112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Either 



or else 



(2)' - /3')7 = 



— C0„" (c/l" + ^2") 



^ 2 di^ d.} 



We will consider this second case. 



W(< should note from (1.3) and (1.4) that 



d{ = u^-D^ - (co - /5(/„)" + 2yD(co - |8?/.o)«i 



^2^ = ?<i-D" - (co - ^ihf - 2jD{o^ - l3uo)ui 



di' + f/o' = 2{u{D' - (co - iSwo)'] 



rfiW = [uiD' + (co - /3;/„)T 



Thus, (1.17) becomes 



(1.16) 



(1.17) 



(1.18) 

 (1.19) 

 (1.20) 

 (1.21) 



(1.22) 

 WD"" + (co - j8mo)2]^ 



If the quantities involved vary sinusoidally with y as cos ru or sin yy, 



-co, 



\u{lf - (co - /3ao)'] 



then 



Our equation becomes 



D' 



-7 



(1.23) 



CO 



P L 



1 + 



CO — jS'Uo 



T^Wi^ 



_ / co - 13^0 Y" 

 \ 7^1 / 



(1.24) 



What happens if we have many transverse velocities? If we refer back 

 to (1.14) we see that we will have an equation of the form 



1 = E - 14 



2^pn 



2 I din + C?2n 



d^d ^ J ^^-^''^ 



"In (fin / 



Here cop„^ is a plasma frequency based on the density of electrons having 

 transverse velocities ±Un . Equation (1.25) can be written 



(co - |(3//o)" "| 



i = E 



A^ 



'M„2 r _ (g, - /3uo)2 -['^ 



L 7-'"n^ J 



(1.2()) 



