92 BELL SYSTEM TECHNICAL JOURNAL 



The dotted curves represent the current in the corresponding 

 smooth Hne. For the smooth Hne, the current, as we have seen, is 

 discontinuous, being identically zero for a time vt = n and having an 

 instantaneous jump to its final value \^ClL at vt = n. The current 

 in the artificial or periodically loaded line differs from that in the cor- 

 responding smooth line in three important respects: (1) the absence 

 of the abrupt discontinuous wave front, (2) the presence of super- 

 posed oscillations, and (3) the absence of a true finite velocity of propaga- 

 tion. It will be observed, however, that the current in any section 

 is negligibly small or even sensibly zero until vt = n, so that the current 

 is propagated with a virtual velocity 1/\/LC per section. The pres- 

 ence of a well marked wave front is also evident although this is not 

 abrupt, as in the smooth line. The effective slope of the wave front 

 becomes smaller as the current wave travels out on the line, decreasing 

 noticeably as the number of sections is increased. When the number of 

 sections becomes large, however, the decrease in the slope is not rapid, 

 being in the 500*^'' section about 60 per cent, of that in the 100*"' section. 



The superposed oscillations are of interest. These are initially 

 of a frequency depending upon and decreasing with the number of 

 sections, n, but in all sections ultimately attaining the frequency 



ir\/LC T 



which is the critical or cut-off frequency of the line, above which 

 steady-state currents are attenuated during transmission and below 

 which they are unattenuated. When vt is larg e com pared with n 

 the amplitude of these oscillations becomes ^/I/tvI so that they 

 ultimately die away and the current approaches the value VC/L 

 for all sections. The current in the loaded line is thus asymptotic 

 to the current in the corresponding smooth line and oscillates about 

 it with diminishing amplitude and increasing frequency. 



Since the abscissas of these curves represent values of 2vt = 2t/\/LC, 

 and the ordinates are to be multiplied by s/ CIL to translate into 

 actual values, the curves are of universal application for all values 

 of the constants L and C. 



The investigation of the building-up of alternating currents in wave 

 filters and loaded lines is very important. It depends for the non- 

 dissipative case on the properties of the definite integrals 



sin WT Jn{r)dT, 







cos WT Jn(j)dT, 



