ELECTRIC OSCILLATIONS AND ELECTRIC WAVES. 259 



along the canal without changing its character, friction being 

 neglected. 



An essential feature of any wave which moves along without 

 changing its shape is that the kinetic energy is equal to the potential 

 energy in the wave at each point. Thus, the kinetic energy of 

 the water wave A, Fig. 190, due to the uniform velocity v of 



gate 

 stationary 



moving "water 



W still water 



gate 



Fig. 190. 



the water in the wave is equal to the potential energy due to the 

 elevation //.* When the potential energy in a wave is equal to 



*The following derivation of the velocity of a water wave in a canal shows the 

 significance of equality of potential and kinetic energy. This discussion is based 

 upon a slight modification of the conditions shown in Fig. 189, as follows : Water of 

 depth x flows along a canal of rectangular section at a uniform velocity (small) of 

 v centimeters per second. A gate is suddenly closed as shown in Fig. 191 ; the 

 moving water, in being brought to rest against the gate, heaps up to a depth x -f- h ; 



and a wave of arrest W, Fig. 191, 

 moves along the canal at a definite ve- 

 locity V. The action involved in Fig. 

 191 is identical to the action involved 

 in Fig. 189. In fact, Fig. 189 can be 

 converted into Fig. 191, by imagining 

 everything in Fig. 189 to be moving to 

 the right at velocity v. The discussion 

 of Fig. 191 is simpler than the discus- 

 sion of Fig. 189 because the potential energy is in one portion of the water and 

 the kinetic energy is in another portion, whereas in Fig. 189 the potential energy and 

 the kinetic energy are both in one portion of the water. Let b be the breadth of the 

 canal. Consider a transverse slice of water one centimeter thick. The volume of 

 this slice is bx cubic centimeters and its mass is dbx grams, where d is the density 

 of the water in grams per cubic centimeter. Therefore the kinetic energy of thisslice 

 of water when it is- moving at a velocity of v centimeters per second is \dbxv*. 

 When the wave of arrest W, Fig. 191, reaches the slice of water under consideration, 

 the slice, as it comes to rest, is squeezed together and increased in depth to x -f- h. 

 The slice is decreased in thickness in proportion to its increase in depth, so that its 



Fig. 191. 



