EXACT, CLOSED-FORM SOLUTION FOR A CYLINDRICAL CAVITY ELECTROMAGNETIC RESONATOR CONTAINING TWO COUPLED ADJOINING DIELECTRICS FILLING THE CAVITY

Continuing from

http://forum.nasaspaceflight.com/index.php?topic=36313.msg1342159#msg1342159Here I post both

a) the expression for “p” (mode shape quantum number for longitudinal direction) in terms of a given frequency f

_{mnp}, where m, n, p are quantum mode shape numbers such that: "m" is the number corresponding to the circumferential-polar-direction, "n" is the number corresponding to the radial--polar-direction and "p" is the number corresponding to the axial-polar-direction.

and

b) the expression for frequency f

_{mnp} in terms of a given “p”

where symbols are defined as follows:

L1 = dielectricLength1

L2 = dielectricLength2

and where L1 + L2 = length where "length" is the total internal length of the cylindrical cavity, such that:

L1 = length - L2; (dielectricLength1 = length - dielectricLength2;)

and where we define the following dimensionless ratios:

dD1 = diameter/L1; (dDielectric1 = diameter/dielectricLength1;)

dD2 = diameter/L2; (dDielectric2 = diameter/dielectricLength2;)

The dimensionless quantity "b" is defined as follows:

b := If[modetype == "TM", xbesselzeros[[m + 1, n]]/Pi, If[modetype == "TE", xprimebesselzeros[[m + 1, n]]/Pi]]

where Pi=3.14159265359...

and where X

_{mn}=xbesselzeros[[m+1,n]] (the zeros of the Bessel function)

xbesselzeros = {{2.40483, 5.52008, 8.65373, 11.7915, 14.9309}, {3.83171, 7.01559,

10.1735, 13.3237, 16.4706}, {5.13562, 8.41724, 11.6198, 14.796,

17.9598}, {6.38016, 9.76102, 13.0152, 16.2235, 19.4094}, {7.58834,

11.0647, 14.3725, 17.616, 20.8269}, {8.77148, 12.3386, 15.7002,

18.9801, 22.2178}, {9.93611, 13.5893, 17.0038, 20.3208,

23.5861}, {11.0864, 14.8213, 18.2876, 21.6415, 24.9349}, {12.2251,

16.0378, 19.5545, 22.9452, 26.2668}, {13.3543, 17.2412, 20.807,

24.2339, 27.5837}, {14.4755, 18.4335, 22.047, 25.5095, 28.8874}}

and where X'

_{mn}=xprimebesselzeros [[m+1,n]] (the zeros of the derivative of the Bessel function)

xprimebesselzeros = {{3.83171, 7.01559, 10.1735, 13.3237, 16.4706}, {1.84118, 5.33144,

8.53632, 11.706, 14.8636}, {3.05424, 6.70613, 9.96947, 13.1704,

16.3475}, {4.20119, 8.01524, 11.3459, 14.5858, 17.7887}, {5.31755,

9.2824, 12.6819, 15.9641, 19.196}, {6.41562, 10.5199, 13.9872,

17.3128, 20.5755}, {7.50127, 11.7349, 15.2682, 18.6374,

21.9317}, {8.57784, 12.9324, 16.5294, 19.9419, 23.2681}, {9.64742,

14.1155, 17.774, 21.2291, 24.5872}, {10.7114, 15.2867, 19.0046,

22.5014, 25.8913}, {11.7709, 16.4479, 20.223, 23.7607, 27.182}}

For example, for mode TE01, b= xprimebesselzeros[[1,1]]/Pi= 3.83171 / Pi = 3.83171 / 3.14159

NUMERICAL EXAMPLE: we take the case used by @aero, containing a dielectric2 of HD PE material with dielectric constant (relative electric permitivity) = 2.3, and a dielectric1 being the empty portion of the cavity, under vacuum:

diameter = 0.08278945 meter;

length = 0.1224489 meter;

dielectricLength2 = 0.027282494103102 meter;

cMedium1 = cVacuum; cVacuum = 299792458 meter/second;

cMedium2 = cVacuum/Sqrt[2.3] ; (relative electric permittivity=2.3;relative magnetic permeability=1)

Results:

modetype = "TE"; m = 1; n = 1; p = 1; root1 = 2.26774 GHz

modetype = "TE"; m = 1; n = 1; p = 2; root1 = 2.93557 GHz

modetype = "TM"; m = 0; n = 1; p = 2; root1= 3.37114 GHz

For this case, root1 is real, there is no need to consider root2. A number of modes are cut-off, for example modes TE011 and TM011 are cut-off