1905 — 6 .] Lord Kelvin on the Growth of a Train of Waves. 413 
§119. As a preliminary (§§119-126) let us consider the energy 
in a uniform procession of sinusoidal waves, in a straight canal, 
infinitely long and infinitely deep, with vertical sides. If the 
water is disturbed from rest by any pressure on its upper surface, 
and afterwards left to itself under constant air pressure, we know 
by elementary hydrokinetics that its motion will be irrotational 
throughout the whole volume of the water : and if, at any 
subsequent time, the surface is brought to rest, suddenly or 
gradually, all the water at every depth will come to rest at the : 
instant when the whole surface is brought to rest. This, as we 
know from Green, is true even if the initial disturbance is so 
violent as to cause part of the water to break away in drops : and 
it would be true separately for each portion of the water detached 
from the main volume in the canal, as well as for the water 
remaining in the canal, if stoppage of surface motion is made for 
every detached portion before it falls back into the canal. 
§ 120. Because the motion of the water is irrotational, we have 
c = dF . <dF 
* dx ’ ^ dz 
(149), 
where F denotes the velocity-potential, F having been taken as 
the displacement-potential (§ 97 above). And by dynamics for 
infinitesimal motion, -as in (64) of § 38 above, 
p^-'Li{x,z,t)+g(z-\+C) . . . (150). 
To express the surface condition, let z= 1 be the undisturbed 
level ; and let denote the vertical component displacement of a 
surface particle of the water, taken positive when downwards 
and let II denote constant surface-pressure, and take — as the 
value of the arbitrary constant, C. Thus (150) gives, at the 
disturbed surface, 
0= + + (151). 
The equality between the second and third members of this 
formula is due to the disturbance being infinitely small, which 
makes —F(x , 1 + , t) - y F(# , 1 , t) an infinitely small quantity 
at at 
