Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Close a circular guide of radius $a$ (Section 7.4) by conducting plates at $z = 0$ and $z = d$ (Figure 7.8.11). The rule of Section 7.8.2 applies unchanged and gives standing waves with $τ = p π / d$ . The cutoff wavenumbers of the circular guide are $h = x_{mn} / a$ for TM modes, with $x_{mn}$ the $n$ -th zero of the Bessel function $J_{m}$ , and $h = x_{mn}^{'} / a$ for TE modes, with $x_{mn}^{'}$ the $n$ -th zero of its derivative $J_{m}^{'}$ . With $k^{2} = h^{2} + τ^{2}$ :

Resonant frequencies of the circular cylindrical cavity

f_{TM_{mn p}} f_{TE_{mn p}} = \frac{v}{2 π} (\frac{x _{mn}}{a})^{2} + (\frac{p π}{d})^{2}, p = 0, 1, 2, \dots = \frac{v}{2 π} (\frac{x _{mn}^{'}}{a})^{2} + (\frac{p π}{d})^{2}, p = 1, 2, \dots (7.8.29)

with $x_{01} = 2.405$ , $x_{11} = 3.832$ and $x_{11}^{'} = 1.841$ , $x_{21}^{'} = 3.054$ , $x_{01}^{'} = 3.832$ .

In plain words

The physics is that of the rectangular box: $m$ counts the periods around $ϕ$ , $n$ the order of the zero across $ρ$ , and $p$ the half guide wavelengths along $z$ . Only the transverse numbers come from Bessel functions instead of sines.

In air it is handy to write $v /2 π = c /2 π = 4.775 GHz cm$ , so that a Bessel zero divided by the radius in centimetres gives the frequency in GHz directly. The lowest resonance is TM $_{010}$ , with $f = x_{01} v / (2 π a)$ , as long as the cavity is not too long. Since TE $_{111}$ has the smaller transverse number ( $1.841 < 2.405$ ) but needs $p = 1$ , it overtakes TM $_{010}$ when $(2.405/ a)^{2} = (1.841/ a)^{2} + (π / d)^{2}$ , that is at

d = \frac{π a}{2.40 5 ^{2} - 1.84 1 ^{2}} = 2.03 a . (7.8.30)

The dominant mode is TM $_{010}$ for $d < 2.03 a$ and TE $_{111}$ for longer cavities.

(b) E_z (orange) and H_\phi (blue). — (a) The cavity.

7.8.11 The circular cylindrical cavity