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In a hollow rectangular waveguide, we may either propagate transverse electric modes (TE$_{mn}$) where we have electric standing waves quantified by m and n in each direction, or transverse magnetic modes (TM$_{mn}$).

The lowest mode possible for TE is (10).

However, my physics notes say that for TM the (10) is not possible in a non-magnetic guide and that (11) is instead the lowest frequency for TM. Why is this the case? Would eg (20) be possible?

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Neither m nor n can be zero in a TM mode. If you look at the TM solutions for a rectangular waveguide, E and H turn out to be zero if either m or n is zero.This is because E_z must vanish on the boundary, which means it must have the x,y dependence ~sin(m\pi x)sin(n\pi y).

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