GROWING WAVES DUE TO TRANSVERSE VELOCITIES 125 



since there are no negative terms in the sum for these ranges and again 

 that the l.h.s. must change sign between the n = and n — I Unes for 

 any k^ in the range < k^ < 1 (since it varies from T oo to ±0°), this 

 information may be combined with that about the immediate vicinity 

 of 5 = k = V^ to enable us to draw a Hue on which the l.h.s. is zero. 

 This is indicated in Figs. 6A and 6B for small z and large z respec- 

 tively. It will be seen that the zero curve and, in fact, all curves on which 

 the l.h.s. is equal to a negative constant are required to have a vertical 

 tangent at some point. This point may be above or below /c^ = ^ (de- 

 pending upon the sign of c or the size of z) but always at a 3^ > ^. For 

 5 < H there are no regions where roots can arise as we can readily see 

 by considering how the l.h.s. varies with k"^ at fixed 5^ For a fixed d^ > }/s 

 we have, then, either for k^ > ]4 or k^ < V^, according to the size of z, 

 a negative minimum which becomes indefinitely deep as 5^ -^ ^. Thus, 

 since the negative terms on the right-hand side are not sensitive to small 

 changes in 5^, we must expect to find, for a fixed value of the l.h.s., two 

 real solutions of (2.28) for some values of 5^ and no real solutions for some 

 larger value of 5 , since the negative minimum of the l.h.s. may be made 

 as shallow as we like by increasing 6". By continuity then we expect to 

 find pairs of complex roots in this region. Rather oddly these roots, which 

 will exist certainly for 5' sufficiently close to V^ + 0, will disappear if 

 5^ is sufficiently increased. 



REFERENCES 



1. L. S. Nergaard, Analysis of a Simple Model of a Two-Beam Growing-Wave 



Tube, RCA Review, 9, pp. 585-601, Dec, 1948. 



2. J. R. Pierce and W. B. Hebenstreit, A New Type of High-Frequency Amplifier, 



B. S. T. J., 28, pp. 23-51, Jan., 1949. 



3. A. V. Haeff, The Electron-Wave Tube — A Novel Method of Generation and 



Amplification of Microwave Energy, Proc. I.R.E., 37, pp. 4-10, Jan., 1949. 



4. G. G. Macfarlg,ne and H. G. Hay, Wave Propagation in a Slipping Stream of 



Electrons, Proc. Physical Society Sec. B, 63, pp. 409-427, June, 1950. 



5. P. Gurnard and H. Huber, Etude E.xp^rimentale de L'Interaction par Ondes 



de Chargd^d'Espace au Sein d'Un Faisceau Electronique se Deplagant dans 

 Des Champs Electrique et Magn^tique Croisfe, Annales de Radio^lectricite, 

 7, pp. 252-278, Oct., 1952. 



6. C. K. Birdsall, Double Stream Amplification Due to Interaction Between Two 



Oblique Electron Streams, Technical Report No. 24, Electronics Research 

 Laboratory, Stanford University. 



7. L. Brillouin, A Theorem of Larmor and Its Importance for Electrons in Mag- 



netic Fields, Phys. Rev., 67, pp. 260-266, 1945. 



8. J. R. Pierce, Theory and Design of Electron Beams, 2nd Ed., Chapter 9, Van 



Nostrand, 1954. 



9. J. R. Pierce, Traveling-Wave Tubes, Van Nostrand, 1950. 



