1903-4.] Lord Kelvin on a Free Procession of Waves. 
325 
similar figure drawn to represent the rear of the finite (two-ended) 
procession which we are now considering. 
§ 30. Direct attack on the problem thus indirectly solved, gives, 
for the case of 1000 wave-crests in the beginning, the following 
explicit solution, 
't=2000 
2(1# (58), 
i= 0 
where if/ is a function found according to the principles indicated in 
§ 4 above, to express the same surface-displacement as our function 
<£ of § 12, and the proper velocities below the surface to give^ in the 
sum, a rightward procession of waves. Our present solution shows 
how rapidly the initial sinusoidality of the head and front of a 
one-ended infinite procession, travelling rightwards, is disturbed in 
virtue of the hydrokinetic circumstances of a procession invading 
still water. Our solution, and the item towards it represented in 
figs. 6 and 7, and in fig. 2 of § 6 above, show how rapidly fresh 
crests are formed. The whole investigation shows how very far 
from finding any definite “group-velocity” we are, in any initially 
given group of two, three, four, or any number, how'ever great, of 
waves. I hope in some future communication to the Royal 
Society of Edinburgh to return to this subject in connection with 
the energy principle set forth by Osborne Reynolds,* and the inter- 
ferential theory of Stokes f and Rayleigh j giving an absolutely 
definite group-velocity in their case of an infinite number of 
mutually supporting groups. But my first hydrokinetic duty, 
the performance of which I hope may not he long deferred, is 
to fulfil my promises regarding ship-waves, and circular waves 
travelling in all directions from a place of disturbance in water. 
§ 31. The following tables show some of the most important 
numbers which have been calculated, and which may be useful 
in farther prosecution of the subject of the present paper. 
* Nature, vol. xvi., 1877, pp. 343-4. 
i t Smith’s Prize Paper, Camb. Univ. Calendar, 1876. 
t Sound, ed. 1, vol. i. , 1877, pp. 246-7. 
[Table I. 
