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?