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# Electromagnetic Waves in a Parallel-Plate Waveguide

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.

### DETAILS

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.

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