TRAVELING-WAVE TUBES 37 



It is easy to make up two-dimensional solutions of Laplace's equation 

 with equipotentials or conductors of approximately circular form, as shown 

 in Fig. 3.9. In case I, the conductors are alternately at potentials — V,-\-V, 

 — V, etc.; and in case II, the potentials are —V, 0, -fF, 0, —V, 0, -\-V, 

 etc. Far away in the x direction from such a series of conductors, the field 

 will vary sinusoidally in the z direction and will vary in the same manner 

 with \- as in the developed helically conducting sheet. Hence, we can make 

 the distant fields of the conductors of cases I and II of Fig. 3.9 equal to the 

 distant fields of developed helically conducting sheets, and compare the 

 E~/(3^P and the impedance for the different systems. Case I would correspond 

 to a helix of approximately two turns per wavelength and case II to four 

 turns per wavelength. 



3.3a Two Turns per Wavelength 



Figure 3.10 is intended to illustrate the developed helically conducting 

 sheet. The vertical lines indicate the direction of conduction. The dashed 

 slanting lines are intersections of the original surface with planes normal to 

 the axis. That is, on the original cylindrical surface they were circles about 

 the surface, and they connect positions along the top and bottom which 

 should be brought together in rolling up the flattened surface to reconsti- 

 tute the helically conducting sheet. 



Waves propagate on the developed sheet of Fig. 3.10 vertically with the 

 speed of light. The vertical dimension of the sheet is in this case taken as 

 X/2, where X is the free-space wavelength. The sine waves above and below 

 Fig. 3.10 indicate voltages at the top and the bottom and are, of course, 

 180° out of phase. As is necessary, the voltages at the ends of the dashed 

 slanting lines, (really, the voltages at the same point before the sheet was 

 slit) are equal. 



A wave sinusoidal at the bottom of the sheet, zero half way up and 180° 

 out of phase with the bottom at the top would constitute along any horizon- 

 tal line a standing wave, not a traveling wave. Actually, this is only one com- 

 ponent of the field. The other is a wave 90° out of phase in both the horizon- 

 tal and vertical directions. Its maximum voltage is half-way up, and it is 

 indicated by the dotted sine wave in Fig. 3.10. The voltage of this com- 

 ponent is zero at top and bottom. It may be seen that these two compo- 

 nents propagating upward together constitute a wave traveling to the right. 

 The two components are orthogonal spatially, and the total power is twice 

 the power of either component taken separately. 



Figure 3.11 indicates an array of wires obtained by developing an actual 



' Section 3.3a is referred to as "two turns per wavelength." This is not quite accurate; 

 it is in error by the difference between the lengths of the vertical and the slanting lines in 

 Fig. 3.10. 



