860 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



tiire straddles the beam rim, by taking the beam area exposed in the 

 first case to be that of a sector of angle 6, and the length of beam sur- 

 face in the second case to be that of the corresponding arc: 



Rf current sample inside of beam ^^ dh'J^ _ h(u — /3i/) -hilSb) 

 Rf current sample at rim of beam 2dbGz 2u/i(/36) 



Here b is the beam radius, and Jz , Gz the longitudinal components of 

 volume and surface current densities, respectively. 7o and /i are modi- 

 fied Bessel functions, (3 is the propagation constant, u the beam veloc- 

 ity, and CO the radian frequency. For the frequencies and Vjeam radii 

 employed in these measurements, this ratio is very much less than unity. 

 Thus the pattern of Fig. 6 is in accord with this mode distribution. The 

 multiple peaks of Fig. 7, however, do not conform to this picture, and 

 are not understood at present. 



As most of the RF power is concentrated near the rim of the beam, 

 the question arises whether the double and triple peaks, in the longi- 

 tudinal distribution patterns of Figs. 2 to 5, are not due to the probe 

 aperture breaking through the beam rim. However, the dip between 

 adjacent noi.se peaks is too great to be explained on the basis of reduced 

 partition noise or weakened gap coupling, assuming the beam diameter 

 there to be less than the gap diameter (0.100 inch). Moreover, double 

 peaks occur even when the beam diameter exceeds the RF gap diameter; 

 for instance, near the last three beam w^aists of Fig. 5. (When all of the 

 beam is transmitted by the 0.100 inch aperture, the collector-current 

 peak is flat-topped.) It seems likely, therefore, that the double and 

 triple peaks correspond to peaks of amplitude over the entire beam 

 cross-section. 



V PROPAGATION ALONG THE RIPPLED BEAM 



To find an explanation for the multiple peaks of space-charge cur- 

 rent, a small-signal, slow-wave analysis of wave propagation along the 

 rippled beam can be made, in which the special features of these experi- 

 ments are exploited: long ripple wavelength, effectively no flux at the 

 cathode, and low frequencies. The first of these features suggests that 

 the propagation constants can be evaluated at each cross-section plane 

 as though the beam were uniform, despite the presence of radial ve- 

 locities. In addition, the space-charge density is assumed constant at 

 each cross-section, and the electron flow laminar. 



With these assumptions, the beam can be regarded as a fluid of mov- 

 ing charge, with a single-valued velocitj^ at each point in space, as fol- 

 lows: 





