536 BELL SYSTEM TECHNICAL JOURNAL 



Exponential Propagation 



While electromotive forces could be applied in such a way that the 

 fields would be of the kind given by (3), in the coaxial transmission 

 line as actually energized the fields are of the circular magnetic type 

 (2) which will claim our special attention in the next few sections. 



In order to solve equations (2), we naturally want to eliminate all 

 variables but one. This purpose can be readily accomplished if Ej 

 and Ep are substituted from the first and the last equations of the set 

 into the second. Thus, we obtain the following equation for the 

 magnetomotive intensity: 



+ -^= C.W,, (6) 



d_ ri djpH^y 



dp Lp dp 

 where 



a^ = gwiii — aj^€ju. (7) 



Adopting the usual method of searching for particular solutions of 

 (6) in the form 



H, = R{p)Z{z), (8) 



where R{p) is a function of p alone, and Z{z) a function of s alone, 

 we get 



l^=p (9) 



1 ^j ^ ^, _ ^,^ 

 P dp J 



1 A 

 Rdp 



where V is some constant about which we have no information for 

 the time being. 



Equation (9) is well known in transmission line theory; its general 

 solution can be written in the form 



Z = Ae^' + Be-^', (11) 



where A and B are arbitrary constants. The solutions of (10) are 

 Bessel functions. Since equation (6) is linear, we may invoke the 

 principle of superposition and add any number of particular solutions 

 corresponding to different values of F. Thus we can form an infinite 

 variety of other solutions so as to satisfy the physical conditions of 

 various practical problems. 



It is seen at once from the first and the last equations of the set (2) 

 that to each H^ of the form (8) there correspond an Ez and Ep of the 

 same form; i.e., there exist circularly symmetric electromagnetic 



