RECTANGULAR WAVE GUIDES 141 



y = C2/C, = -[(r: + d+ya)/{yl + e+Y,)U,, 



r = [y'/yUv, = [(rl + yY',)/{ya + Tn)].-«. (3-11) 



ie = [(^ + r)/(r - r)]..„, exp -ikv, - 2 j^ {h - ic) dv\ 



where the arguments of Ya{v) and Yb{v) have been omitted for brevity. 



If // should change from a positive imaginary quantity to a positive real 

 quantity in {vi , v^) and remain greater than some fixed positive number for 

 z) > z;2 it may be shown that | i? | = 1 (7 and T are real and Im ^ = Im d~, 

 Real ^ = —Real ^ at d = vi). This complete reflection is to be expected 

 .from physical consideration. 



2. An exact expression for the reflection coefficient which holds when h 

 satisfies the conditions following (3-1) (in particular it must not pass through 

 zero anywhere in — co < v < qo ) is 



R = i(ic)-^ r e-^y{v) ^, h-^ dv (3-12) 



J- M dv- 



where ^ is given by (3-5). Before this integral for R may be evaluated 

 y{v), and hence R itself, must be known. Nevertheless, when R is small a 

 useful approximation may be obtained by using the WKB approximation 



y{v) = (ic/hyh-^ (3-13) 



Thus 



R^- r e-'^h"' ^h-Uv 

 2 J- 00 dv- 



= 2,L' U^ ( 



5 ,,-5/2 (dK\ 1 -,-3/2 d K 



Til 1 _ t 



(3-14) 



dv / 4 dv 



dv 



in which K = — li\ 



The expression (3-12) for R is obtained by letting vo^ — ^ in the integral 

 equation 



y{ro) = {ic/hfe-^' - f_^ Ga(vo,v)yiv)h^~^Jt-Uv, 



, ^ {e^-^\ V < ro ^ ^ 



Ga(ro,r)=-P^^/rM ^ ^ (3-l:>) 



^0 - ^ = / lidv, ho = //(z'o), ?o = ^(I'o). 

 Go(^o , "v) is the approximate Green's function suggested by (3-13). The 



