194 BELL SYSTEM TECHNICAL JOURNAL 



Now, at y = we have just above the boundary 



E, = -jlHoe-^^' (4.11) 



coe 



The fields in the particular slot just below the boundary will be in phase 

 with these (we specify this by adding a factor exp —j^z to 4.10) and hence 

 will be 



coe 

 From (4.11) and (4.12) we see that we must have 



^oh tan ^oh = yh 



10 



(4.12) 



(4.13) 



/3h 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 



Fig. 4.2 — The approximate variation of the phase constant j3 with frequency (propor- 

 tional to Poll) for the circuit of Fig. 4. 1 . The curve is in error as p( approaches x, and there 

 is a cutoff at B( = tt. 



Using (4.8), we obtain 



^h = 



cos |So h 



(4.14) 



In Fig. 4.2, I3h has been plotted vs |Qo//, which is, of course, proportional to 

 frequency. This curve starts out as a straight line, /? = /3o ; that is, for low 

 frequencies the speed is the speed of light. At low frequencies the field falls 

 off slowly in the y direction, and as the frequency approaches zero we have 

 essentially a plane electromagnetic wave. At higher frequencies, /? > /3o , 

 that is, the wave travels with less than the speed of light, and the field falls 

 off rapidly in the y direction. According to (4.14), /3 goes to infinity 

 at ^oh = ir/2. 



As a matter of fact, the match between the fields assumed above and below 

 the boundary becomes increasingly bad as jS^ becomes larger. The most rapid 



