References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Problem 9.30.
We can work out the theory of a transverse magnetic (TM) wave in a rectangular wave guide of dimensions in the direction and in the direction, in the same way as for the TE wave. In a TM wave the component parallel to the axis of the wave guide is zero, so we have only a single wave equation to solve.
As this is identical to the equation for TE waves, we have the same solution:
The boundary conditions are different here, however, as we require
If the wave guide is a perfect conductor, then inside it, so at all boundaries of the guide. In particular, for and . This means , so
Because uses sines rather than cosines, the integers and must both be non-zero in order for the wave to exist at all, so the lowest TM mode is .
The wave number has the same form as in the TE case:
The phase and group velocities are the same as for TE waves
The ratio of lowest cutoff frequencies is