Reflection from Corners in Rectangular Wave Guides - Conformal Transformation

T HE propagation of electromagnetic waves around a rectangular corner has been studied in two recent papers, one by Poritsky and Blewett1 and the other by Miles2. 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 published. Of these, we mention one due to Elson3 who gives values of reflection coefficients 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 4 who studied the vibration of a membrane of irregular shape by transforming it into a rectangle. After the transformation the density (analogous to the propagation 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, M a s s .