Snapshot 1: single peak is seen in the fields of
mode (TM mode with
(double peaks appear for
Snapshot 2: three peaks are seen in the fields of
(number of peaks reduces to one for
Snapshot 3: four peaks are seen in the fields of
(number of peaks reduces to two for
The periodic solution of the wave equation satisfying the boundary conditions
takes the following forms:
Other field components are derived accordingly. Here,
is the angular frequency and the propagation constant
is given by
root of the following equations:
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
being the instantaneous field values. The average Poynting vector is given by
, which is always in the
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.
 D. K. Cheng, Field and Wave Electromagnetics
, 2nd ed., New York: Addison-Wesley, 1989.