636 BELL SYSTEM TECHNICAL JOURNAL 



Now. from (14.17), (14.19) and (14.20) 



H. - ;a;6(6,/e) tanh Ke,/ey'\f - /3^)'''y] 



(14.22) 



But 



Hence 



E, {^x/^y'%^' - 3,'y" 



H. -./VeAKciA)'"/?.. tanh [{e./eViS' - ^iVyl 



E, (J- - /.V)"- 



At V = </, (14.5) must apply. From (14.24) we can write 



(14.24) 



„_ {.,'.)'" i-^ix^h [{.,'. y''\d' - diY"] ,.,... 



^ - (02 - eiY'-' ^ ^^ 



Here is given by (14.14) 



e^ = ^o<^ = {o3/c)d (14.26) 



and P is given by 



P = B/MV^ = B/doV^ (U.27) 



Thus, 00 expresses J in radians at free-space wavelength and P is a measure 

 of the wall reactance, the susceptance rising as B rises. 



14.2 Waves esj the Absence of Electrons 



In this section we will consider (14.25) in the case in which there are no 

 electrons and ei/e = 1. Li this case (14.25) becomes 



_ tanh (r - dlY" . 



P - - (e2 - eg)i-2 (i-i-28) 



Suppose we plot tlae right-hand side of (14.28) vs 6 for real values of di 

 corresponding to unattenuated waves. In Fig. 14.3 this has been done for 

 do = 1/ 10. For do > 7r/2 the behavior near the origin is dilYerent, but in 

 cases corresponding to actual traveling wave tubes do < tt/I. 



Intersections between a horizontal line at height P and the curve give 

 values of 6 representing unattenuated waves. We see that for the case 

 which we have considered, in which do < ir/2 and do cot do> 1, there are 

 unattenuated waves if 



P > - tan do/do (14.29) 



For P = — 00 (no slot depth and no wall reactance) the system for do < ir/l 

 constitutes a wave guide operated below cutoff frequency for the type of 



