RESUMEN

The classical problem of homogenization of elliptic operators in arbitrary domains with periodically oscillating coefficients is considered. As the period goes to zero, an asymptotic analysis of the corresponding sequence of operators is performed with the help of this new method which we call in a natural way the Fourier homogenization method, since it is based on the standard Fourier transform. This method offers an alternative way to view the classical results in homogenization. It works in the Fourier space and thus in a framework dual to the one used in most of the mathematical approaches to this subject.

The Fourier homogenization method is then used to derive an expression for the effective speed of sound for an acoustic wave that propagates through a background flow made up of a periodic array of vortices, in the limit of wavelength large compared with the lattice spacing. The main result is an effective speed of sound that depends on the relative orientation between wave vector and lattice. Examples in two and three dimensions are provided.

RESUMEN

We study the scattering of an incident wave by a composite medium with periodic microstructure in a slab geometry. We compute the expansion on the wave field in powers of the parameter $\epsilon=d \lambda$, where d is the characteristic scale of the microstructure and $\lambda$ is the wavelength. For $\eps=0$ this corresponds to homogenization theory. Here, we focus on the correction terms for $\eps>0$ and compute the frequency-dependent impedance and the energy reflection coefficient as functions of $\epsilon$, the material properties, and the volume fraction. This is used to describe the shape and amplitude of a transmitted or reflected pulse signal.