Bessel functions, zeros, and frequencies

Question

How can we find the first three zeros $k_{mn}$ of each of the Bessel functions J0, J1, J2, and J3.

And how to find the first 6 frequencies of a vibrating circular membrane?

So, I was thinking maybe Wolfram alpha, but need some clarification. thanks

Answer 1

The nontrivial zeroes of the Bessel functions of the first kind are not expressible in terms of standard elementary functions and are typically found numerically. You can look up the first few in a table. The DLMF has everything you want to know about these zeroes, except their values.

The frequency of the $m$th mode associated with the $n$th Bessel, i.e., the angular pattern of the mode is $\cos{n \phi}$, is given by $j_{m,n} \omega/a$, where $\omega$ is the angular temporal frequency of the vibration and $a$ is the radius of the membrane.