Reflection from Corners in Rectangular Wave Guides — 

 Conformal Transformation* 



By S. O. RICE 



A conformal transformation method is used to obtain approximate expressions 

 for the reflection coefficients of sharp corners in rectangular wave guides. The 

 transformation carries the bent guide over into a straight guide filled with a non- 

 uniform medium. The reflection coefficient of the transformed system can be 

 expressed in terms of the solution of an integral equation which may be solved 

 approximately by successive substitutions. When the corner angle is small and 

 the corner is "not truncated the required integrations may be performed and an 

 exphcit expression obtained for the reflection coefficient. Although appUed here 

 only to corners, the method has an additional interest in that it is applicable to 

 other types of irregularities in rectangular wave guides. 



Introduction 



THE propagation of electromagnetic waves around a rectangular corner 

 has been studied in two recent papers, one by Poritsky and Blewett^ 

 and the other by Miles-. Poritsky and Blewett make use of Schwarz' 

 "alternating procedure" in which a sequence of approximations is obtained 

 by going back and forth between two overlapping regions. Miles derives 

 an equivalent circuit by using solutions of the wave equation in rectangular 

 coordinates. Several papers giving experimental results have been pub- 

 lished. Of these, we mention one due to Elson^ who gives values of reflection 

 coeflScients for various types of corners. 



Here we shall deal with the more general type of corner shown in Fig. 1 

 by transforming, conformally, the bent guide (in which the propagation 

 "constant" of the dielectric is constant) into a straight guide in which the 

 propagation "constant" is a function of position— its greatest deviation 

 from the original value being in the vicinity of points corresponding to the 

 corner. This type of corner has been chosen for our example because it 

 possesses a number of features common to problems which may be treated 

 by the transformation method. 



The essentials of the procedure used are due to Routh* who studied 

 the vibration of a membrane of irregular shape by transforming it into a 

 rectangle. After the transformation the density (analogous to the propaga- 

 tion constant in the guide) was no longer constant but this disadvantage 

 was more than offset by the simplification in shape. 



Until this paper was presented at the Symposium I was unaware of any 



* Presented at the Second Symposium on Applied Mathematics, Cambridge, Mass. , 

 July 29, 1948. 



1 See list of references at end of paper. 



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