How are modes classified in rectangular waveguides?

[Roodles01]

Homework Statement

How do I classify modes? If there is a rectangular waveguide how can I show it is operating in TE₁₀, TE₂₀ or TE₁₁ mode.

Homework 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 realized 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.