Bloch-wave homogenization for a spectral problem in fluid-solid structures

This paper is concerned with the study of the vibrations of a coupled fluid-solid periodic structure. As the period goes to zero, an asymptotic analysis of the spectrum (i.e., the set of eigenfrequencies) is performed with the help of a new method, the so-called Bloch-wave homogenization method (which is a blend of two-scale convergence and Bloch-wave decomposition). The limit spectrum is made of three parts: the macroscopic or homogenized spectrum, the microscopic or Bloch spectrum, and the boundary-layer spectrum. The two first parts are completely characterized: The homogenized and the Bloch spectra are purely essential, and have a band structure. The boundary-layer spectrum is shown to be empty in the special case of periodic boundary condition.