84 BELL SYSTEM TECHNICAL JOURNAL 



where K is the characteristic impedance of the artificial hne (at mid- 

 series position). Hence by (223) and (222) 



11 1 



K Z2(l-e-r)+2p 



(224) 

 ^ 1 1 ^ 1 1 



222\/p + p2 V2l32 Vl+P 



By aid of the preceding the direct current wave can be written as 



^^^ Vo_Wl+p-V'pr (225) 



■\/ziZ2 Vl+p 



This formula is not so physically suggestive as its equivalent 



in ^e 



but is useful when we come to the solution of the operational equation. 

 Before proceeding with the operational equation, and the investi- 

 gation of transient phenomena in artificial lines, it will be of interest 

 to deduce from the foregoing the unique and remarkable properties 

 of wave filters in the steady state. For this purpose we return to 

 equation (221) 



cosh r = l + 2p. 



Now suppose that the series impedance Zi is an inductance L and 

 the shunt impedance 22 a capacity C, so that, symbolically, 



. ^ 1 co-LC 



and 



cosh r = l-|co2LC. (226) 



Now let us write Y = iQ, where i = \/ — 1; the preceding equation 

 becomes 



cos = 1-1 w^LC (227) 



and the ratio of currents in adjacent sections is e~'^. Consequently ij 

 6 is a real quantity the ratio of the absolute values of the currents in ad- 

 jacent sections is unity, and the current is propagated without attenuation. 

 Inspection of equation (227) show^s that d is real provided the right 

 hand side lines between -j-1 and —1: or that w lies between and 

 2/\/LC. Consequently this type of artificial line transmits, in the 

 steady state, sinusoidal currents of all frequencies from zero to 

 l/iry/TC without attenuation. It is known as the low-pass filter. 



