426 BELL SYSTEM TECHXICAL JOURNAL 



Field lujiialions 



The equations for the fields arc easily obtained from section 6.12 of 

 Stratton's book, and are given in Table II, which is self-explanatory, except 

 for the quantity c, which we now proceed to discuss. 



Resonaiil Frequencies 



The ellij)tic c\linder has the major diameter, 2a, and the focal distance; 



2c[) . The equation of its surface is then cx{)ressed bv ^ = ^ — a. On 



this surface, £, must vanish. This requires that '"J f{c, a) ~ for TE 



modes and that '"J/ic. a) = for TM modes. The series expansions are 



in terms of c^ as variable. Let ca ~ rf,n or r^,,, be the roots of the above 



^ r . 



equations. Then — = - (dropi)ing the subscripts f, m). Xow, in working 



out the solution of the differential equations, it turned out that c — Coki. 



, f 



Here ^i is one component of the wave number, kj. Hence ^i = - . Further- 



a 



more, the eccentricitv is e = — = - . The indicated procedure is: 1) choose 



a r 



a value of c; 2) laid the various values of r for which the radial function or 



its derivative is zero; 3) then calculate the corresponding eccentricity and 



resonant frequency. Notice that for a given value of c, the values of r 



will depend on the mode, and hence so will the eccentricity. 



We now wish to express our results in terms of the ellipticity and the 



average diameter. To convert eccentricity to ellipticity, we use 



£ = 1 - Vf ^^-• 



The perimeter of the ellipse is given by P = ■iaE(e) where E(e) is the com- 

 plete elliptic integral of the second kind.tt 

 In terms of the average diameter we find 



*-l 



2r£(e) "[ 



2s 

 or calling the C[uantity in brackets s, A'l = -— . This is now in the same form 



as ki for a circular cylinder of diameter D. The quantity 5 is the recipro- 

 cal of Chu's ■^. 



t It is recalled that 



2ir / , r tiTT 



^ = _ =, V)fe2 + k^ ; ki = - ; k, =— , 



X 1 ' a L 



tt This is tabulated as E(a) in Jahnke & Emde, p. 85, with a = sin-^e. 



