Images in Some Problems of Surface Waves. 274 
Fig. 2 shows the corresponding curves for a vertical doublet, calculated from 
(14) for the case (16); the doublet is at the point C. Here, again, the broken 
curve shows the cosine term of the solution to which the disturbance 
approximates. 
We may also regard this as an approximate solution for the flow of a stream 
] ] I T I T | 
———— 
! n i 
Fig. 2. 
over a bed of a certain form. This is obtained by taking the zero stream-line 
for the combination of the uniform stream and a vertical doublet at C under the 
conditions given in (16) ; the equation of this curve is 
(y + 2a) {a? + (y + 2a)%} + ax = 0, (17) 
and its form is shown in the figure. Fig. 2 may be compared with a graph given 
by Wien* for the ease of a sudden small rise in the bed of a stream. 
It is interesting to note the general similarity of the surface elevation in the 
two cases shown in figs. 1 and 2; although the regular waves are given by a sine 
curve in one case and a cosine curve in the other, that is only because of the 
different position of the origin relative to the general form of the obstacle. 
Second Approximation for Circular Cylinder. 
4. We may now carry out further approximations for a circular cylinder in 
a uniform stream by the method of successive images. Reference may be 
made to fig. 3, which is not drawn exactiy to scale. 
The image of the stream in the circle is a horizontal doublet M at the centre 
C. The image of M in the free surface is a doublet — M at the image point Cy 
* W. Wien, ‘ Hydrodynamik,’ p. 206. 
271 
