260 Lord Rayleigh on Waves. 



turbed level, b the breadth at that level. Then, as before, 



(A + bh)u=Ait Q 

 if h be small ; whence 



2 2 2M 2 



Now by dynamics 



w »— u 2 =2gh 



if the upper surface be free ; and thus 



which gives the velocity of propagation. In the case of a 

 rectangular section we have the same result as before, since 

 A=R 



The energy of a long wave is half potential and half kinetic. 

 If we suppose that initially the surface is displaced, but that 

 the particles have no velocity, we shall evidently obtain (as in 

 the case of sound) two equal waves travelling in opposite direc- 

 tions, whose total energies are equal, and together make up 

 the potential energy of the original displacement. Now the 

 elevation of the derived waves must be half of that of the ori- 

 ginal displacement, and accordingly the potential energies less 

 in the ratio of 4 : 1. Since therefore the potential energy of 

 each derived wave is one quarter, and the total energy one 

 half of that of the original displacement, it follows that in the 

 derived wave the potential and kinetic energies are equal. 



We may now investigate the effect on a long wave of a gra- 

 dual alteration in the breadth of the canal and the area of the 

 section. The potential energy of the wave varies directly as 

 the length, breadth, and square of the height ; and, by what has 

 been proved above, the same is true of the total energy. Now 

 the length of the wave in various parts of the canal is obvi- 

 ously proportional to the velocity of propagation, viz. a / _ . 

 and we may therefore write 



E a a/^. (height) 2 . 6. 



But when the alteration in the canal is very gradual, there is 

 no sensible reflection and the energy of the wave continues 

 constant ; so that 



height ocA~*&~*. 

 In the case of a rectangular section, 



height ozl~~*b~K 



