CONFORM A /, 7'A'. 1 XSl-ORMA TION 105 



other wave guide work based on conformal transformations (as described 

 above) except that of Krasnooshkin'. At the meeting I learned that 

 the transformation method had also been discovered (but not yet published) 

 by Levine and by Piloty independently of each other. Levine has studied 

 the same corner, see Fig. 1, as is done here. However, his method of 

 approach is quite different in that he obtains expressions for the elements 

 in the equivalent pi network representing the corner, whereas here the 

 reflection coefficient is considered directly. This is discussed in more 

 detail at the beginning of Section 6. Piloty's work is closely related to the 

 material presented in a companion paper and is discussed in its introduction. 



In this paper the partial differential equation resulting from the trans- 

 fomiation, together with the boundary conditions, is converted into a rather 

 complicated integral equation. Numerical work indicates that satisfactor>' 

 values of the reflection coefficient, in which we are primarily interested, 

 may be obtained by solving this integral equation by the method of succes- 

 sive substitutions. However, the question of convergence is not investigated. 



Although they are here applied only to corners, the equations of Sections 3, 

 4 and 5 are quite general. In order to test their generality they were used 

 to check the expression^ for the reflection coefficient of a gentle circular 

 bend in a rectangular wave guide, E being in the plane of the bend. The 

 work has been omitted because of its length. It was found that the essential 

 parts of the transformation may be obtained by regarding the inner and 

 outer walls of the guide system as the two plates of a condenser, solving the 

 corresponding electrostatic problem (using series of the Fourier type), and 

 utilizing the relation between two-dimensional potentials and the theory of 

 conformal mapping. 



When the angle of the corner is small we may obtain the series (7-5) 

 and (7-11) for the reflection coefficients corresponding to simple (i.e. not 

 truncated) E and H corners, respectively (a corner having the electric 

 intensity E in the plane of the bend will be called an E corner or an electric 

 corner. H corners are defined in a similar manner). When the angle of 

 the general E corner shown in Fig. 1 is small we may use the series (7-18). 



The series (7-5) and (7-11) giving the reflection from small angle corners 

 are related to the series giving the reflection coefficients for gentle circular 

 bends. In fact, if the radii of curvature of the latter be held constant 

 while the angle of bend is made small, the series for the circular bends 

 reduce to those for the corners. 



As for the limitations of the method, note first that it can be used onl}- 

 for wave guide systems in which the dimension normal to the plane of 

 transformation is constant throughout. Moreover, the integral equations 

 of the present paper, except for the work of Appendix III, are derived 

 on the assumption that the dimensions of the guide approach constant 



