Wave Propagation Mathematical Study and Simulation

This work is devoted to the resolution of the Helmholtz equation $-(\mu u')' - \rho \omega^2 u = f$ in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation.

(não se adequa)

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameterε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly inε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniquesà laGrisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence inεin all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

While spoof plasmons have been proposed in periodic arrays of sound-hard inclusions, we show that they also exist when inclusions are penetrable. Moreover, we show that their wavelength can be tuned by the impedance mismatch between the inclusion material and the surrounding medium, beyond the usual effect of filling fraction in the array. It is demonstrated that sound-soft materials increase the efficiency in the generation of sub-wavelength plasmons, with much lower wavelengths than sound-hard materials and than a homogeneous slab. An application to the generation of acoustic spoof plasmons by an ultra compact array of air/polydimethylsiloxane inclusions in water is proposed with plasmon wavelength tunable up to deep sub-wavelength scales.

We consider the scattering of time-dependent acoustic waves by some bounded obstacles located in a homogeneous and non-dissipative medium. We propose a far-field model of time reversal mirror that reverses the asymptotic wave profile of a scattered wave. Following the idea developed for time-harmonic waves in the so-called decomposition of the time reversal operator in French (DORT) method, we define a time reversal operator (TRO) in the time domain, which is related to the scattering operator involved in scattering theory. We show how this operator is related to the family of single frequency TROs involved in the DORT method, which clarifies the link between time-dependent and time-harmonic waves in the time reversal process. As an application of this relation, we prove that iterative time reversal generally converges to a single frequency time-harmonic wave.