  The lowest eigenmode of thin axisymmetric shells is investigated for two
physical models (acoustics and elasticity) as the shell thickness (2$\epsilon$)
tends to zero. Using a novel asymptotic expansion we determine the behavior of
the eigenvalue $\lambda$($\epsilon$) and the eigenvector angular frequency
k($\epsilon$) for shells with Dirichlet boundary conditions along the lateral
boundary, and natural boundary conditions on the other parts. First, the scalar
Laplace operator for acoustics is addressed, for which k($\epsilon$) is always
zero. In contrast to it, for the Lam{\'e} system of linear elasticity several
different types of shells are defined, characterized by their geometry, for
which k($\epsilon$) tends to infinity as $\epsilon$ tends to zero. For two
families of shells: cylinders and elliptical barrels we explicitly provide
$\lambda$($\epsilon$) and k($\epsilon$) and demonstrate by numerical examples
the different behavior as $\epsilon$ tends to zero.
