WAVE RESISTANCE OF A CYLINDER 328 
of the indentation at uw) according as wu) > or < g*t/2é, that is, according 
as € > or < ct. In this manner, it is found that as far as the regular 
waves are concerned, (9) gives 
2rkyare-F sinkyx (€ > dct), 
—2rk,areFsinkyx (€ < dct). (10) 
Similarly if € < 0 the line of steepest descent is the line w = g!t/2é+re?*7 
and the corresponding contribution is —27K) a?e-*/ sin Ky x. 
Summing up this outline of an analysis, the surface elevation at any 
time is made up of three parts: (i) the local symmetrical disturbance 
travelling with the cylinder given by the first and third terms in (6); 
(ii) a regular train of waves 47x, a?e-"/ sin kx behind the cylinder extend- 
ing from x = 0 tox = — et; (iii) the part given by the remaining integrals, 
representing a disturbance which spreads out in both directions and 
diminishes in magnitude as time goes on. 
The second part agrees with the general description using the idea of 
group velocity. The third part has not been examined in detail, but an 
asymptotic expansion suitable for large values of € and t may be found 
from the transformed integrals indicated in the previous discussion. For 
large positive values of € and (gt/2&)—u,é?, the first term in such an 
expansion is 1/2,,1/2 
ra 1 
= cos( zn — gt¥*2). (11) 
REWAE = z ct) 
For € = ct, that is, at a point over the centre of the moving cylinder, 
this reduces to 
2 nfl 
act enete/sé 2 
ar( 72) tnt cos 1(m—k,y ct), (12) 
a result which can be obtained directly from the integrals in (7) by using 
the method of stationary phase. After a sufficient time, (12) gives approxi- 
mately the departure of the motion over the cylinder from the quasi- 
steady state consisting of the local symmetrical disturbance and the regular 
train of waves to the rear. If Aj (= 27/k,) is the wave-length in the regular 
train, the wave-length of the disturbance near the cylinder is 4), the wave- 
length for which cis the group velocity. The usual direct solution for motion 
with uniform velocity leads to the surface elevation (6) with regular waves 
in advance as well as to the rear. The so-called practical solution is then 
obtained by superposing a free infinite wave train cancelling out those in 
advance and doubling the amplitude to the rear. Another well-known 
method of obtaining this practical solution directly is to use the frictional 
coefficient introduced by Rayleigh. In the present analysis we have not 
used this frictional coefficient, the values of the integrals being interpreted 
539 
