GENERALIZED TELEGRAPHIST'S EQUATIONS 785 



Maxwell's equations. From the physical point of view this method 

 represents a conversion of Maxwell's equations into generalized "tele- 

 graphist's equations." 



Thus it is already known that Maxwell's partial diflfcreutial eciuations 

 and the boundary conditions alojig a waveguide are convertible into a 

 set of independent ordinary differential equations, each resembling tele- 

 graphist's equations for electric transmission lines. ^ Each ecjuation de- 

 scribes a "mode of propagation" in terms of concepts well known in 

 electric circuit theory. A waveguide can be considered as an infinite 

 system of transmission lines. If the medium inside the waveguide is 

 homogeneous and isotropic and if the surface impedance of the boundary 

 is zero, the method of separating the variables enables us to obtain a set 

 of "normal", that is, uncoupled modes of propagation. Any irregularity 

 or "discontinuity" in the waveguide provides a coupling between 

 some, or all, modes of propagation. The irregularity may be in a boundary 

 of the waveguide or in the dielectric within it. A heterogeneous dielectric 

 may be considered as a homogeneous dielectric with distributed irregu- 

 larities. Similarly a heterogeneous non -isotropic dielectric may be con- 

 sidered as a homogeneous isotropic dielectric with distributed irregu- 

 larities. Such irregularities provide a distributed coupling between the 

 ^'arious modes appropriate to homogeneous isotropic waveguides. Our 

 problem is to calculate the coupling coefficients. The generalized tele- 

 graphist's equations, obtained in this manner, are very complicated in 

 that they represent an infinite number of coupled transmission modes. 

 They are useful, however, in suggesting a physical picture of w^ave 

 propagation under complicated conditions, and can be used in approxi- 

 mate analysis when we can ignore all but the most tightly coupled 

 modes. For example, this picture was successfully employed })y Alber- 

 sheim in studying the effect of gentle bending of a waveguide on propa- 

 gation of circular electric waves. In this case it was important to consider 

 the coupling between only two modes, TEoi and TMn , which have the 

 same cutoff frequency in a straight waveguide. More recently, Stevenson 

 obtained exact equations for waves in horns of arbitrary shape.* His 

 equations express the propagation of the axial components of E and //. 

 The various modes are coupled through the boundary of the horn. In 



1 S. A. Schelkunoff, "Transmission Theory of Plane Electromagnetic Waves," 

 Proc. Inst. Radio Engrs., Nov. 1937, pp. 1457-1492. 



2 S. A. Schelkunoff, "Electromagnetic Waves," D. van Nostrand Co., (1943), 

 pp. 92-93. 



^ W. J. Albersheim, "Propagation of TEoi Waves in Curved Waveguides," 

 Bell System Tech. J., Jan. 1949, pp. 1-32. 



* A. F. Stevenson, "(general Theory of Electromagnetic Horns," J. Appl. 

 Phys., Dec. 1951, pp. 1447-1460. 



