140 BELL SYSTEM TECHNICAL JOURNAL 



the conditions. For example, if h — ic/{\ -\- exp v) then h —^ ic exp {—v) 

 as z) ^ 00 , and the solutions of (3-1) behave like Bessel functions of order 

 zero and argument c exp {-v). It may be verified that these solutions do 

 not satisfy {i-2>). Again, condition {2>-i) may be satisfied without y having 

 much resemblance to an outgoing wave at i) = oo . Thus if /? — > ia/v as 

 z; — )■ oo , y inc eases like v" whe e )i' — )i — a — 0. When < a < 1/2 both 

 values of n He between and 1, and both solutions satisfy {S-3). Despite 

 these sho-tcomings it still seems best to etain (3-3) to specify the behavio" 

 oi y Bit V — 00 . 



It should be mentioned that P. S. Epstein^ has obtained the reflected 

 wave by transforming the hypergeometric differential equation into the 

 form (3-1). This method has been extended by K. Rawer^ who gives a 

 number of references in which the approximation (3-4) is used to study 

 propagation in a medium having a variable dielectric "constant". An 

 interesting paper on the general subject of reflection in non-uniform trans- 

 mission lines has been written by L. R. Walker and N. Wax*. 



1. When most of the reflection occurs in a short interval, say near v = 0' 

 R may be obtained by numerical integration of (3-1). One method is to 

 start at z) = with the initial conditions y = 1, y' = and work outVvards 

 in both directions. Let Ya(v) denote this solution and Vb{v) the solution 

 obtained by starting with y = 0, y' = 1. The general solution is 



y = C^Yaiv) + C^Ybiv). (3-6) 



Ci and C2 are to be determined by the conditions 



y — (constant) lr^'-e~^ , v > V2 (3-7) 



y = (/c//?)'/-[e-f + Re^ , V < vi ' (3-8) 



where I'l and V2 are large negative and positive values, respectively, of v. 

 These conditions lead to equations for Ci ,€2 , R- 



[y'+ ^y],.„, = 



[y' - e-y + 2(khy' e-~^l=,, = (3-9) 



[y' + ry - 2(ichy'Re^],^,, = 



in which ^ is given by (3-5) and 



e^ = hzL h'/(2h). ■ (3-10) 



The required value of R is obtained by letting di -^ — 00 , ^jg ^ °° in the 

 expressions, which follow from (3-9), 



