2Co ELEMENTS OF ELECTRICITY AND MAGNETISM. 



the kinetic energy, we have what is called a pure wave, and when 

 the potential energy in a wave is not equal to the kinetic energy 

 the wave is called an impure wave. 



The behavior of an impure wave pulse in a canal may be stated 

 by considering an extreme case of an impure wave as follows : 

 Consider an elevated portion of still water in a canal as shown 



still water 

 still water [-..'... ; -.1 still water 



Fig. 192. 



Fig. 193. 



thickness is reduced to x/(x-}-A) or to (l hjx) of a centimeter, h being very 

 small. Therefore, the decrease of thickness is h] x of a centimeter. The force 

 acting to reduce the thickness of the slice is to be considered as that force which is 

 due to the increase of pressure in the water produced by the increasing depth h. 

 This increase of pressure is equal to hdg dynes per square centimeter when the slice 

 has reached its greatest depth, so that the average increase of pressure due to increas- 

 ing depth is \hdg, which produces over the face of the slice a force equal to 

 \hdg X bx, and the product of this force and the decrease of thickness of the slice 

 gives the work done in decreasing its thickness. This work must be equal to the 

 original kinetic energy of the slice, so that 



\dbxv* = \dbh*g 



Consider the instant t seconds after the closing of the gate in Fig. 191. The wave 

 of arrest W has reached the distance Vt from the gate, and the excess of water that 

 is represented by the raising of the water level (= Vty^h~y^b cubic centimeters) is 

 the amount of water supplied by the flow of the canal in / seconds (=bxvt cubic 



centimeters'). Therefore 



Vthb = bxvt 



Substituting the value v from equation (i) in equation (ii),we have 



V= Vgx (iii) 



Therefore the velocity of progression of a wave in a canal is equal to the velocity 

 gained by a body in falling freely through the distance xJ2. 



