414 T. H. Havelock 
to that of the incident waves at the points x = 1, a, 0, —a, —1 to be 1-05, 
1-08, 1-09, 0-99 and 0-95 respectively. 
The alteration in amplitude at bow and stern would be greater for a fuller 
model, and especially for a bluff-ended form. Nevertheless, these approxi- 
mate calculations confirm the view that for a fine model the modification 
caused by the reflexion of the incident waves may be treated as a second 
order correction. It should also be noted that these results are for a model 
of infinite draft; it may be presumed that the effect would be much smaller 
for one whose draft is small compared with the wave-length. 
4. For a vertical obstacle of infinite draft, we may readily transfer 
results from other diffraction problems. The effect of a cylinder of elliptic 
section would be of special interest, but the analytical solution does not 
lend itself to computation when the wave-length is of the same order as 
the length of the axis. It is, however, worth while examining briefly two 
other cases from the present point of view. 
Let the cylinder be circular, its water plane section being the circle r = a. 
For plane waves of amplitude h moving in the negative direction of Oz, 
the complete solution is given by 
@ = (ghia) ete**\ Jy(kr) +2 y i".J_, (Kr) cos nd| 
1 
—(gh/c) cite bp Hr) +23 7"b, H® (kr) cos nh aay liz) 
1 
where o?= gk, and 6b, = J,(ka)/H®” (ka). 
Putting r = a in the expression for the surface elevation, and reducing by 
means of relations for the Bessel functions, we obtain on the cylinder 
2hert C = nC 9 
=— 4 7 On S 
G a ( ot x |, COS N \; (18) 
where C,, = 1/H®” (ka). 
Computation from this expression, which involves tabulation of J/?+ ¥/?, 
can be carried out without much difficulty except when xais large. A detailed 
study might be of interest, but for the present purpose the following results 
suffice to show the variation of amplitude round the cylinder. The numbers 
in table 1 give the ratio of the amplitude at each point to the amplitude of 
the incident waves; 4 = 0° corresponds to the bow and 6 = 180° to the 
stern. 
475 
