## Section: New Results

### Wave propagation in complex media

#### Enriched Homogenization in presence of boundaries or interfaces

Participants : Clement Beneteau, Sonia Fliss.

*This work is done in the framework of the PhD of Clement Beneteau and is done in collaboration with X. Claeys (Sorbonne & EPI Alpines).*

This work is motivated by the fact that classical homogenization theory poorly takes into account interfaces or boundaries. It is particularly unfortunate when one is interested in phenomena arising at the interfaces or the boundaries of the periodic media (the propagation of plasmonic waves at the surface of metamaterials for instance). To overcome this limitation, we have constructed an effective model which is enriched near the interfaces and/or the boundaries. For now, we have treated and analysed the case of simple geometries: for instance a half-plane with Dirichlet or Neumann boundary conditions or a plane interface between two periodic half spaces. We have derived a high order approximate model which consists in replacing the periodic media by an effective one but the boundary/transmission conditions are not classical. The obtained conditions involve Laplace- Beltrami operators at the interface and requires to solve cell problems in periodicity cell (as in classical homogenization) and in infinite strips (to take into account the phenomena near the boundary/interface). We establish well posedness for the approximate model and error estimates which justify that this new model is more accurate. From a numerical point of view, the only difficulty comes from the problems set in infinite strips. The method has been implemented using Xlife++.

This approach has been extended to the long time homogenisation of the wave equation. It is well known that the classical effective homogenized wave equation does not capture the long time dispersive effects of the waves in the periodic media. Since the works of Santosa and Symes in the 90’s, several effective equations (involving differential operators of order at least 4) that capture these dispersive effects have been proposed, but only in infinite media. In presence of boundaries or interfaces, the question of boundary/transmission conditions for these effective equations was never treated. We have first results in that direction.

#### Transmission conditions between homogeneous medium and periodic cavities

Participant : Jean-François Mercier.

*In collaboration with A. Maurel (Langevin Institute), J. J. Marigo (LMS) and K. Pham (Imsia).*

We have developed a model for resonant arrays of Helmholtz cavities, thanks to a two scale asymptotic analysis. The model combines volumic homogenization to replace the cavity region by a homogeneous anisotropic slab and interface homogenization to replace the region of the necks by transmission conditions. The coefficients entering in the effective wave equation are simply related to the fraction of air in the periodic cell of the array. Those involved in the jump conditions encapsulate the effects of the neck geometry.

In parallel, this effective model has been exploited to study the resonance of the Helmholtz resonators with a focus on the influence of the neck shape. The homogenization makes a parameter $B$ to appear which determines unambiguously the resonance frequency of any neck. As expected, this parameter depends on the length and on the minimum opening of the neck, and it is shown to depend also on the surface of air inside the neck. Once these three geometrical parameters are known, $B$ has an additional but weak dependence on the neck shape, with explicit bounds.

#### Mathematical analysis of wave propagation in fractal trees

Participants : Patrick Joly, Maryna Kachanovska.

We have continued our work (in collaboration with A. Semin (TU Darmstadt)) on wave propagation in fractal trees which model human lungs. One of the major results of this year is a complete analysis of such models. In particular, provided Sobolev spaces ${H}_{\mu}^{1}$, ${L}_{\mu}^{2}$ (which generalize weighted Sobolev spaces on an interval to the case of fractal trees) we clarified the following questions for a range of parameters of the trees not covered by the previous theory: existence of traces of ${H}_{\mu}^{1}$-functions on fractal trees; approximation of ${H}_{\mu}^{1}$-functions by compactly supported functions; compact embedding of ${H}_{\mu}^{1}$ into ${L}_{\mu}^{2}$.

#### Hyperbolic Metamaterials in Frequency Domain: Free Space

Participants : Patrick Ciarlet, Maryna Kachanovska.

In this project we consider the wave propagation in 2D hyperbolic metamaterials [Poddubny et al., Nature Photonics, 2013], which are modelled by Maxwell equations with a diagonal frequency-dependent tensor of dielectric permittivity $\epsilon $ and scalar frequency-independent magnetic permeability. In the time domain, the corresponding models are well-posed and stable. Surprisingly, in some regimes in the frequency domain, when the signs of the diagonal entries of $\epsilon $ do not coincide, the problem becomes hyperbolic (and hence the name). The main goal of this project is to justify the well-posedness of such models in the frequency domain, first of all starting with the case of the free space. We have obtained partial results in this direction: radiation condition, which ensures the well-posedness of the problem, mapping properties of the resolvent (with refined estimates on the propagation of singularities in these models). We are currently working on the limiting absorption and limiting amplitude principles.