146 



BELL SYSTEM TECHNICAL JOURNAL 



Although the WKB approximation has the same form as (3-30) in the 

 region where v is finite, we regard (3-30) and (3-31) as being the exact limiting 

 forms of y. Hence, g^ may differ from /"+. 



Letting I'o — ^ — oc in (3-29) and comi)aruig the result with (3-30) gives the 

 exact result 



J— 00 



(3-32) 



which leads to an approximation for the reflected wave when y{v) is known 

 approximately. 



The integral equation (3-29) may be obtained by premultipiying both 

 sides of y = Ay by the transpose of the approximate Green's matrix 



Ga(vo , V) = 



— 2^6" ~ So , 



V < Vi 





— |5e"° ~ So , V > Vo ■ 



and integrating by parts twice. It is seen that each column of Ga(vo , v) 

 is an approximate solution of y = Ay, in which the columnar constants of 

 integration are the columns of -So , and represents a wave traveling away 

 from Vo in both directions. Gaii'o , ^') is continuous at v = Vo and 



^ Ga(Vo , V) 



dv 



-~ Ga(Vo , V) 



dv 



= So^oSo = I 



Thus the Hth column of Ga(vo , v) gives the approximate values of yi(v), 

 y-ii"!-'), ■ ■ • , yn(v), subject to the conditions that all these and all of their first 

 derivatives are continuous at t; = Vq except y„{v) which has the jump 

 y„(i'o+0) - y„ivo-0) = 1. 

 The presence of 



2*^' + $5' = *5' - ^S'SS~' 



= ^{S'S - S'S)S~' 



in (3-29) and (3-32) makes the X variable case somewhat different from the 

 case xV = 1. 



vS. When Z,„,i and I^„„ are slowly varying functions of v the approximate 

 solution of the transmission line equations 



dv 

 dJr, 



= -T z J 



.V 

 = - Z Vmn Vn 



(3-32) 



