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The modes in a rectangular waveguide are denoted by TE$_{mn}$/TM$_{mn}$, where $m$ and $n$ are the eigen numbers along the larger and smaller dimensions of the waveguide, respectively. Which one of the following statements is true?

1. The TM$_{10}$ mode of the waveguide does not exist
2. The TE$_{10}$ mode of the waveguide does not exist
3. The TM$_{10}$ and the TE$_{10}$ modes both exist and have the same cut-off frequencies
4. The TM$_{10}$ and the TM$_{01}$ modes both exist and have the same cut-off frequencies

Correct answer: The TM$_{10}$ mode of the waveguide does not exist

Solution

In a rectangular waveguide, TE modes can have one index zero, but TM modes require both field components to satisfy boundary conditions that make the $TM_{10}$ mode impossible. The $TE_{10}$ mode is the dominant and fundamental mode, so it does exist.

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