Reflections from Circular Bends in Rectangular 

 Wave Guides — Matrix Theory 



By S. O. RICE 



A method of computing reflections produced by circular bends in rectangular 

 wave guides is presented. The procedure employs the theory of matrices. Al- 

 though the matrix equations are quite simple, a considerable amount of calculation 

 is necessary before quantitative results may be obtained. Fortunately, the ap- 

 proximate formulas pertaining to gentle bends hold surprisingly well for rather 

 sharp bends. These formulas are obtained by a limiting process from the matrix 

 equations. The approximate formula for reflection from an H-bend (in which the 

 magnetic vector lies in the plane of the bend) generalizes an earlier result due to 

 R. E. Marshak. The corresponding formula for the E-bend appears to be new. 



Introduction 



A NUMBER of investigators have studied the propagation of electro- 

 -^ ■*' magnetic waves in a bent pipe of rectangular cross-section, the bend 

 being along an arc of a circle. H. Buchholz\ S. Morimoto^ and W. J. 

 Albersheim have employed Bessel functions to express the field in the bend. 

 The form assumed by the field when the radius of curvature of the bend 

 becomes large has been obtained by K. Riess* and R. E. Marshak^ who use 

 approximations suited to this case. Marshak also obtains expressions for 

 various reflection and transmission coefficients. A discussion of the subject 

 using rather simple but approximate analysis is given on pages 324-330 of 

 a text book by S. A. Schelkunoff. The Bessel function approach is also 

 sketched in the same section. 



Here we study the disturbance produced when a wave goes around a 

 circular bend (of some given angle) in a rectangular wave guide, the guide 

 being straight on either side of the bend. Especial attention is paid to the 

 dominant mode reflection coefficients gTo and doi corresponding to H-bends 

 and E-bends, respectively. As equations (4.2-6) and (4.4-4) show, these 

 reflection coefficients (which are of the nature of voltage rather than power 

 reflection coefficients) vary inversely as the square of the radius of curva- 

 ture of the bend when the bend is gentle. The substance of (4.2-6) has 

 been given by Marshak** for the important case in which only the dominant 

 mode is propagated and the angle of the bend not too small. 



When the bend is so sharp that the formulas mentioned above do not 

 apply the reflection coefficients may be computed from the rather simple 

 looking matrix expressions (2.3-3) together with (2.3-4). However, their 

 appearance is deceptive and, as is shown by the numerical work in Part V, 

 considerable labor is necessary to obtain an answer. 



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