14 Lord Kelvin on the 



(3) The Initiation and continued Growth of a Train of Twp- 

 Dimensional Waves due to the Sadden Commencement 

 of a Stationary, SinusoidaUy Varying, Surface-Pressure. 

 §§ 118-158. 



§ 118. A forcive consisting of a finite sinusoi dally varying 

 pressure is applied, and kept through all time applied, to the 

 surface of the water within a finite practically limited space 

 on each side of the middle line of the disturbance. In the 

 beginning the water was everywhere at rest and its surface 

 horizontal. The problem solved is, to find the elevation or 

 depression of the water at any distance from the mid-line of 

 the working forcive, and at any time after the forcive began 

 to act. 



§ 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 hydro- 

 kinetics 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 

 Mater 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 e^ery detached portion before it falls 

 back into the canal. 



§ 120. Because the motion of the water is irrotational, we 

 have 



• d¥ ■ r/F 



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=-f k F{x,; : t)+ ff (;-l + C) . . (150). 



To express the surface condition, let z — 1 be the undisturbed 

 ^vel : and let & denote the vertical component displacement 

 of a surface particle of the water, taken positive when 



I eve 



