329 T. H. HAVELOCK 
as principal values wherever necessary. The chief point of the discussion 
is that there is no ambiguity when the motion starts from rest. The motion 
which is gradually established as time goes on is the practical solution for 
the steady state, with regular waves only to the rear of the cylinder; this 
result is in fact associated with the group velocity being less than the wave 
velocity. 
3. The Wave Resistance (Revised, 1959). 
The velocity potential (2) is sufficient for the surface elevation to 
the usual approximation; but, in order to calculate the forces on the 
cylinder from the fluid pressure, it is necessary to add a further ap- 
proximation so as to satisfy the correct boundary condition on the 
surface of the cylinder with 2 = x + zy and V = (g/x)!/?, the complex 
potential of which (2) is the real part is given by 
t (os) 
nips aca a d 
P Oban sino TA 
0 0 
te ~ix(e-V) (t-7)_p-(e+V) C2: xe EKzZ 2K RAK , (13) 
We may expand this in the neighbourhood of the cylinder in the form 
2 
+ Ey Age” (14) 
Hence the required form for the complex potential is 
ca’ CawAL* 
W=—-+2A ar +d (15) 
gm 
valid near the cylinder, the asterisk denoting the conjugate complex 
quality. 
If X and Y are the horizontal and vertical forces on the cylinder, 
we have from 
2 
: (V/A . 0 
X -iY = 5 pi|(32) dz — pia w*da* (16) 
the integrals being taken round a small contour surrounding the origin 
From (14) and (15) we get, to the first order in the co-efficients A, 
X — ty = 4zpca*A, — Qmpa” ue A, (17) 
*EDITOR’S NOTE: In preparing this 1959 revision of Section 3, pages 329, 331, and 332 of 
the original paper were modified and page 330 was deleted completely. 
540 
