200 ADVANCED ELECTRICITY AND MAGNETISM. 



dy 



is the increase of -7- from a to b, and this increase is 



d-y 

 equal to -7-^ dx. This is evident when we consider that 



d 2 y dy 



is the rate of increase of j- with respect to x. There- 



d 2 y d 2 y 



fore, substituting dx for tan 6' tan 0, we get Tj^ dx 



as the value of the net upward force acting on the portion ab 

 of the string, and this net upward force is equal to the product 

 of the mass, m-dx, of the portion and the upward acceleration, 



d 2 y 



, of the portion. Therefore we have: 

 at 



or 



dit^mdx 2 



This equation is identical with equation (n) of Art. 114 and its 

 general solution is: 



y = F(x - vt) + f(x + vt) (2) 



where 



Fr 



(3) 



The term F(x vt) represents a wave (a bend of any shape) 

 traveling along the string from left to right, and the term 

 f(x + vt) represents a wave (a bend of any shape) traveling along 



If 



the string from right to left; the velocity of travel being V-* 



nt 



cencimeters per second in each case. 



Another point of view. It is evident from Art. 114 that travel, 

 pure and simple, is about the only thing that is established by 

 the solution of equation (i) above. Therefore one might expect 

 to obtain a perfectly clear and simple insight into the motion 

 of a stretched string by introducing the idea of travel at the 



