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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?

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Could you clarify your question? Are you interested in sequences w.r.t. $k$ with $\lambda$ fixed, or w.r.t. $\lambda$ with $k$ fixed?. It seems that you have two parameters. Explain perhaps in what sense the question is reduced to ordering zeroes of $J_k$ and $Y_k$. –  Athanagor Wurlitzer Sep 24 '13 at 10:47
    
I just would like to establish the frequencies in an icreasing order, and the frequencies are connected to the zeroes of Bessel and Neuman functions. Everything is fixed, $k$ is fixed, $\lambda$ is fixed, we still have $2$ countable sets of zeroes of $J_k$ and $Y_k$. –  Olga Sep 27 '13 at 10:11
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Formula 10.21.3 tells you that there is exacty one zero of $Y_k$ between two zeros of $J_k$, and if you forget the origin, the $n$-th zero of $Y_k$ is before the $n$-th positive zero of $J_k$. –  Athanagor Wurlitzer Sep 28 '13 at 18:35
    
This is what I want, thank you very much. Although I can not access the site you are referring to. –  Olga Oct 7 '13 at 11:38
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Your question is an example of the funny things that happen when names get transliterated to Russian and back again. The name is Neumann. –  Michael Renardy Oct 8 '13 at 15:21
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1 Answer

up vote 1 down vote accepted

This is just to mark this question as answered, as the comments were enough.

Zeros of Bessel functions are the subject of Chapter XV in Watson's Treatise on the theory of Bessel Functions (1922).

Many older results are found there, together with new ones (at the time).

It seems that what you were asking is the following (page 487):

  • $j_k$ is the $k$-th strictly positive real zero of the Bessel function of the first kind $J_k$,
  • $j^\prime_k$ is the $k$-th positive real zero of the derivative of $J_k$
  • $y_k$ is the $k$-th positive real zero of the Bessel function of the second kind $Y_k$, then

$...<j_{k-1} < j^\prime_{k} < y_{k} <j_{k}<...$

The reason of this exclusion of $0$, which is the first zero for all $J_n$, $n\geq1$, is that otherwise two different rules should be given for $n=0$ and otherwise. Watson attributes this result to Schafheitlin, Journal für Math. CXXII, 1900, pp. 317-321.

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