ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 543 



propagation constant r„ is either real or purely imaginary; it vanishes 

 if o = w(X/2), that is, if the spacing between the planes is a whole 

 number of half wave-lengths. When the propagation constant is 

 real, the longitudinal impedance is purely imaginary, and vice versa, 

 when the propagation constant is purely imaginary, the longitudinal 

 impedance is real. In the former case, no energy is transmitted 

 longitudinally but merely surges back and forth, and in the latter 

 case we have a true transmission line. The transverse impedance is 

 purely imaginary at all frequencies and, hence, the energy merely 

 fluctuates to and fro. 



If the frequency is sufficiently low, all of these higher order propaga- 

 tion constants are real and all the energy is transmitted in the principal 

 mode described by equations (21) to (29). The role of the higher 

 propagation constants consists in redistributing the energy near the 

 sending terminal, ^^ that is, in terminal distortion. But as the fre- 

 quency gets high enough to make the wave-length less than 2a, the 

 next transmission mode may become prominent, and so forth up the 

 infinite ladder of transmission modes. 



Imperfect Coaxial Conductors ^^ 

 We shall now suppose that the conductors are not perfect; i.e., 

 the conductivity instead of being infinite, is merely large. Assuming 

 that our solutions are continuous functions of conductivity (this can 

 be proved), we conclude: first, there exists an infinite series of propaga- 

 tion constants approaching the values given in the preceding section 

 as the conductivity tends to infinity; second, one of these propagation 

 constants, namely that approaching ^'coVe^t, is very small unless the 

 conductivity is too small. In the immediately succeeding sections we 

 shall be concerned only with electromagnetic fields corresponding to 

 this particular propagation constant. 



Let us now prove that the simple expression for the magnetomotive 

 intensity in the dielectric between perfectly conducting cylinders is 

 still true for all practical purposes, even if the conductors are merely 

 good, and even when there are more than two of them. Since the 

 lines of force are circles, coaxial with the conductors, and since H^ is 

 independent of <p, the total magnetomotive force acting along any 

 one of the circles equals H^ times the circumference of the circle {2irp). 

 This M.M.F. also equals the total current / passing through the area 

 of the circle. Therefore, the magnetomotive intensity is (7/2 xp) 

 amperes/cm. This expression is true at any point in the conductors as 



" And near the receiving terminal as well, if the line is finite. 

 1* The general theory of wave propagation in a multiple system of imperfect 

 coaxial conductors is amply covered by John R. Carson and J. J. Gilbert [2, 3]. 



