CONFORM. I L TRA NSFOR.UA TION 1 3 1 



chosen to be moderately large positive and negative numbers, respectively. 

 It turns out that, when ki and k are very small, this is related to the "excess 

 capacity" localized at the irregularity whose effect must be added to that 

 of the mismatch, indicated by (A3-6). 



When the entering and leaving guides are of the same size it is still possible 

 to use the formulas of this appendix. N~(vn) may be replaced by an expres- 

 sion which now has for its limiting value 



,V-(^) = 1 + i(k/2) f i(v) dv (A3-8) 



// •/;/ Plane of Irregularity 



Let the figure corresponding to the irregularity be Fig. 3 with b and bi 

 replaced by a and ai , respectively. In addition to the quantities c and k 

 defined by equations (5-2) we define 



Ki = 2ai/Xo , c, = {k\ - 1)''' (A3-9) 



where we assume k and ki to lie between 1 and 2. At u = — =0 P{v^ 0) 

 still consists of the unit incident wave plus the reflected wave given by the 

 first of equations (5-4) and g{v, 6) is still zero. However, now, a.t v = x , 



P{v, 6) - Tac'^''" sin e 



|(.x>) - kik"' - 1 = K~\c\ - c") (A3-10) 



The integral equation for P{v, 6) and Th is 



P{v, , do) - e-'"" sin ^0 + ;^ dv 



Zw •'—00 ^0 



■dd{g(T, e)P{v, e) - g{i)T„er''''' sin e\G{v, , e, ; V, d) 



+ Til sin doFni^'o) 

 in which 



Fh(vo) = e-''^"'g(To)/g(^) - e-'-^ATivo) - e''"> M"- (z'o) 



J— 00 



(A3-11) 



dv 



(A3- 12) 



-{-c)v 



M^ivo) = ^(2cr\c + c:)-' [ i'{i)e-''''^"' dv 



J Da 



First approximations are 



ri" = l/M-(oo), R*-^^ = -M+(-oo)/M-(oo) (A3-13) 



