References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Problem 9.28.
The possible TE modes in a rectangular wave guide are given by
where and are integers and and are the dimensions of the rectangle, with . These values are related to the wave number by
so any modes where
are not allowed, as they would give an imaginary value of . The frequency
is therefore the cutoff frequency for the mode .
As an example, suppose we had a wave guide with , and a frequency of . Then the condition for a real is
Since at least one of and must be non-zero, the allowed modes are , , and (where the first subscript is and the second is ).
If we wish to allow only one mode, this would have to be (since that gives the lowest value on the LHS). The angular frequency range that would do this is between and , that is
For the values above
The corresponding frequencies and wavelengths are