786 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



the present paper we shall consider waveguides of cotistant cross-section 

 and conical horns of arbitrary shape filled with a heterogeneous and 

 non-isotropic dielectric and derive the equations for propagation of the 

 generalized voltages and currents representing the transverse field com- 

 ponents. The various modes are coupled through the medium. It is very- 

 likely that our ecjuations can ])e generalized to include the coupling 

 through the boundary. 



To understand the mechanism of coupling between the various modes 

 through the medium consider Maxwell's equations 



curl E = -jo,B, curl H = 'J -\- jc,D, (1) 



where "^J is the density of conduction current while the other letter 

 symbols have the usual meanings. In the most general linear case the 

 components of B and D are linear functions of the components of H 

 and E respectively, with the coefficients depending on the coordinates. 

 These equations can always be rewritten as follows 



curl E = ->m// - M, curl H - jcceE + ./, (2) 



ion 



where M and ./ are the densities of magnetic and electric polarizati 

 currents. 



M = MB - uH), J = "J + MD - eE), (3) 



and n, e are constants (not necessarily those of vacuum). If M and / 

 were given, they would act as sources exciting various modes of propa- 

 gation in a homogeneous, isotropic waveguide. If M and J are functions 

 of H and E, they can still be considered as the sources, acting on power 

 borrowed from the wave, of the \'arious modes. Thus ^1/ and J will 

 provide the coupling between the modes existing in a homogeneous, 

 isotropic waveguide. 



Thus in order to derive the generalized telegraphist's equations we 

 shall first consider the various modes of propagation in a homogeneous 

 isotropic wave guide. Each mode is described by a trans^'erse field distri- 

 bution pattern*^ T(u, v), where u and v are orthogonal coordinates of a 

 point in a typical cross-section. This function is a solution of the follow- 

 ing two-dimensional partial differential ecjiiation 



AT = — 



6162 



"a {^^ ^ ] -\- 1 (^2^ 



_du V Ci du / dv Ke-i dv 



= -X% (4) 



* See Reference 2. 



6 S. A. Schelkunoff, "Electromagnetic Waves," D. van Xostrand Co. (1943), 

 Chapter 10. 



