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BELL SYSTEM TECHNICAL JOURNAL 



Electroma.i:jnetic waves cannot be freely transmitted in dielectric 

 wires or hollow conductors at all frequencies but only when the wave- 

 length is less than a certain value set by the material of the guide and 

 its dimensions. There is, therefore, for a given guide a critical fre- 

 quency below which waves may not be propagated. We refer to this 

 as the cut-ofT frequency. In a similar way we have for a given fre- 

 quency, critical or cut-off diameters. These critical frequencies de- 

 pend not only on the diameter id) of the guide but on the dielectric 

 constant (k) of the medium as well. Also they are, in general, different 

 for the different types of waves. For guides enclosed by a metallic 

 conductor the cut-off wave-length is such that the circumference of 

 the guide measured in wave-lengths is equal to the roots of certain 

 Bessel's functions. These in turn result from solution of the Max- 

 well equations expressed in cylindrical coordinates. These relations 

 are shown more fullv in Table I. 



As a simple numerical example let us assume the Hi type of wave 

 having a frequency of 3000 mc. (X = 10 cm.) being propagated in a 

 hollow metallic pipe. The critical diameter turns out to be 5.85 cm. 

 or roughly 2.v30 inches. If the space were filled with a material having 

 a dielectric constant of, say 5, this would have been reduced to a diam- 

 eter of roughly one inch. For higher frequencies or for materials 

 having still higher dielectric constants these critical dimensions would 

 obviously be still further reduced and would be comparable in size to 

 the larger conductors used in ordinary electrical practice. The critical 

 dimensions for the other types of waves are of course larger. 



Referring again to Fig. 1 we see that in the so-called En type of 

 wave, a line of electric force originating at a point a on the inner surface 

 of the wall of the guide passes radially toward the center then axialh* 

 and again radially to a corresponding point b on the inner wall roughly 

 one-half wave-length farther along. The entire wave front as seen 

 in cross section cut through cd consists of a symmetrical arrange- 

 ment of these radial lines. The magnetic field associated with 

 this wave consists of a series of coaxial circles shown as dotted lines not 



