visualizing three index cylindrical modes logic

[yefj]

Hello, From rectangular waveguide i know that mnl means we have half wave behvior in each direction.
How do i imagine the shape of a nml mode in cylindrical resonator by its indexes?
is there any visual intuitions?
Thanks.

 

[PlanarMetamaterials]

It's essentially the same concept in both coordinate systems; integer multiples of a half wavelength along each of the cylindrical axes (axial, radial, azimuthal). The radial and azimuthal fields vary as Bessel functions rather than sinusoidal functions, but they're basically very similar. See the nice figures here.

 

[yefj]

Hello PlanarMetamaterials,a circular waveguide was built and i got the modes shown bellow.
I understand the idiea thats its similar to rectangular waveguide were we count the number of half waveguides.
i saw that mode 1 is similar to TE11 but where is the half wavelength issue.
how can i see it in the photo?
Also how can i recognize mode 4 shown bellow?
Thanks.

mode 1 cutoff 12.8Ghz
E-field:

mode 1 H-field:

mode 4 electric field fc=20.46:

mode 4 magnetic field:

 

[PlanarMetamaterials]

The number of wavelengths around the azimuth gives you the azimuthal mode index, whereas the number of half-wavelengths along a diameter gives you the radial index. So, the first mode is the TE11 that you have accurately identified, the second mode looks like a TE21.