[Reprinted from the PRocEEDINGS oF THE Royat Soctery, A, Vol. 115.] 
The Method of Images in Some Problems of Surface Waves. 
By T. H. Havetock, F.R.S. 
(Received May 26, 1926.) 
Introduction. 
1. When a circular cylinder is submerged in a uniform stream, the surface 
elevation may be calculated, to a first approximation, by a method due originally 
to Lamb for this case, and later extended to bodies of more general form: the 
method consists in replacing the cylinder by the equivalent doublet at its 
centre and then finding the fluid motion due to this doublet. In discussing 
the problem some years ago,* I remarked that if the solution so obtained were 
interpreted in terms of an image system of sources, we should then be able to 
proceed to further approximations by the method of successive images, taking 
images alternately in the surface of the submerged body and in the free surface 
of the stream. This is effected in the following paper for two-dimensional fluid 
motion, and the method is applied to the circular cylinder. It provides, theoreti- 
cally at least, a process for obtaining any required degree of approximation, 
but, of course, the expressions soon become very complicated. It is, however, 
of interest to examine some cases numerically so as to obtain some idea of the 
degree of approximation of the first stage. 
An expression is first obtained for the velocity potential of the fluid motion 
due to a doublet at a given depth below the surface of a stream, the doublet 
being of given magnitude with its axis in any direction. A transformation of this 
expression then gives a simple interpretation in terms of an image system. 
This system consists of a certain isolated doublet at the image point above the 
free surface, together with a line distribution of doublets on a horizontal line 
to the rear of this point ; the moment per unit length of the line distribution 
is constant, but the direction of the axis rotates as we pass along the line, the 
period of a revolution being equal to the wave-length of surface waves for the 
velocity of the stream. The contribution of each part of the image system to 
the surface disturbance is indicated. 
Before proceeding to the circular cylinder, two cases are worked out in some 
detail, namely, a horizontal doublet and a vertical doublet. To a first approxi 
mation these give the surface disturbance of a stream of finite depth with an 
obstruction in the bed of the stream ; in the first case the bed of the stream is 
plane with a semi-circular ridge, and in the second case it has a more com- 
* «Roy. Soc. Proc.,’ A, vol. 93, p. 524 (1917). 
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