[Reprinted from the PROCEEDINGS OF THE RoyaL Society, A. Vol. 93] 
Some Cases of Wave Motion due to a Submerged Obstacle. 
By T. H. Havetock, F.R.S. 
(Received May 14, 1917.) 
1. As far as I am aware, only one case of wave motion caused by a 
submerged obstacle has been worked out in any detail, namely the two- 
dimensional motion due to a circular cylinder; for this case, Prof. Lamb has 
given a solution applicable when the cylinder is of small radius and is at a 
considerable depth.* The method can be extended to bodies of different 
shape, and my object in this paper is to work out the simplest three- 
dimensional case, the motion of a submerged sphere. 
The problem I have considered specially is the wave resistance of the 
submerged body. In the two-dimensional case, this is calculated by considera- 
tions of energy and work applied to the train of regular waves. But for a 
moving sphere the wave system is more complicated, like the well-known 
wave pattern for a moving point disturbance, and similar methods are not so 
easily applied ; I have therefore calculated directly the horizontal resultant 
of the fluid pressure on the sphere. Before working out this case, the 
analysis for the circular cylinder is repeated, because it is necessary to carry 
the approximation a stage further than in Prof. Lamb’s solution in order to 
verify that the resultant horizontal pressure on the cylinder is the same as 
the wave resistance obtained by the method of energy. 
The stages in approximating to the velocity potential may be described in 
terms of successive images; the first stage ¢; is the image of a uniform 
stream in the submerged body, the second stage do is the image of @ in the 
free surface, the third 3 is the image of ¢2 in the submerged body, and so 
on. In order to keep the integrals convergent, a small frictional coefficient is 
introduced in the usual manner ; after the calculations have been carried out, 
the coefficient is made zero. Further, the solution for uniform motion is 
built up so that expressions can be found for the velocity potential at any 
time after the starting of the motion, although only the final steady state has 
been studied in detail. The wave resistance of a sphere is found to have the 
form const. x «/?¢~4/? W,,(«), in which « is 2¢f/c, with f the depth of the 
sphere and ¢ its velocity; W1,1(«) is a confluent hypergeometric function. 
In order to graph the wave resistance as a function of the velocity, 
expansions have been found for this particular variety of the function 
*H. Lamb, ‘Ann. di Matematica,’ vol. 21, p. 237; also ‘Hydrodynamics,’ 
6th edn. (1932) p. 410. 
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