ys} T. H. Havelock. 
The series is sufficiently simple for calculation, though in some of the cases it- 
was necessary to take a large number of terms. 
For both the horizontal and vertical doublets we take 
N= er, f= My; Kof = 4. (16) 
This means that we take the velocity to be such that the wave-length of the 
regular waves is dxf. We are assuming, in each case, a given doublet at depth 
f below the surface of deep water. The only restrictions so far are the general 
ones due to neglecting the square of the fluid velocity at the free surface, and 
the consequent limitation to waves of small height. From this point of view 
the data of (16) are rather extreme; but, this being understood, it may be 
permissible to use them for a comparison of the two cases. With the values 
in (16), the calculations are comparatively simple, and lead to graphs which can 
be drawn suitably on the same scale throughout; these are shown in figs. | 
and 2, where the unit of length is the quantity a. 
In fig. 1, there is a horizontal doublet at C; the arrow shows the direction 
of the stream assuming the doublet to be stationary, and Oz is in the undis- 
turbed surface. The surface elevation was calculated from (12) for the case 
(16). The broken curve shows the regular sine waves to which the disturbance 
approximates as we pass to the rear. This solution is also the first approxi- 
mation for a submerged cylinder of radius a; or, again, to the same order, it 
gives the effect caused by a semicircular ridge on the bed of a stream of depth 
twice the radius. rom this point of view the diagram may be compared with 
that viven by Kelvin* for a small obstruction on the bed of a stream of finite 
depth. 
* Kelvin, ‘Math. and Phys. Papers,’ vol. 4, p. 295. 
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