I would like a simple explanation of what, in a physical problem, determines the order of the Bessel functions that describe the solutions. What are dimensional(?) parameters of the system that induce a certain order of the resulting Bessel function order?

I know that Bessel equations occur in cylindrical/spherical problems. For instance, take the example in this paper [https://aapt.scitation.org/doi/full/10.1119/1.4976698][1] about bessel modes in cylindrical waveguides. 
Equation 5 reads: 
[![enter image description here][2]][2]

$\rho$ being the distance to center, R radial part of the field, m the parameter that describes the mode and the resulting order of the Bessel function solution, and k the propagation constant (here $k_\rho$ in radial direction).

I short circuit a few equations to give the resulting field form (eq 8) for the fundamental mode : 
[![enter image description here][3]][3]

H being a Hankel function, J the Bessel function of first kind.
Higher modes have corresponding higher m.

So here the order of the function involved is related to the propagation mode, hence the number of zeros in the field. Now I know the question of zeros in Bessel functions is can get quite tricky, and I must admit i haven't done much research into that... but firstly aren't they supposed to be infinite? because this contradicts my idea that a mode is linked to the number of nodal points of the field.



  [1]: https://aapt.scitation.org/doi/full/10.1119/1.4976698
  [2]: https://i.sstatic.net/hKk8W.png
  [3]: https://i.sstatic.net/tyVdt.png