526 
The Forces on a Circular Cylinder Submerged in a 
Uniform Stream 
By T. H. HAvELOcK, F.R.S. 
(Received 18 August, 1936) 
1—Although many investigations have been made on the wave resistance 
of submerged bodies, no case has been solved completely in the sense of 
taking fully into account the condition of zero normal velocity at the 
surface of the body. The simplest case is that of the two-dimensional 
motion produced by a long circular cylinder, with its axis horizontal and 
perpendicular to the stream, submerged at a certain depth below the 
upper free surface. This problem was propounded many years ago by 
Kelvin, and it was solved later, as regards a first approximation, by 
Lamb; in that solution the cylinder was replaced by a doublet, and the 
effect of the disturbance at the surface of the cylinder was neglected. 
Applying the method of images, I examined a second approximation, 
and also by the same method obtained a first approximation for the 
vertical force on the cylinder. 
Although the problem is not in itself of practical importance, it seems 
of sufficient interest to obtain a more complete analytical solution, and 
this is given in the present paper. The solution contains an infinite 
series, whose coefficients are given by an infinite set of linear equations; 
expansions are given for the coefficients in terms of a certain parameter, 
and corresponding expressions obtained for both the wave resistance and 
the vertical force. Numerical calculations have been made from these 
for various velocities and for different ratios of the radius of the cylinder 
to the depth of its axis. These confirm the general impression that the 
first approximation is a good one over a considerable range. The effect 
of the complete expressions appears in an increase in the wave resistance 
at lower velocities and a slight decrease at high velocities; this may be 
described as due largely to a shifting of the maximum of resistance 
towards the lower velocities, an effect which might have been anticipated. 
The similar three-dimensional problems of the submerged sphere, or 
spheroid, are of more practical interest, as the first approximations which 
I have given for these cases have had certain applications in ship resis- 
t ‘Proc. Roy. Soc.,’ A, vol. 115, p. 268 (1926). 
t ‘Proc. Roy. Soc.,’ A, vol. 122, p. 387 (1928). 
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