THE RESISTANCE OF A SUBMERGED CYLINDER 
IN ACCELERATED MOTION 
By T. H. HAVELOCK (King’s College, Newcastle-on-Tyne) 
[Received 8 March 1949] 
SUMMARY 
The problem considered is the resistance to motion of a circular cylinder at a 
constant depth below the free surface, in particular when the motion starts from 
rest and has uniform acceleration. The resistance is expressed as the sum of two 
terms ; one corresponds to the wave resistance for uniform velocity, and the other 
may be taken as giving an effective inertia coefficient, the variation of which during 
the motion is of special interest. The expressions are carried to the second order 
of approximation and have been reduced to forms suitable for numerical computa- 
tion. Curves are given showing the variation of both parts of the resistance during 
the motion, for various values of the acceleration. 
1. For the steady motion of a submerged body with velocity c parallel to 
Ox, the condition at the free surface of the water is 
02074 /éx?-+-g éd/ey = 0, 
where ¢ is the velocity potential, Ox is horizontal, and Oy upwards. For 
small values of c this becomes formally equivalent to 0¢6/éy = 0, while for 
large velocities the corresponding limit may be taken as d = 0. The same 
effect may be seen if we consider the expressions for, say, a moving point 
source at a given depth below the surface; it is easily seen that in the limit 
the image system becomes a point source for small velocities, while it 
approximates to a sink for large velocities. Some discussion has arisen as 
to the appropriate surface condition to use when estimating the effective 
inertia of submerged or floating bodies; but any argument based on steady 
motion assumes a state which has been uniform for a long time, and 
cannot be applied directly to accelerated motion or motion started at a 
given instant. In a previous paper (1) expressions were given for resistance 
in accelerated motion, but no case has hitherto been worked out. It can 
be seen from equation (30) of that paper that, if we proceed only as far 
as the first approximation, the total resistance separates into two parts, 
the wave resistance and the inertia resistance; further, the latter part is, 
to that approximation, the same as for motion under a free surface 
neglecting gravity and thus corresponding to the surface condition ¢ = 0. 
To obtain a more accurate result it is necessary to proceed further in 
the approximation to the solution. In the present paper we consider the 
problem of the circular cylinder moving at constant depth below the 
surface, examining, in particular, motion with uniform acceleration starting 
(Quart. Journ. Mech. and Applied Math., Vol. II, Pt. 4 (1949)] 
545 
