancel out those due to the other, and the wave resistance 
is zero; the corresponding ideal form of ship has been called a wave-free 
pontoon. Such cases of perfect interference do not occur in three-dimensional 
problems; the graph showing the variation of wave resistance with velocity 
has the humps and hollows which are characteristic of the resistance curves 
of ship models. 
Although the main object is to show how to calculate the wave resistance 
for assigned surface pressures of considerable generality, it is of interest to 
interpret some of the results in terms of a certain related problem. With 
certain limitations, the waves produced by a travelling surface pressure are 
such as would be caused by a submerged body of suitable form. The expres- 
sion for the wave resistance of a submerged sphere, given in a previous 
paper, is confirmed by the following analysis. It is alsoshown how to extend 
the method to a submerged body whose form is derived from stream lines 
obtained by combining sources and sinks with a uniform stream; in par- 
ticular, an expression is given for the. wave resistance of a prolate spheroid 
moving in the direction of its axis. 
2. Take axes Ox and Oy in the undisturbed horizontal surface of deep 
water, and Oz vertically upwards, and let ¢ be the surface elevation. For an 
‘initial impulse symmetrical about the origin, that is for initial data 
pho = F(a); c=0; 
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