modes cylindrical waveguide

as is well known, a waveguide (take rectangular waveguide for example) is able to support several modes at a single frequency if this operating frequency is beyond the cutoff frequency of these modes.
Here my question is, what determines the very mode that exists in the waveguide if only one mode can actually exist in the waveguide at one time? or what the E and H field distribution should be if those possible modes exist in the waveguide simutaneously?
Thanks!

perhaps , you could compute several cutoff frequencies with different modes , and compare them with your operating freqency . Kc of TE10 determine the propagating mode . That is The dimention decide the Kc of TE10. as for the distribution of E or H , the simple method is through simulation tools like HFSS , you could observe the different mods distribution .