WOLFRAM|DEMONSTRATIONS PROJECT

Electromagnetic Waves in a Cylindrical Waveguide

​
mode
TM
TE
mode numbers
n
0
1
2
3
4
p
1
2
3
4
5
waveguide radius a (m)
0.25
0.5
0.75
1.
frequency f (GHz)
0.5
observation pl. angle θ (°)
90
time in ω t (°)
0
f >
f
c
= 0.37 (GHz) ⟶ wave passes through.
displayed fields
electric field
magnetic field
Poynting vector
(power density)
values
1000 V/m
2.654 A/m
2000 W/
2
m
displayed
0.2 m length
in red arrow
0.2 m length
in blue arrow
1 m length in
horizontal graph
Electromagnetic waves can propagate through cylindrical waveguides. This Demonstration shows the associated fields, energy distributions, and energy transport. These waveguides support transverse magnetic (TM) and transverse electric (TE) modes. Given the mode numbers
n,p
, radius
a
, and frequency
f
, the instantaneous fields and energy density distribution are displayed for the designated time
ωt
on the plane with angle
θ
through the cylindrical axis. The maximum electric field is fixed at 1000 V/m for all cases. The frequency has to be higher than the cut-off frequency
f
c
, which is determined by
b
and the mode numbers.
Using cylindrical coordinates and taking the wave propagation direction as the
z
axis, the fields are a function of
r
,
ϕ
,
z
, and
t
. The energy flows along the channel in the positive
z
direction. The electric and magnetic fields are shown by the red and blue arrows, respectively. The energy density is displayed by varying colors. The energy transport or power density is equal to the averaged Poynting vector, whose magnitude depends on
r
and
ϕ
; it is shown by the curve in the observation plane on the right. The fields are described in the table.