80 BELL SYSTEM TECHNICAL JOURNAL 



presence of inductance in the line, and shows its extreme importance 

 in the propagation of alternating waves and the reduction of transient 

 distortion. 



It should be pointed out, however, that if the line is very long and 

 the attenuation is very high, the integral terms of (214) and (215) 

 are not negligible unless the applied frequency is correspondingly very 

 high. For example, on a long submarine cable, the a.c. attenuation 

 is so large that the first terms of (214) and (215) are very small, and 

 /(/) is very large compared with ^/C/L e""*. Consequently here 

 there is very serious transient distortion and alternating currents 

 are therefore not adapted for submarine telegraph signalling. 



This discussion may possibly be made a little clearer, without 

 detailed analysis, if we recall the discussion of alternating current 

 propagation in the non-inductive cable of the preceding chapter. 

 From that analysis it follows that, when the applied frequency aj/27r 

 is sufficiently high, the integral term of (214) becomes approximately 



-J' it) 



03 



and the complete current wave is 



4 



^e-"' sin co(/-x/z;)+— /'(/) (216) 



Li (^ 



and similarly the voltage wave is 



e---smco(_t-x/v)+-W'{t). (217) 



CO 



Now if the total attenuation ax is large the last terms of (216) and 

 (217), before they ultimately die away, may become very large com- 

 pared with the first terms, which represent the ultimate steady-state. 



Appendix to Chapter VII. Derivation of Formula (211) 



The only troublesome question involved in deriving (211) from 

 (207) and (209) is that we have to differentiate with respect to x, in 

 accordance with (207), the discontinuous function F{t). To accom- 

 plish this we write (209) in the form 



F{t) =<j){t-x/v)e~P'Io{<TVt'-x'/v') (209-a) 



where 0(0 is defined as a function which is zero for t<x/v and unity 

 for f^x/v. Clearly this is equivalent to (209) and permits us to deal 



