Generalized Telegraphist's Equations for 

 Waveguides 



By S. A. SCHELKUNOFF 



(Manuscript received April 30, 1952) 



In this paper Maxwell's partial differential equations and the boundary 

 conditions for waveguides filled with a heterogeneous and non-isotropic 

 medium are converted into an infinite system of ordinary differential equa- 

 tions. This system represents a generalization of ^'telegraphisVs equations" 

 for a single mode transmission to the case of multiple mode transmission. 

 A similar set of equations is obtained for spherical waves. Although such 

 generalized telegraphist's equations are very complicated, it is very likely 

 that useful results can be obtained by an appropriate modal analysis. 



From a purely mathematical point of view the problem of electro- 

 magnetic wave propagation inside a metal waveguide reduces to obtain- 

 ing that solution of Maxwell's eciuations which satisfies certain boundary 

 conditions along the waveguide and certain terminal conditions at the 

 ends of the waveguide. If the medium inside the wa\eguide is homo- 

 geneous and isotropic and if the cross-section of the waveguide is either 

 rectangular or circular or elliptic, the desired solution is obtained by the 

 method of separating the variables. The method works for some other 

 special cross-sections. It works also if the medium inside a rectangular 

 waveguide consists of homogeneous, isotropic strata parallel to one of 

 its faces. Similarly, it works if the medium inside a circular waveguide 

 consists of coaxial, homogeneous, isotropic layers. But in general if the 

 medium is either nonhomogeneous or non-isotropic or both, the method 

 does not work. The mathematical reason for this is that the solution is 

 of a more complicated form than a simple production of functions, each 

 depending on a single coordinate. Any function that one usuall}^ en- 

 counters in physical problems, and therefore a solution of Maxwell's 

 equations, may be expanded in a series of orthogonal functions. Sets of 

 such functions are pro\-ided by the solutions for waveguides filled ^\'ith 

 homogeneous media. Such functions already satisfy the proper boundary 

 conditions and the problem is to obtain series which also satisfy 



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