The spectral problem of thin elastic shells in membrane approximation does not satisfy the classical properties of compactness and so there exists an essential spectrum. In the first part, we propose to determinate this spectrum and the weakness directions in the shell. We particularly study the case of homogeneous and isotropic shells with some examples. In the second part, we consider an elementary model problem to study the propagation of singularities and their reflections at the boundary of the domain. In the last, we study the problem of propagation for an isotropic cylindrical shell and we show that the equation of propagation does not depend on the Poisson coefficient.
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