1904 - 5 .] Lord Kelvin on Deep Sea Ship-Waves. 1069 
groups, given by Stokes in his Smith’s Prize examination paper, 
published in the Cambridge University Calendar for 1876 : and 
for rejecting it for the case of any single group of waves. In 
reality the front of a group, left to itself, travels with accelerated 
velocity exceeding the velocity of periodic waves of the given 
wave-length, instead of with half that velocity. 
§ 77. Fig. 29 shows the steady motion , symmetrical in front and 
rear of a single travelling forcive, which is a solution of our problem; 
but it is an unstable solution (as probably are the solutions of the 
problem of § 45 above, shown in figs. 13, 14, 15). If any large 
finite portion of the water is given in motion according to fig. 29, 
say, for example, 50 wave-lengths preceding 0 (the forcive) and 
50 wave-lengths following 0, the front of the whole procession, to 
the right of O, will become dissipated into non-periodic waves 
travelling rightwards and leftwards with increasing wave-lengths 
and increasing velocities ; and the approximately steady periodic 
portion of it will shrink backwards relatively to the forcive. 
Thus before the forcive has travelled fifty wave-lengths, the 
periodic waves in front of it are all gone : but there is still 
irregular disturbance both before and behind it. After the forcive 
has travelled a hundred wave-lengths, the whole motion in advance 
of it, and the motion for perhaps 30 wave-lengths or more in its 
rear, will have settled to nearly the condition represented by fig. 
26, in which there is a small regular elevation in advance of the 
forgive, and a regular train of approximately sinusoidal waves in 
its rear ; these waves being of double the wave-height given 
originally. This motion, as said above in § 68, will go on, leaving 
behind the forcive a train of steady periodic waves, increasing in 
number ; and behind these an irregular train of waves, shorter 
and shorter, and less and less high the farther rearward we look 
for them (see R in fig. 10 of £§ 26, 27 above). It is an interesting, 
but not at all an easy problem, to investigate the extreme rear 
(with practically motionless water behind it) of the train of waves 
in the wake of a forcive travelling uniformly for ever. I hope to 
return to this subject when we come to consider the work done by 
the travelling forcive. 
§ 78. Pass now to the investigation of the formulas by the 
calculation of which figs. 26, 27, 28, 29, 30 have been drawn, and 
