High Frequency Oscillations of First Eigenmodes in Axisymmetric Shells as the Thickness Tends to Zero

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2e) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue λ(e) and the eigenvector angular frequency k(e) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts.