ELECTROMAGNETIC FIELD ANALYSIS 497 



also equal and opposite currents in the shield which, however, are not equal 

 to the corresponding currents in the wires. In the other mode, the currents 

 in the wires are equal and similarly directed, the return path being through 

 the shield; this mode is similar to the coaxial mode since the wires act in 

 parallel, effectively as one conductor. In the case of n wires there are n 

 distinct modes of transmission. Each mode is characterized by the ratio 

 of currents in the wires and by the field pattern that goes with it. 



In all these modes the longitudinal current paths are conductive; but 

 there is no reason whatsoever why the circuit closure should not take place 

 through the dielectric. Even in those modes of transmission in which all 

 longitudinal current paths are conductive, we have to depend on the dielectric 

 for completion of the circuit; this should prepare us for the idea that con- 

 ductors are not essential for wave transmission. If we include the dielectric, 

 the number of possible longitudinal tubes of flow becomes infinite and so 

 does the number of possible transmission modes; but as the cross-section 

 of each individual tube decreases the longitudinal capacitance also de- 

 creases, and these modes will participate in the transfer of power over 

 substantial distances only at correspondingly higher frequencies. It is 

 not merely that at low frequencies the longitudinal impedance becomes 

 very high; it is capacitive and causes high attenuation. The effect is 

 analogous to the attenuation in high-pass filters below the cutoff. 



The mathematical analysis which lends quantitative substance to these 

 ideas is similar to that involved in the cavity resonator problem. Once all 

 the modes of transmission have been found, the next problem is that of 

 the excitation of these modes by a given source, that is, of coupUng of the 

 source to various modes. 



To summarize: A physical transmission line or a wave guide has always 

 an infinite number of transmission modes either independent or substan- 

 tially independent of each other. It is as if we had a system of single-mode 

 transmission lines without couplers. For each transmission mode the 

 structure behaves as a high-pass filter. If n is the number of conductors, 

 there are n — \ transmission modes with the cutoff frequency equal to 

 zero. Since the lowest non-zero cutoff frequency corresponds to a wave- 

 length comparable to the transverse dimensions of the guide, it is clear that 

 in systems with two or more conductors we have a certain finite number of 

 transmission modes which are well separated on the frequency scale from 

 all the rest. For this reason we may ignore all the higher modes when we 

 are concerned with transmission of low freouencies only, by ''low" meaning 

 the frequencies well below the frequency equal to the velocity of light 

 divided by the largest transverse dimension of the transmission line. 



Analysis of waves in free space proceeds along similar lines. An electric 



