334 BELL SYSTEM TECHNICAL JOURNAL 



In Fig. 6.5-1 (c) and again in Fig. 6.5-1 (d) we find successive positions 

 of these same wave fronts as they have moved forward in the guide. We 

 may, if we like, think of these fronts as discrete waves moving zig-zag 

 through the guide or as a single large wave front folded repeatedly back 

 upon itself. Fixing our attention for the moment on Fig. 6.5-1 (d), we 

 observe that the velocity v at which any point of incidence of the wave front 

 (say at point 5) moves along the guide is given by the relation 



V 



sm Q 



This particular velocity z'a is the phase velocity of the wave as seen by a 

 myopic observer located near a lateral wall of the guide. 



Referring again to Fig. 6.5-1 (d) and fixing our attention on the geometri- 

 cal relation between the wavelength X and the width of the guide c, we 

 may construct a right triangle with X/2 and a as sides and show that 



cos = ;^ (6.5-1) 



la 



and since 



sin Q = Vl - cos- 8 (6.5-2) 



" = 1/1 - {ij (6.5-3) 



and 



m 



(6.5-4) 



This says that for very large guides, that is, X < 2a, Vg = v, but as X ap- 

 proaches 2a, Ve approaches infinity. The particular case where \ = 2a 

 and Vz = CO is referred to as the cut-of condition. At cut-off, it would appear 

 that the individual waves approach the wall at perpendicular incidence 

 and a kind of resonance between opposite walls prevails. At wavelengths 

 greater than cut-off no appreciable amount of power is propagated through 

 the guide. 



The particular value of wavelength measured in air, corresponding to 

 cut-off, is referred to as the critical or cut-ojf wavelength and is designated 

 thus: X,; = 2a. The corresponding frequency is similarly known as the 

 critical or cul-ojf frequency and it is designated thus : /« = v/ X,-. It is sometimes 

 convenient to designate the ratio of the operating wavelength to the 

 critical wavelength by the symbol p. From Equation 6,5-4 it follows that 



