WAVEGUIDE TRANSMISSION 337 



Since l/vl — v' is the ratio of the apparent wavelength in the guide to 

 that in free space and since for hollow pipes it is greater than unity, it is 

 sometimes referred to as the stretching factor. It appears frequently in 

 quantitative expressions relating to waveguides. Since velocity is equal 

 to the number of waves passing per second times the length of each wave, 

 we have 



(6.5-9) 



This is equivalent to the relation shown as Equation 6.5-5. 



A matter of special interest is the rate at which energy is propagated 

 along the guide. For present purposes, it is convenient to regard a moving 

 Hne of force and its associated magnetic force as a unit of propagated energy. 

 A knowledge of the path followed by such a line of force will therefore 

 shed light on the rate at which energy is propagated along a waveguide. 



It was pointed out in connection with Equation 6.4 2 that, when a wave 

 is incident obliquely upon a metal surface, the apparent phase of the wave 

 progresses at a velocity v, greater than the velocity of light v, but that the 

 energy actually progresses parallel to the interface at a velocity v' less than 

 the velocity of light. It was pointed out, too, that v' = v ?>\n 6 = v, sin- d. 

 Because of multiple reflections between opposite walls of a waveguide, its 

 phase velocity is identical with v^. Also, because of these multiple reflections, 

 energy being carried by these component plane waves follows a rather 

 devious zig-zag path and will therefore progress along the axis of the guide 

 at a relatively slow rate. This velocity which is known as the group lelocity 

 is idential with v' above. From relations already given, it will be seen that 



v' = v\/\ - v' (6.5-10) 



also 



v' = v,{l - v") (6.5-11) 



It will be apparent from this relation that, at cut-off, where v = I, 

 energy is propagated along the guide with zero velocity. This is consistent 

 with the idea already set forth that, at cut-off, energy oscillates back and 

 forth between opposite faces of the guide. As we leave cut-off and progress 

 toward higher frequencies (shorter waves), the group velocity v' increases 

 as the phase velocity z'j decreases, until, at extremely high frequencies, 

 both approach the velocity v characteristic of the medium. This relation- 

 ship is made more evident by Fig. 6.5-2. 



Reviewing again the simple analysis just made, we find that the wave 

 configuration that actually progresses along a conventional rectangular 

 waveguide may be regarded as the result of interference of ordinary uni- 



