Wolfram Demonstrations Project

Electromagnetic waves can propagate through a parallel-plate waveguide under appropriate conditions. This Demonstration determines the corresponding fields, energy distributions, and energy transport. The parallel plates support transverse magnetic (TM) and transverse electric (TE) waves. Specifying one of those modes, the mode number

, channel width

, and frequency

, the instantaneous fields and energy density distribution are determined for the designated time or phase

. The maximum electric field is fixed at 1000 V/m in all cases. The frequency has to be higher than the cut-off frequency

, which is determined by

and

. Taking the wave propagation direction as the

axis, the fields are function of

, and

. The energy flows along the channel (in the positive

direction).

The electric and magnetic fields are shown by red and blue arrows, respectively. The energy density is represented by color variation. The energy transport or power density equals the averaged Poynting vector, whose magnitude depends on the

position, as indicated by the curve on the right. The magnitudes of the fields are shown, as described in the table.

Contributed by: Y. Shibuya

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Snapshot 1: the energy density has a maximum at the conductor surface in

mode (TM mode with

)

Snapshot 2: the energy density oscillates three times along the

axis in

mode

Snapshot 3: the directions of the electrical and magnetic fields mode are interchanged in

mode

Periodic solutions of the wave equation satisfying the boundary conditions

take the following forms, for

modes:

where

is the angular frequency,

is the permittivity of air (approximately equal to its vacuum value), and

is the propagation constant given by

. The constant

becomes pure imaginary if the frequency is higher than a certain value, the cut-off frequency. A similar solution is derived for TE modes. The propagation constant and the cut-off frequency are the same.

The energy density can be calculated by

, where

and

are the instantaneous field values. The average Poynting vector is given by

, which is always in the

direction.

In the case

mode is constructed by

and

with

. This means the transverse electromagnetic (TEM). However,

mode does not exist. If this is selected, no fields are displayed.

It is possible to select the frequency below the cut-off frequency without error. If selected, it will be noted that the energy density is no longer periodic along the

axis, but decays with distance.

The electromagnetic field in parallel plate wave guides can be represented as the superposition of two plane waves reflected at the upper and lower conductor surfaces.

Reference

[1] D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Reading, MA: Addison-Wesley, 1989.

Electromagnetic Wave Incident on a Perfect Conductor (Wolfram Demonstrations Project)

Y. Shibuya

"Electromagnetic Waves in a Parallel-Plate Waveguide"
http://demonstrations.wolfram.com/ElectromagneticWavesInAParallelPlateWaveguide/
Wolfram Demonstrations Project
Published: December 2, 2012