546 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XIII. 



Then for 



x # 2 > at or #! x > at 



all the functions occurring have the value zero, therefore V is zero. 

 This at once distinguishes the solution from that for the case of 

 the cable, for the disturbance does not arrive at x until the time 



fp J _ f*\ 



t = - on the right, 



Cv 



or t = ^-? on the left. 



a 



When x x > at > x # 2 



we have 



This represents the disturbance while the wave is passing over, 

 for a point on the right. In like manner for a point on the left, 

 for #j x < at < # 2 x, 



(33) V 



Finally, at later instants, 



x x l < at or # 2 x < at, 



abt rxx l / % 



(34) V= -YJ^ F (a - X) (b + | 



This represents the disturbance after the wave, travelling with 

 velocity a, has passed on. Accordingly the solution, while repre- 

 senting a wave travelling with the velocity a, as in free space, 

 differs from that case in that there remains a residue, or tail to the 

 wave, which does not fall to zero however great the time. The 

 exponential factor shows that the disturbance, both in the wave 

 proper, and in the tail, is continually becoming attenuated. 



Thus when successive impulses are transmitted, each leaves 

 a tail, which interferes with all the succeeding waves, and the 

 possibility of telephonic speaking depends not only on the attenua- 

 tion and distortion with the distance, but on the magnitude of 

 the tail of the wave. The tail also explains the discrepancies 

 that existed between the results of the attempts made to determine 



