TE modes in EM waveguides

Roodles01

1. The problem statement, all variables and given/known data
How do I classify modes? If there is a rectangular waveguide how can I show it is operating in TE₁₀, TE₂₀ or TE₁₁ mode.


2. Relevant equations
Could someone help clarify how the 10, 11 or 20 comes about?
Is it due to the dimensions of the guide or something else?

Roodles01

OK Wikipedia has helped.
Could someone confirm this please?

"In rectangular waveguides, rectangular mode numbers are designated by two suffix numbers attached to the mode type, such as TEmn, where m is the number of half-wavelengths across the width of the waveguide and n is the number of half-wavelengths across the height of the waveguide."

Also realised that the EM waves travel within the cavity, through the dielectric (air) & losses (attenuation) are through interaction with the wall & from skin effects.
Generally at the boundary the electric & magnetic fields are zero.

I'm not quite sure as to why the E & H fields should be zero though, except that when there is no E field, then H will also be zero.