In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of thelower Bloch spectrum. For the three dimensional linear elasticity system,the first eigenvalue is degenerate of multiplicity three and henceexistence of such a regular Bloch spectrum is not guaranteed. Theaim here is to develop all necessary spectral tools to overcome thesedifficulties. The existence of a directionally regular Bloch spectrum isproved and isused in the homogenization. As a consequence an interesting relation betweenhomogenization process and wave propagation in the homogenized medium isobtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequenceit is proved that higher modes apart from the first three do not contribute to the homogenization process.

A new asymptotic approach is suggested for studying spectra of linear differential operators with periodic coefficients. The resulting formal recurrent procedure and its rigorous justification allow us to prove a classical theorem on the density of states in one dimension.

AMS 2000 Mathematics subject classification: Primary 34L20. Secondary 74J99; 78M35

This article is divided into two chapters. Theclassical problem of homogenization of elliptic operators withperiodically oscillating coefficients is revisited in thefirst chapter. Following a Fourier approach, we discuss someof the basic issues of the subject: main convergence theorem,Bloch approximation, estimates on second order derivatives,correctors for the medium, and so on. The second chapter isdevoted to the discussion of some non-classical behaviour ofvibration problems of periodic structures.