34 BF.Ll. SYSTEM TKCHNICAI. JOURNAL 



are relative radius, i.e., p/R; the ordinates are relative magnitude referred 

 to the maximum value. The drawings also give r = ttD/Xc for each mode, 

 where Xc is the cutoff wavelength in a circular guide of diameter D. Values 

 for any point of the surface of the end plate can be calculated by using these 

 curves in Conjunction with equation (4). 



In general, for each mode there are certain radii at which the current 

 flow is entirely radial, (/« =0). At these radii, which correspond to zeros 

 of Jt(x) or Jf(x), the annular cuts mentioned in the introduction are quite 

 effective. However, the maxima of Ip do not coincide with the zeros of 

 fe; and a more sophisticated treatment gives the best radius as that which 

 maximizes pip-. X'alues of the relative radius for this last condition are 

 given in Table IV. 



Contour lines of equal relative current intensity are obtained by setting 

 H^ constant in (4), which then expresses a relation between x and 6. The 

 easiest and quickest way to solve (4) is graphically, by plotting H vs. x for 

 different values of 6. 



End Plate: Current Streamlines 



It is easy to show that the equations of the current streamlines are given 

 by the solutions of the differential equation 



Ie^~'Hp- ^^^ 



In the case of the TE modes, (7) is easily solved by separation of the vari- 

 ables, leading to the final result: 



J((x) cos fd = C (8) 



in which C is a i)arameter whose value depends on the streamline under 

 consideration. In the TE modes, the £-lines in the interior of the cavity 

 also satisfy (8), hence a {)lot of the current streamlines in the end plate 

 serves also as a plot of the E lines. 



In the case of the TM modes, (7) is not so easily solved. Separation of 

 the variables leads to: 



f f-J({x) 

 -logsm^^ = j ^j'^^dx. (9) 



The right-hand side of (9) can be reduced somewiiat, yielding 



-log sin te = log [xJt{x)\ + \ i/, dx (10) 



J Jf(x) 



but no further reduction is possible. The remaining integral represents a 

 new function which must be tabulated. Its ev^aluation is discussed at 



