DAMAK, M. Mekki (2024) Propagation des ondes dans un milieu à micro-structure aléatoire en temps long PRE - Research Project, ENSTA.
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Abstract
The objective of this project is to study the propagation of waves in various random heterogeneous media and compressibility module. We consider media that exhibit a random microstructure, meaning that density variations occur on a scale much smaller than the wavelength. Specifically, we examine two cases of one-dimensional random media: piecewise homogenized independent and identically distributed \textit{i.i.d.} media, and media with constant impedance. In both cases, each segment \([i,i+1]\), \(i \in \mathbb{Z}\), has constant, strictly positive coefficients randomly drawn independently of the other segments according to given probability laws. In the \textit{i.i.d.} case, the laws of density and compressibility module are independent. In the constant impedance case, the ratio between density and compressibility module is constant throughout the medium. We prove in both cases that in finite time, the solution can be approximated by a solution of an equation with constant coefficients, called the homogenized solution. We also establish an error estimate based on the size of the microstructure and the propagation time. But what happens over longer times? Over what horizon is the homogenized solution a good approximation of the exact solution? These are the questions we will study here. We have carried out simulations of wave propagation in such media and in the corresponding homogenized media over short and increasingly long times. We have thus calculated the error convergence rates over different time horizons in both cases to determine how long homogenization remains valid. Studying these two cases will help us better understand how random variations in coefficients influence wave propagation in heterogeneous media. By obtaining a homogenized wave equation, we can simplify calculations while maintaining a good approximation of the wave behavior in the original medium. This approach is particularly useful in fields where the material properties are not perfectly known or are subject to random variations.
Item Type: | Thesis (PRE - Research Project) |
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Uncontrolled Keywords: | Equation des ondes, Homogénéisation en temps long, Homogénéisation stochastique. |
Subjects: | Mathematics and Applications |
ID Code: | 10182 |
Deposited By: | Mekki DAMMAK |
Deposited On: | 28 août 2024 18:16 |
Dernière modification: | 02 sept. 2024 16:16 |
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