CONFORM A L TRANSFORMATION 123 



where 



00 



B - J2 n-'ie-'"' - 2(-)"e-"'] 



n=l 



^„ = COS 2kl - [2cos kt - (-)"r''."']^ 



Equation (7-18) is an approximation, to within terms of order a-, for the 

 reflection coefficient of a truncated corner which turns through a small 

 angle lira. The electric vector lies in the plane of the bend. When t = 0, 

 (7-18) reduces to (7-5) by virtue of 2a — /3. 



APPENDIX I 



CoNFORM.'VL Transformation of Truncated Corner 



We shall use a Schwarz-Christofifel transformation* to carry the guide of 

 Fig. 1 into the straight guide of Fig. 2. The first step is to transform the 

 interior of Fig. 1 into the upper half of an auxiliary complex plane which we 

 shall denote by f . Let the points 21,22,23,24, Z5 in Fig. 1 correspond to 

 the points — h, h, 1, <x) ^ —I in the f plane. A suitable transformation 

 is then 



z = D-\- e[ (t-]- hy'^ir - h)--{T - 1)-i(t + Vj-^dr (A 1-1) 

 Jo 



where D, E and h are to be determined from the geometry of Fig. 1. Because 

 of the symmetry of our transformation about the line joining Zo and 24 

 it follows that 2 = 2o corresponds to f = 0. Hence D = Zq . As f travels 

 from 1 — € to 1 + e, € being very small and positive, along a semicircular 

 indentation above ^ = 1, z as given by (A 1-1) increases by 



2\-o 



£(1 - h')-2-' I \r ~ \r' dr ^ ^— (1 - h') 

 Ji-( 2 



while, according to F'ig. 1, it increases from x -|- /O to ^ + ib. Hence we 

 set the real part of E equal to — 2^~'(1 — //-)". We have tacitly assumed 

 the factors in (Al-1) to have their principal values at t = 1 + e and also 

 that < ^ < 1. As 2 goes from 21 to 2-2 , i" goes from —h to -\-h. In this 

 range arg(r -\- h) = and arg(T — h) = w. 

 Consequently, if | 2-2 — 2i | = f, then 



22 — 2i — 



J-h 



* See, for example, S. A. Schelkunoff, Electromagnetic Waves, New York (1943) 

 pp. 184-187. 



