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 the paper Am. J. Phys. 85, 341 (2017), about Bessel modes in cylindrical waveguides. Equation 5 reads:
$$\frac{d^2}{d\rho^2} R(\rho)+ \frac{1}{\rho} \frac{d}{d\rho} R(\rho) + \left(k_\rho^2 - \frac{m^2}{\rho^2}\right) = 0,$$$$\frac{d^2}{d\rho^2} R(\rho)+ \frac{1}{\rho} \frac{d}{d\rho} R(\rho) + \left(k_\rho^2 - \frac{m^2}{\rho^2}\right)R(\rho) = 0,$$
$\rho$ 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 :
$$ E(\rho,z) = \left[H_0^{(1)}(k_\rho\rho)+H_0^{(2)}(k_\rho\rho)\right] e^{ik_zz} = 2 J_0(k_\rho\rho) e^{ik_zz} $$
$H_0^{(j)}$ being a Hankel function (of order zero), $J_0$ the Bessel function of first kind (and order zero). 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.
