642 



BELL SYSTEM TECHNICAL JOURNAL 



propagation. The roots of (14.32) are modilied by the introduction of the 

 electrons. To show this effect, let $„ be a solution of (14.32), and_;($„ + b) 

 be a solution of (14.25), The waves considered will thus vary with distance 

 as 



^((*„+6)/d]z 



We see that we must have 



vl/2 



2n1/2 



(6,/e)"^ ($; + 6t>y" cot ($1 + el) 



,2x1/2 



,2x1/2 



1/2/ 



= {{^n + by + e'oY" cot l(6i/6)"^((*„ + bY + e'.Y'-'] 



(ei/e)''^ = 1 



(14.43) 



(14.44) 



(14.15a) 



X {Be - j^n + by J 



As ^ <<C 1, it seems safe to neglect b in (14.15a) and to expand, writing 



(,,/e)i/2 = 1 - « 

 A _ A[{el - $1) + 2jde^,] 



2(de-j^ny 



If I 6 I <C $„ , we may also write 



(($« + bY + dlY" - 



life + ^\f 

 ^nb 



(14.46) 

 (14.47) 



{< + ^0^) 



2^/2 + ^^\ + ^o)'" (14.48) 



We thus obtain, if we neglect products of 5 and a 



(1 - a) cot ($; + doY" = 1 + 



(i 



(4>„ + ^o) 



^nb 



cot (<!>; + ^o) 



2\l/2 



(14.49) 



^($; + eW' ~ V ''' ^^' + ^"^ 

 Solving this for 5, we obtain 



(*l + dlY" Tcos ($1 + di)'" + CSC (4>l + dlY" 



2\l/2 



5 = - 



5 = 





+ i 



LCOS i^l + e^)^'^ - CSC ($1 + ^o)''^J 



a (14.50) 



L*„(e! + *;)^ ' ' (el + $;)^ 



csc^ (*; + 0^)'^^ + cos (*l + 0^)^'n /K^o + ^lY" 



Lcsc « + elY" - cos (ci>; + elY^'J 



(14.51) 



As the waves vary with distance as exp [(± <J>„ + 5)'S/<^]) this means that all 

 modified waves travel in the —z direction, and very fast, for the imaginary 

 part of b, which is inversely proportional to tlie pliase velocity, will be small. 



