Shape of waves in waveguide
The walls of the waveguide place some requirements on what kinds of waves can exist within the guide.  These are requirements on the value of the fields and how the fields are changing with distance in the guide. It becomes a math problem to then figure out what shapes are compatible with the shape of the guide.  

For example: The walls of the waveguide short out any electric field on them ( the are conductive). Considering single frequency sine waves, long wavelengths will not have zero voltages at the walls. These frequencies are below the “cutoff” for the guide. The frequency must be high enough so there are some zero crossings at the walls. Only these higher frequencies can exist within the guide.