1905 - 6 .] Lord Kelvin on the Growth of a Train of Waves. 415 
§ 124. We are going to compare this with the total energy, 
kinetic and potential, K + P, per wave-length. In the first place 
we shall find separately the kinetic energy, K, and the potential 
energy, P. We have (the density of the water being taken as 
unity) 
K = dz{p + $) .... (162); 
Jo J i 
P = f gfdxZl (163), 
J 0 
where £ x denotes the surface displacement. 
By (149) and (152) we find 
£ = — cos mix — vt) . . . (164); 
£ = mhe~ m(z ~ 1) sin m(x - vt) .... (165); 
£ T = - cos m(x - vt) .... (156) repeated. 
Hence, 
f 1 1 
K = J j dx ~2m = l mJc2X ~ 2^ 2 ‘ * • ( 2 66 ) 1 
(167) > 
where v 2 is eliminated by (157). 
§ 125. Thus we see that the kinetic energy per wave-length, 
and the potential energy per wave-length, are each equal to the 
work done per period by the water on the negative side, upon the 
water on the positive side, of any vertical plane perpendicular 
to the length and sides of the canal. Thus we arrive at the 
remarkable and well-known conclusion that in a regular pro- 
cession of deep-sea waves, the work done on any vertical plane 
is only half the total energy per wave-length. This is only 
half enough to feed a regular procession, advancing to 
infinity with abruptly ending front, travelling with the ivave- 
veloeity v. It is exactly enough to feed an ideal procession of 
regular periodic waves, coming abruptly to nothing at a front 
travelling icith half the “ wave-velocity ” v\ which is Osborne 
