Details

Snapshot 1: single peak is seen in the fields of mode (TM mode with , ) for (double peaks appear for )

Snapshot 2: three peaks are seen in the fields of for (number of peaks reduces to one for )

Snapshot 3: four peaks are seen in the fields of for (number of peaks reduces to two for )

The periodic solution of the wave equation satisfying the boundary conditions takes the following forms:

(TM modes),

(TE modes).

Other field components are derived accordingly. Here, is the angular frequency and the propagation constant is given by , where is the root of the following equations:

(TM modes),

(TE modes).

The constant is purely imaginary if the frequency is higher than a certain value, the cut-off frequency. The propagation constant and the cut-off frequency are not the same for TM and TE modes.

The energy density can be calculated by , and being the instantaneous field values. The average Poynting vector is given by , which is always in the direction.

It is possible to select a frequency below the cut-off value, in which case the energy density is no longer periodic along the axis, and the Poynting vector is zero.

Reference

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