doubt in waveguide propagation modes

hi..
in the link below, about waveguides

Rectangular & Circular Waveguide: Equations & Fields Formulas - RF Cafe

it is mentioned something like this..
m=Number of ½-wavelength variations of fields in the "a" direction
n=Number of ½-wavelength variations of fields in the "b" direction

can u please explain what this "½-wavelength variations of fields" means? please help..
thanks in advance..

The half-wave variation of the field intensity in a waveguide means that the field pattern repeats in half-wave periods.

If you see a sinusoid, you can see the first half of wave rising and then falling to "zero"; then it rises and falls in negative sense.
A field intensity is considered always "positive", the peaks of the sinusoid have the same "polarity".

Look into a textbook where the equations and also field patterns are presented; the pictures are the best to illustrate what it looks like.

Also in waveguide resonators, the waves are reflected back and forth from the walls (or air-dielectric boundaries), creating a standing wave. Such standing wave is also only one half-wavelength long. Therefore the smallest waveguide size for a resonator is one half-wavelength.

thank you!!!