## Homogenization of a couple problem for sound propagation in porous media.(English)Zbl 1261.35014

The authors describe an homogenization result for the sound propagation in porous media. They consider a 3D connected domain $$\Omega$$ which is splitted in a solid part $$\Omega ^{s}$$ and a fluid part $$\Omega ^{f}$$. They write the displacement field $$U(x,t)=u_{f}(x)e^{i\omega t}$$ in $$\Omega ^{f}$$ and $$U(x,t)=u_{s}(x)e^{i\omega t}$$ in $$\Omega ^{s}$$ and the pressure $$P(x,t)=p(x)e^{i\omega t}$$ in $$\Omega ^{s}$$. In the fluid part, they write the incompressible Stokes equation $$-\rho _{f}\omega ^{2}u_{f}-i\omega \Delta u_{f}+\nabla p=f^{f}$$ with $$\nabla \cdot u_{f}=0$$. In the solid part, they consider a linear elastic medium of Lamé type $$-\rho _{s}\omega ^{2}u_{s}-(\lambda +\mu )\nabla \nabla \cdot u_{s}-\mu \Delta u_{s}=f^{s}$$. Here $$f^{f}$$ and $$f^{s}$$ are applied forces which will finally be neglected. The authors impose mixed boundary conditions. They introduce a microstructure in $$\Omega$$ depending on a small parameter $$\varepsilon$$. They first prove uniform estimates on the solution of this problem. The main part of the paper describes the asymptotic behaviour of the solution, the authors using the two-scale convergence tools and the solutions of cell problems.

### MSC:

 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)