CON FORM A L TRA NSFORMA TION 1 20 



APPENDIX III 



Integral Equation When Guides Entering and Leaving Irregularity 

 Are of Different Sizes 



Here we shall indicate how the integral equation method may be extended 

 to cover the case mentioned in the above title. It is supposed that only the 

 dominant mode is propagated freely in both guides. 



E in Plane of Irregularity 



Let the notation for the guide carrying the incident wave be the same as 

 for the ^-corner, b denotes the narrow dimension of the guide and the 

 quantities k and 7^ are given by (2-1). Both guides have the same wide 



INCIDENT WAVE 







Fig. 3 



dimension a. The narrow dimension of the guide shown on the right of 

 Fig. 3 is ii . We introduce the new quantity 



ki = [(26i/Xo)- - (b,/ayf" (A3-1) 



to correspond to k. Since, by assumption, only the dominant mode is 

 freely propagated in both guides both k and ki are real positive quantities 

 less than unity. 



Let z = f{w) carry the system of Fig. 3 into a straight guide of width tt 

 in the w = v -\- id plane (see Fig. 2), and let g{i', 6) be defined by 



^ + giv,e) = |/'(ziO|-. 



The behavior of g{v, 6) at infinity is shown by the table 



V dz/dw g{v, &) 



— <x> b/ir 



+ 00 bi/r kikT — 1 



where bi/b = ki/k has been used. It is convenient to introduce the ap- 

 proximation g{v) to g{v, 6). g(v) may be chosen at our convenience subject 

 only to the conditions that it be differentiable, g{— 00) = 0, and |(°o) = 

 klk-'- - 1. 



When we define G by equation (3-3) so that, as before, it is the Green's 



