ELECTRIC OSCILLATIONS AND ELECTRIC WAVES. 197 



is x = x' + vt so that x' = x vt. Therefore, substituting 

 this value of x' in equation (i) we have: 



y = F(x vt) (2) 



and this is the equation of the traveling -curve cc referred to the 

 stationary origin 0. 



moving 



Fig. 135. 



Similarly it may be shown that 



y=f(x + vt) (3) 



is the equation of a curve traveling to the left at velocity v with 

 respect to a stationary origin. 



The two equations (2) and (3) satisfy one of the most important 

 of the differential equations of physics, namely, the differential 

 equation of wave motion for the case in which no energy is dissi- 

 pated by friction or resistance; and it is helpful to derive this 

 differential equation from equation (2) or (3) as follows: 



Let the quantity x vt be represented by the single letter z. 



That is: 



z = x vt (4) 



and equation (2) becomes: 



y = F(z) (5) 



dy d^y 



Let -r- be represented by F'(z) and let be represented by 

 az cLz 



F"(z). Then, according to the rule for differentiating a function 



of a function, we have: 



dy _dy dz 

 dx~ dz ' dx 



dz 



but -- = I, so that 

 ox 



(6) 



