332 BELL SYSTEM TECHNICAL JOURNAL 



and 



v'v, = v" (6.4-1) 



Group velocity also plays an important part in waveguide transmission. 



6.5 Waveguide Transmission 



It was pointed out in an earlier chapter that each' of the various con- 

 figurations observed in waveguides may be considered as the resultant of a 

 series of plane waves each traveling with a velocity characteristic of the 

 medium inside, all multiply reflected between opposite walls. In the case 

 of certain of these waves, this equivalence may not be readily obvious, but 

 for the dominant mode in a rectangular guide, which is one of the more 

 important practical cases, it is relatively simple. It also happens that the 

 analysis of such waves throws considerable light on the nature of guided 

 waves, and furthermore it enables us to deduce many of the useful relations 

 used in waveguide practice — relations that might otherwise call for rather 

 complicated mathematical analysis. 



It is assumed in Fig. 6.5-1 that we are viewing, in longitudinal section 

 and at successive intervals of time, a hollow rectangular pipe having 

 transverse dimensions of a and h measured along the x and y axes respec- 

 tively. In this case the illustration is in the xz plane. It is further assumed 

 that the electric force lies perpendicular to the larger dimension a and is 

 consequently perpendicular to the plane of the illustrations. We assume 

 in Fig. 6.5-1 (a) a particular plane wave front 1, perhaps a crest, that has 

 recently entered the guide from below. Let us say that its velocity is 

 1) — i>a/\^ iJ-rir ^nd that is it so directed as to make an angle d with the left- 

 hand wall as shown. ^^ Reflection at the left-hand wall will therefore be 

 identical with that already shown in Fig. 6.4-2. A portion of the wave front 

 that has just previously undergone reflection is shown immediately below 

 at 2 in Fig. 6.5-1 (a). We assume further that this front is made up of lines 

 of electric force perpendicular to the illustration together with associated 

 lines of magnetic force lying in the plane of the illustration. It will be 

 obvious presently that, like the case of reflection from a single conduct- 

 ing sheet discussed in the previous section, we may obtain two rather 

 different pictures of what takes place within the guide, depending on 

 whether we fix our attention on the configuration as a whole or on some 

 particular line of force which we may identify and follow through a con- 

 siderable interval of time. We shall first consider the configuration as a 

 whole. 



'^ II is to 1)C noted that the angle which the wave front makes with the metal wall is 

 ccjual to the angle which the wave-normal (ray) makes with the perpendicular to the metal 

 wall. 



