Propagation of acoustic wave with random sources.
Actually, one part of my post-doctoral work is to implement resolution of acoustic wave, first in homogeneous media, then in heterogeneous media, witha given number of random sources that are correlated by a given function of time and space. The implementation is done in the code Montjoie (see the softwares page to have a brief description of this code).
Propagation of acoustic waves in fractal trees.
The second part of my Ph.D. thesis deals with this ''exotic'' topic, which derives from the works of Bertrand Maury et al about modeling the respiratory system. We are interested in studying the wave propagation on a tree with many generations (we consider a tree as a network with the additional notion of successive generations, as a genealogical tree), seen as an infinite tree, the idea being that the infinite tree should be easier than the large finite tree! The mathematical model consists in solving the 1D wave equations on each branch of the tree, these being coupled by node conditions of the Kirchhoff type and can be reinterpreted as a weighted wave equation on the tree involving a positive weight function μ. This work is also closely connected (although independent) to a series of work by Yves Achdou et al about PDE's in fractal domains. In some particular cases, we can truncate the infinite tree by giving Dirichet-to-Neumann conditions. These conditions have been implemented in the code Netwaves.
Propagation of acoustic waves in junction of thin slots.
The first part of my Ph.D. thesis deals with this topic. It follows works of Sébastien Tordeux and Xavier Claeys on asymptotic analysis. In this part, we study time harmonic (or time domain) wave propagation in thin domains that are junctions of thin slots whose thickness ε is small with respect to the wave length λ and converges, when ε tends to 0, to a 1-dimensional graph.Intuitively, one expects that the solution of the original model converges to a 1D function defined on the limit graph. The homogeneous Neuman boundary condition is considered. The limit model is known for a long time : the limit solution satisfies the 1D time harmonic wave equation (namely the Helmholtz equation) and the so called Kirchoff conditions (in electricity) at each node of the graph : the solution is continuous at the node of the graph and the sum of fluxes at this node vanishes. A natural question is to look for more accurate approximate models, i.e. models that would permit to identify not only the limit solution but also its first order (or higher order) expansion with respect to ε. We derived from the analysis an improved 1D model defined on the 1D limit graph, but depending on ε.
