327 T. H. HAVELOCK 
The first two terms in (5) give 
2a*f 
= — 27K, a2e-"oF sin Ky xv — 
1 af? 0 0 
oa . 
P sin —kK COS 
— 2k) a | £6 f = Fore dic (x > 0), 
Ko ko 
0 
2a2f 
— —— Q7K, a2e-Kof sin ky X— 
1 Bape 0 0 
oa) é 
: sin kf —k cos 
—%k,a2 | © eal a Force dic (27 <0). (6) 
k°-+ ko 
0 
The expressions for 7, represent a steady state relative to the moving 
cylinder and symmetrical fore and aft of it. With x+ct = € = distance 
from a fixed origin at the starting-point, the last two terms in (5) represent 
the surface elevation at any time due to an initial displacement and velocity 
which is the negative of that given by (6); this must be the case in the 
present problem and it can be directly verified. With a change of variable, 
and with wi = xo, the last two terms in (5) are given by the real part of 
oo 
F etléur—g'u) - i - etlEu?+g*lu) é : 
Ns, = —2a?2 | ——— we! du—4a? wet du. (7) 
J U—U U+ Up 
=o 0 
The limiting value of 7, as t becomes infinite is derived from the principal 
value of the first integral in (7); taking the real part, we find that 
No > 2K, ae—T singe as t—+> +00. (8) 
n 
Turning to (6), we see that ultimately (8) cancels out the regular waves 
in advance of the cylinder and doubles the amplitude of those in the rear. 
Without examining the surface elevation in detail, we may specify more 
closely the part which at any time consists of a regular train of waves 
accompanying the moving cylinder. It is clear from the form of the integrals 
in (7) that the oniy contribution to such a train comes from the first 
integral, or from 
o 2) 
¢ (En2—ghu) 
© Dy 2 
—2a7UG [ Se UAC (9) 
U—Upg 
we) 
and is due to the pole at w= w. Regarding wu as a complex variable, the 
path is along the real axis indented at w= w). There is a saddle-point at 
u = g't/2&. First suppose € > 0. The path of integration may be rotated 
round the saddle-point to the line of steepest descent, namely, the line 
u = g't/2é+ret'7, the contribution of the circular arcs required to complete 
the closed contour being zero in the limit. We have also to take account 
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