Limiting behaviour of a spectral problem in fluid-solid structures

The aim of this paper is the asymptotic analysis of a spectral problem which involves Helmholtz' equation coupled with a nonlocal Neumann boundary condition on the boundary of a periodic perforated domain of R². This eigenvalue problem represents the vibrations (eigenfrequencies and eigenmotions) of a tube-bundle immersed in a perfect compressible fluid. Our analysis of convergence shows that the vibrations of this fluid–solid structure with a large number of tubes are close to the spectrum of an unbounded operator in a Hilbert space. Using the method of Bloch and exploiting the periodic structure of the problem, we derive the spectral family of this limit operator and we prove that its spectrum can be completely determined by only computing local eigenvalue problems in the basic cell representing the periodic structure in the problem.