22 BELL SYSTEM TECHNICAL JOURNAL 



If the transmission line is non-uniform, that is, if Z and Y are func- 

 tions of X, then the solutions of (2) are usually more complicated. 

 In any case, however, there are two linearly independent solutions 

 I'^{x) and I^{x) in terms of which the most general solution can always 

 be expressed 



I{x) = AI+{x) + BI-{x). 



These independent solutions may represent either progressive waves 

 in two opposite directions or certain convenient combinations of such 

 waves. 



The corresponding F-functions are found by differentiation from 

 (2) ; thus 



^^ Y dx ' ^ ^^ Y dx • 



The impedance of the F+, /+-wave is then 



^o-(x) =^^= -——= --- (log I-). 



Similarly the impedance of the V~, /"-wave is 



7 ~( \ V~{x) \ dl- \ d . . 



The negative sign in (5) is merely a matter of convention: the "posi- 

 tive" and the "negative" directions of the transmission line are so 

 defined that the real parts of Zo+ and Zq~ are positive. 



In general, Zo+ and Z^r are not equal to each other. Moreover, 

 there is a considerable amount of arbitrariness in our choice of the 

 basic solutions /+ and /~. Thus, we are brought face to face with 

 the fact that we must regard the impedance as an attribute of the 

 wave as well as of the transmission line. This point of view will 

 become even more prominent when we come to deal with the wave 

 transmission in three-dimensional media. There even progressive 

 waves may have different characters (they may be plane, cylindrical, 

 spherical, etc.) and the impedances of the same medium to these waves 

 will be different. And naturally, it goes without saying that the 

 impedances to like waves in different media may also be different. 

 One could, perhaps, take the position that geometrically similar 

 waves in different media are not really alike if the corresponding 

 "force/velocity" ratios are not equal and that under all circumstances 

 the "impedance" is the property of a wave. However, "intrinsic im- 



