. The Bell System technical journal . We will consider this second case. W(< should note from () and () 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, () becomes () () ()()()() ()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 () CO P L 1 + CO — jSUo T^Wi^ _ /co - 13^0Y\ 7^1 / () What happens if we have many transverse velocities
. The Bell System technical journal . We will consider this second case. W(< should note from () and () 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, () becomes () () ()()()() ()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 () CO P L 1 + CO — jSUo T^Wi^ _ /co - 13^0Y\ 7^1 / () What happens if we have many transverse velocities? If we refer backto () 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 havingtransverse velocities ±Un . Equation () can be written (co - |(3//o)| i = E A^ M„2 r _ (g, - /3uo)2-[^ L 7-n^ J (()) GROWING WAVES DUE TO TRANSVERSE VELOCITIES 113. (u;-/3Uo Fig. 1 Suppose we plot the left-hand and the right-hand sides of () versus(co — ^Uo)- The general appearance of the left-hand and right-hand sidesof () is indicated in Fig. 1 for the case of two velocities Un . Therewill always be two unattenuated waves at values of (w — /3wo) > y Ugwhere Ue is the extreme value of lu; these correspond to intersections 3and 3 in Fig. 2. The other waves, two per value of Un , may be unat-tenuated or a pair of increasing and decreasing waves, depending on thevalues of the parameters. If CO pn -yhir? > 1 there will be at least one pair of increasing and decreasing waves. It is not clear what will happen for a Maxwellian distribution of veloci-ties. However, we must remember that various aberrations might give avery different, strongly peaked velocity distribution. Let us consider the amount of gain in the case of one pair of transversevelocities, ±i/i . The equation is now 2 27 Ui C0„2 [ 1 + CO — |3wo )•] [ ■ - (^OI (1
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