ELECTRIC CIRCl'IT THEORY 79 



/(/) and Wit) respectively, while ihc first component of (210-1)) and 

 (211-b) introduce merely a delay. Thus, if the e.m.f. impressed 

 at time / = is/(/), the corresponding waves for t'>x/v or t'>x, are 



/" (212) 



+ / j\t-t,)J'{tMtu 



V=e-"'f{t-x/v) 



r (213) 



+ / f(t-l,)W'(tr)dt„ 

 Ux/v 



where J'{t)=^J{t) and W\t) ='^r,W{t). 

 at at 



The integrals of (212) and (213) can be computed and analyzed 

 in precisely the same way as discussed in connection with the non- 

 inductive cable problem, and are of very much the same character 

 as the alternating current waves of the cable. In the total waves, 

 however, as given by (212) and (213), a very essential difference is 

 introduced by the absence of the first terms, which represent undis- 

 torted waves propagated with velocity v. Thus, if the impressed 

 e.m.f. is sin w/, (212) and (213) become 



\C 

 I = ^^e "'^ sin iv(t — x/v) 



+ / sin co(/-/i)/'(/i)rf/i, for/>.r> 



Jx/v 



x/v 



7=g-"v sin wit — x/v) 



-I- / sin a)(/-/i)ir(/i)r//i, for/>x/z'. 



tJx/v 



(214) 



(215) 



Now the first terms of (214) and (215) are simply the usual steady- 

 state e.xpressions for the current and voltage waves when the fre- 

 quency is sufificiently high to make the steady-state attenuation 

 constant equal to a and the phase velocity equal to v. Furthermore 

 the integral terms become smaller and smaller eis the applied fre- 

 quency w/27r is increased. It follows, therefore, that for high fre- 

 quencies the waves assume substantially their final steady value 

 at time t = x/v, and that the tails of the waves, or the transient 

 distortion, becomes negligible. This is a consequence entireh' of the 



