École Nationale Supérieure de
Techniques Avancées

Abstract

The goal of this project is to study the wave equation in different heterogeneous media. First, we started by studying the case of a medium with periodic microstructure. We then showed that in the framework of such a medium, the wave equation is well-posed. Then, we carried out a formal development to find a homogeneous equation approximating the wave equation in a heterogeneous medium. We subsequently demonstrated the convergence of the heterogeneous solution to the homogeneous solution. Next, we numerically implemented both the homogeneous and the heterogeneous solutions, initially for short times and then for long times. In the latter case, the homogeneous solution does not sufficiently approximate the heterogeneous solution. Following this, we revisited the formal calculations to determine a finer approximation which we in turn implemented. Finally, we adapted our algorithm to the case of a stationary random microstructure.