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BELL SYSTEM TECHNICAL JOURNAL 



In an axially symmetrical electrode system, if the modulation coefficient 

 on the axis is /3o , the modulation coefficient at a radius r is 



/3. = ^0 h (yr) 



(bl6) 



Here In is a modified Bessel function. 



It is easy to see why (bl5) and (bl6) must be so. The field in the gap can 

 be resolved by means of a Fourier integral into components which vary 



0.5 1.0 1.5 0.2 2.5 0.3 3.5 0/J 45 0.5 5.5 0.6 6.5 0.7 



TRANSIT ANGLE, yd, IN RADIANS 



Fig. 1 19. — Modulation coefficient for fine parallel grids vs transit angle across the gap in 

 radians. 



^ = I sin {yd/2) /{yd/2) \, y = w/uo = 3nO/xV\^ . 



sinusoidally along the axis and as the hyperbolic cosine (in the two-dimen- 

 sional case) of the same argument normal to the axis or as the modified 

 Bessel function (in the axially symmetrical case) of the same argument 

 radially. When the integration of (b9) is carried out, only that portion of 

 the Fourier integral representation for which the argument is 7.V con- 

 tributes to the result, and as that part contains as a factor cosh 7.V or 

 Io{yx) (bl5) and (bl6) are established. 



The simple theory of velocity modulation presented in Appendix III 

 makes no provision for variation of modulation coefficient across the beam. 

 If we confine ourselves to very small signals, we find that the factor which 

 appears is /3^. We may distinguish two cases: If the distance from the axis 



