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\begin{document}



\title{ASYMPTOTICALLY DOUBLE EIGENVALUES
IN THE PROBLEMS OF SHELLS DYNAMICS AND BUCKLING}
\author{P.E.Tovstik}{St.-Petersburg University}

The equations of the two-dimensional shells theory
contain the geometric small parameter $\mu$ which is connected
with the relative shell thickness.
When we construct the eigenfunctions in linear
boundary value problems of shells dynamics and buckling there are
cases in which the eigenfunctions are localized near some
lines of the shell middle surface and exponentially decrease away
from these lines. The eigenfunctions are similar to the
functions of the "jumping ball" type in acoustics [1].

The typical form of eigenfunctions, which are localized
near the line $x=0$ may be represented in the form
of the asymptotic series [2,3]
$$w(x,y,\mu)\simeq\sum_{k=0}^{\infty}\mu^{k/2}w_k(\xi,y)
\exp\left[i(\mu^{-1/2}p_0\xi+a\xi^2)\right],\quad x=\mu^{1/2}\xi,\eqno(1)$$
where $w_k(\xi,y)$ are polynomials in $\xi$, the values $p_0$ and $a$
are constant, $p_0$ is real, and the condition ${\rm Im}\,a>0$
gives the decreasing of function (1).

If $p_0\ne 0$ then function (1) is complex. We put $w=w^{(1)}+
iw^{(2)}$.  According that the initial equation is real
as the function $w^{(1)}$ so the function $w^{(2)}$ formally satisfy
to this equation. But only two linear combinations
$C_1w^{(1)}+C_2w^{(2)}$ are the asymptotic expansions of the exact
eigenfunctions and in general case the different eigenvalues
$\lambda^{(1)}$ and $\lambda^{(2)}$ correspond to them. The values
$\lambda^{(1)}$ and $\lambda^{(2)}$ have the coinciding asymptotic
expansions
$$\{\lambda^{(1)},\,\lambda^{(2)}\}\simeq\lambda_0+\mu
\lambda_1+\mu^2\lambda_2+\cdots\quad{\rm at}\quad\mu\to 0. \eqno(2)$$

We study two examples in which we compare the eigenvalues
$\lambda^{(k)}$ obtained by the numerical solution with the
approximate explicit expression of the two first terms of series (2)
$\lambda_0+\mu\lambda_1$. When we numerically calculate the values
$\lambda^{(k)}$ we essentially use that the eigenfunctions are even
or odd.

The small free low-frequency vibrations
of the thin shell of the form of an elliptical cylinder
with the freely supported edges is examined. The vibration modes are
localized near the generatrices which cross the smaller
semi-axis of the ellipse. According that there are two such generatrices the
eigenvalues are asymptotically fourfold so we compare
four close eigenvalues $\lambda^{(k)},\ k=1,\,2,\,3,\,4$.

In the problem of the initial membranous
stress state buckling of a circular cylindrical shell
under the non-uniform axial compression the buckling mode is also
localized near the generatrix and the eigenvalues are
asymptotically double [4]. The influence of the wave length in the
longitudinal direction on the eigenvalues is discussed.

\begin{thebibliography}{99}

\bibitem{1} {\it V.M.Babich and V.S.Buldyrev.} Asymptotic methods in short
wave diffraction problems. -- "Nauka", Moscow, 1972, 456\,p. (Russian)

\bibitem{2} {\it P.E.Tovstik.} Stability of thin shells. --
"Nauka", Moscow, 1995, 320\,p. (Russian)

\bibitem{3} {\it V.B.Lidsky and P.E.Tovstik.} Spectra in the theory of shells
// Adv. in Mech., 1984, v.\,7,\char'362\,2, 25--54.(Russian)

\bibitem{4} {\it G.V.Timofeeva.} Loss of stability by a cylindrical shells
upon bending by force and moments // Vestnik Leningrad Univ. Mat.
Mekh. Astron (1984), \char'362\,4, 99--101. English transl.
Len. Univ. Mech. Bull. (1985), \char'362\,4, 72--76.


\end{thebibliography}
\end{document}
