218-220] ENERGY. 379 



and equal to ^gpa l \. We may express this by saying that the 

 total energy per unit area of the water-surface is {gpa?. 



A similar calculation may be made for the case of progressive 

 waves, or we may apply the more general method explained in 

 Art. 171. In either way we find that the energy at any instant 

 is half potential and half kinetic, and that the total amount, per 

 unit area, is ^gpa?. In other words, the energy of a progressive 

 wave-system of amplitude a is equal to the work which would be 

 required to raise a stratum of the fluid, of thickness a, through a 

 height a. 



220. So long as we confine ourselves to a first approximation 

 all our equations are linear; so that if </>!, </> 2 , ... be the velocity- 

 potentials of distinct systems of waves of the simple-harmonic 

 type above considered, then 



<#>=4>i + < 2 + ........................... (1) 



will be the velocity-potential of a possible form of wave-motion, 

 with a free surface. Since, when is determined, the equation of 

 the free surface is given by Art. 216 (5), the elevation above the 

 mean level at any point of the surface, in the motion given by (1), 

 will be equal to the algebraic sum of the elevations due to the 

 separate systems of waves <f> lt <f> 2) ... Hence each of the latter 

 systems is propagated exactly as if the others were absent, and 

 produces its own elevation or depression at each point of the 

 surface. 



We can in this way, by adding together terms of the form given in 

 Art. 218 (12), with properly chosen values of a, build up an analytical ex- 

 pression for the free motion of the water in an infinitely long canal, due to any 

 arbitrary initial conditions. Thus, let us suppose that, when = 0, the equa- 

 tion of the free surface is 



*=/(*), ....................................... (i), 



and that the normal velocity at the surface is then F(x\ or, to our order of 

 approximation, 



(ii). 



The value of is found to be 



