640 BELL SYSTEM TECHNICAL JOURNAL 



This will occur only if 



.l/2//,2 n2\I/2 •/ I TT 



{ey/er\f - dlY'-' = j(nrr + ^ 



(€i/e)(r - 0^) = -(WTT + 2 



(14.36) 



Let 



6 = ti -\- jw (14.37) 



From (14.37), (14.36) and (14.15) 

 A 



If we separate the real and imaginary parts, we obtain 



1 



((;/ + jwf - el) = -Lw + fj (14.38) 



2 (14.39) 



- AAuW^de - U) - [{de - UY + w'-] I UW + '" " 



[{A - l){ee - U)' - {A + l)w-]iH- - W- - do~) 



W{u[{d, - Uf + W-] - A[{de - Uf - W-\ + {de " «) (m' - W - dl)) = 



(14.40) 



The right-hand side of (14.39) is always positive. Because always A < I, 

 the first term on the left of (14.39) is always negative if w > (w + ^o), 

 which will be true for slow rates of increase. Thus, for very small values 

 of w, (14.39) cannot be satisfied. Thus, it seems that there are no waves 

 such as we are looking for, that is, slow waves {u « c). It appears that 

 the increasing waves must disappear or be greatly modified when P ap- 

 proaches — =o . 



So far we have considered only four of the waves which exist in the 

 presence of electrons. A whole series of unattenuated electron waves exist 

 in the range 



de - \/Z < e < de -\- VZ 



In this range (ei/e)^'^ is imaginary, and it is convenient to rewrite (14.25) 

 as 



P - i-^i/ey"te.nK-er'ey'\d'-eiy''] ,., ... 



ie' - dlY" ^ ^ 



The chief variation in this expression over the range considered is that due 1 

 to variation in (— ei/e)''^. For all practical purposes we may write 



(dl - elf 



