90 



Generation, Growth and Propagation of Waves 



energy, travels with the wave (see p. 22); the kinetic energy, on the contrary, 

 is constantly gathered at the forward edge of the wave and left behind at 

 the rear edge. This can be illustrated by considering a parallelepiped of unit 

 width, extending to a depth below which wave motion is negligible and whose 

 forward and rear edges travel beneath two adjacent wave crests (see Fig. 42). 



Fig. 42. Energy changes of an individual wave of length L (A in text) travelling from left 



to right with a velocity c. 



At the forward edge of the moving parallelepiped energy is gained at the 

 rate of 



E , . 8 ( E 

 C 2^ X Vx\ C 2 



and at the rear edge energy is lost at the rate c\E. The total energy budget 

 is therefore 



d(EX) 

 dt 



R * ±R «+U4 



(IV. 36) 



The rate at which the wave length increases and therefore at which the 

 parallelepiped "stretches", is determined by the difference in speed between 

 the two adjacent wave crests: 



dk dc , 8c 1 dk , gX 



~r = — X or — = T -=- or again as c- = ^~ 

 dt 8x 8x X dt 6 2n 



this equation can be written 



8c _ 2 dc 

 8x c dt 



