I realize the abundance of the literature on this theme which is a disservice in this case, because there are too many formulas in all the books I've found and no explicit answers. Maybe someone could give me a good reference, the question is quiet classical.

Let us consider a Bessel equation $$ r^2 R'' + r R' + (r-k^2) R = 0 $$

with the conditions on the boundaries $R(\sqrt{\lambda})=R(2\sqrt{\lambda})=0$. This comes from the equation of oscillation of an annulus with fixed boundaries. The question is *to find the frequencies of oscillation in an increasing order*, that is just find such $\sqrt{\lambda}$ that the system above holds.

We know that the general solution of the equation is the linear combination between Bessel and Neyman functions $J_k$ and $Y_k$, $k \in \mathbb{N}\cup \{0\}$ so the question is reduced to ordering the zeroes of $J_k$ and $Y_k$. Is there an answer in a form of explicit sequence or, at least, how many first terms do we know?