742 BELL SYSTEM TECHNICAL JOURNAL 



Let us consider an example. Suppose 



X = 7.5 cm 



D = .03 



These values are chosen because there is a longitudinal field tube which oper- 

 ates at 7.5 cm with a value of C (which corresponds to D) of about .03. The 

 table below shows the ratio of the maximum value of Xi to the maximum 

 value of xi for no magnetic field. 



A field of 50 to 100 gauss should be sufficient to give useful focusing action. 

 Thus, it may be desirable to use magnetic focusing fields in deflection travel- 

 ing wave tubes. This will be more especially true in low-voltage tubes, for 

 which D may be expected to be higher than .03. 



5. Mixed Fields 



In tubes designed for use with longitudinal fields, the transverse fields far 

 off the axis approach in strength the longitudinal fields. The same is true 

 of transverse field tubes far off the axis. Thus, it is of interest to consider 

 equation (41) for cases in which a is neither very small nor very large, but 

 rather is of the order of unity. 



If the magnetic field is very intense so that /5o is large, then the term con- 

 taining a^, which represents the effect of transverse fields, will be very small 

 and the tube will behave much as if the transverse fields were absent. 



Consideration of both terms presents considerable difficulty as (41) leads 

 to 5 waves (5 values of 5) instead of 3. The writer has attacked the problem 

 only for the special case oi b = d = 0. In this case we obtain from (41) 



^=-i^C^[^ + ^]. (55) 



In work which is given as an appendix, Dr. L. A. MacCoU has shown that 

 the two "new" waves (waves introduced wheno: = 0) are unattenuated and 

 thus unimportant and uninteresting (unless, as an off-chance, they have 

 some drastic effect in fitting the boundary conditions). 



Proceeding from this information, we will find the change in 5 as jSo is in- 

 creased from zero. From (51) we obtain 



,. .^3^3 r 2d8 labdh 2ad^o "1 ,- >. 



d6 = -j^ c !_-- - ^^.-^^^ - (^r+^.J- (56) 



