Fourier Homogenization Method and the Propagation of Acoustic Waves Through a Periodic Vortex Array

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.