198 ADVANCED ELECTRICITY AND MAGNETISM. 



g - rv> (7) 



Differentiating again, we get: 



&y d(F'(z)} dz 

 a? ~ dz ' fe 





but = I and = F"(z), therefore, we have: 



ox 02 



S = F "v (9) 



By differentiating twice with respect to / we get: 



S 2 v 



^ = *F"(z) (10) 



and by combining equations (9) and (10) we get: 



This same result would be obtained from equation (3) by pro- 

 ceeding as above. Indeed equations (2) and (3) both satisfy 

 the differential equation (n) and the general solution of (n) is: 



y = F(x -vt) +f(x + vt) (12) 



This equation represents two curves of any shape superposed 

 upon each other, one traveling to the right at velocity v, the 

 other traveling to the left at velocity v. 



115. Equation of motion of a stretched string. It is very 

 helpful to consider wave motion on a stretched string before 

 considering the equations of electric wave motion on a trans- 

 mission line. When a stretched string is in equilibrium it is, of 

 course, straight. Let us choose this equilibrium position of the 

 string as the #-axis of reference as shown in Figs. 136 and 137, 

 and let us set up the differential equation which expresses the 

 mode of motion of the string while the string is vibrating or while 

 a bend is traveling along the string as a wave. We will assume 

 that each part of the string moves only up and down (at right 



