[Reprinted from the PROCEEDINGS OF THE ROYAL Soctery, A, Vol. 122.] 
The Vertical Force on a Cylinder Submerged in a Uniform Stream. 
By T. H. Havetocg, F.R.S. 
(Received November 28, 1928.) 
1. The horizontal force on a circular cylinder immersed in a stream is 
familiar as an example of wave resistance. The following note supplies a 
similar calculation for the resultant vertical force. The problem was sug- 
gested in a consideration of the forces on a floating body in motion, the hori- 
zontal and vertical forces and the turning moment ; but the case of a partially 
immersed body presents great difficulties. It seemed, however, of sufficient 
interest to compare the resultant horizontal and vertical forces for a simple 
case of complete immersion for which the calculations can be carried out. 
The horizontal force, or wave resistance, has usually been obtained indirectly 
from considerations of energy, but a different method is adopted here for both 
components of force and the turning moment. In a former paper the method 
of successive images was applied to the problem of the circular cylinder, taking 
images alternately in the surface of the cylinder and in the free surface of the 
stream. Using these results to the required stage of approximation, the com- 
plete force on the cylinder is now obtained as the resultant of forces between 
the sources and sinks within the cylinder and those external to it. The same 
method can be applied to any submerged body for which the image sytems are 
known, and the resultant force and couple calculated in the same manner. 
The proposition used in this method is that for a body in a fluid, the motion 
of which is due to given sources and sinks, the resultant force and couple on 
the body are the same as if the sources and their images attract in pairs accord- 
ing to a simple law of force, inverse distance for the two-dimensional case and 
inverse square of the distance for point sources. This fairly obvious proposition 
follows directly from a contour integration in the two-dimensional case ; and, 
in view of the application, the extension is given in § 2 when the flow is due to 
a distribution of doublets. In §3 the horizontal and vertical force on a 
circular cylinder are obtained by this method, the former agreeing with the 
usual expression for the wave resistance. The different variation of the two 
components with velocity is of interest, and the expressions are graphed on the 
same scale. The additional vertical force due to velocity changes direction at 
a certain speed, and is clearly associated more with the surface elevation 
immediately over the centre of the cylinder. In § 4 reference is made to the 
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