prefab for circular modes?

[xoxos]

hi,
i'm a software developer who is generally quite happy to muddle along with
my own devisings.

there is one information i have sought for almost a decade. i don't want to
install octave but i will if there's decent chance i can find what i'm
looking for.

given the number of diametric and radial nodes, i want to be able to
determine the frequency (or ratio to fundamental) for the vibrational mode
of an ideal circular membrane, to a 'fair' number of decimal places.
examples are here -
http://hyperphysics.phy-astr.gsu.edu/hbase/music/cirmem.html#c4

i haven't been able to apply chladni's law, or anything. i am comfortable to
trig. i have, quite honestly now, spend solid weeks of my life staring at
notation for bessel functions et al. without understanding or being able to
use them to any degree. teh internets have yet to realise any sensible to
myself expression of methodology for doing so, yet there are dozens of
matlab and mathematica owners who merrily post animations of vibrating
circlular modes.

i will pay for the formula, okay? i enjoy making software instruments, i
believe physical modeling audio synthesis to be beneficial to users..
sensitising, enlivening, instead of dependent on samples. if it weren't for
people enjoying round percussion so much, i'd happily continue with
abstractions.

thank you,
xoxos.

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