WIRE TRANSMISSION THEORY 279 



To formulate the retarded potentials ot the system under consideration 

 we have recourse to the Sommerfeld integral 



^'= r/„(,xv-..-.'.V^^^L=. (4a) 



r Jo VX^ - /3- 



where p = V/ + -^^ and Jo is the Bessel function in the usual notation. 

 Applying this integral to the system of currents and charges under 

 consideration, and remembering that they are surface currents and 

 charges at p = a and p = b respectively, we get without difficulty, 

 for X — 0, 



$ = (2o /o(pX)[/o(flX) - Mb\)2e 

 Jo 



-xyjy--^--^ 



\d\ 



Jo(pX)[/o(aX) - /o(6X)>-VX^^ -VX^^=^ 



- _ 



which reduces to 



4> = 2Qoe-'^^ I* /o(pX)[/o(aX) - /o(6X)]^ 



(5a) 



-QoJ^ -^o(pX)[/o(aX)-/o(&X)]-;^P=^|-— ^^^^^— ^ 



(6a) 



Since the currents are entirely axial, we have also Ay = Az = 0, and 



from (2a) 



A, = $. (7a) 



The first integral in <!> represents a potential wave propagated 

 along the X axis in precisely the same way as the current and charge; 

 it will therefore be termed the homogeneous potential wave. We find 

 further that the field derivable from the homogeneous potentials is 

 precisely the principal wave field, as given by the particular solution 

 of Maxwell's equation, corresponding to 7 = i^. 



The second integral in $ represents a potential wave propagated 

 in an entirely different manner, and dying away for sufficiently large 

 values of x. The corresponding field may be called, for want of a 

 better term, the heterogeneous field, since its mode of propagation is 

 quite different from that of the current and charge. It is this field 



