Speaking of Bessel fns, when I was providing instrumentation support
to the Molten Salts Electrochemistry Lab at MIT we developed a system
to measure conductivity using coaxial cylindrical electrodes. It
works very well so long as the melt conductivity is a couple of decimal
orders of magnitude lower than that of the electrode metal. Some
of the melts of interest were conductive enough that the perfect
equipotential approximation for the electrodes was no longer valid.
So we looked at a transmission line analysis, finite resistivity
for the electrodes and both conductivity and dielectric across
the melt, with inductive coupling. Complex Bessel (actually Hankel)
fns all over the solutions, of course, from solving the magnetic
(vector) diffusion equation in cylindrical coords.