(Reprinted from the PRocEEDINGS oF THE Rovat Soctety, A, Vol. 121.] 
The Wave Pattern of a Doublet in a Stream. 
By T. H. Havetock, F.R.S. 
(Received September 18, 1928.) 
1. The following paper is a study of the surface waves caused by a doublet 
in a uniform stream, and in particular the variation in the pattern with the 
velocity of the stream or the depth of the doublet. In most recent work on 
this subject attention has been directed more to the wave resistance, which 
can be evaluated with less difficulty than is involved in a detailed study of the 
waves ; in fact, it would seem that it is not necessary for that purpose to know 
the surface elevation completely, but only certain significant terms at large 
distances from the disturbance. Recent experimental work has shown con- 
siderable agreement between theoretical expressions for wave resistance and 
results for ship models of simple form, and attempts have been made at a 
similar comparison for the surface elevation in the neighbourhood of the ship. 
In the latter respect it may be necessary to examine expressions for the surface 
elevation with more care, as they are not quite determinate ; any suitable 
free disturbance may be superposed upon the forced waves. For instance, it 
is well known that in a frictionless liquid a possible solution is one which gives 
waves in advance as well as in the rear of the ship, and the practical solution 
is obtained by superposing free waves which annul those in advance, or by some 
equivalent artifice. This process is simple and definite for an ideai point 
disturbance, but for a body of finite size or a distributed disturbance the 
complete surface elevation in the neighbourhood of the body requires more 
careful specification as regards the local part due to each element. It 
had been intended to consider some expressions specially from this point of 
view, but as the matter stands at present it would entail a very great amount 
of numerical calculation, and the present paper is limited to a much simpler 
problem although also involving considerable computation. 
A horizontal doublet of given moment is at a depth f below the surface of 
a stream of velocity ¢; the surface effect may be described as a local dis- 
turbance symmetrical fore and aft of the doublet together with waves to the 
rear. Two points are made in the following work. One is the variation of the 
local disturbance with the depth of the doublet, or rather with its relation to 
the velocity. Roughly, it may be said that the local surface effect changes 
from a depression to an elevation at a certain speed, which we might have 
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