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.