Spectral asymptotics of the Helmholtz model in fluid-solid structures

A model representing the vibrations of a coupled uid{solid structure is considered. This structure consists of a tube bundle immersed in a slightly compressible uid. Assuming periodic distribution of tubes, this article describes the asymptotic nature of the vibration frequencies when the number of tubes is large. Our investigation shows that classical homogenization of the problem is not su cient for this purpose. Indeed, our end result proves that the limit spectrum consists of three parts: the macro-part which comes from homogenization, the micro-part and the boundary layer part. The last two components are new. We describe in detail both macro- and micro-parts using the so-called Bloch wave homogenization method.