Team organization may help swarms of flies to become invisible in closed waveguides

We are interested in a time harmonic acoustic problem in a waveguide containing flies. The flies are modelled by small sound soft obstacles. We explain how they should arrange to become invisible to an observer sending waves from $-\infty$ and measuring the resulting scattered field at the same position. We assume that the flies can control their position and/or their size. Both monomodal and multimodal regimes are considered. On the other hand, we show that any sound soft obstacle (non necessarily small) embedded in the waveguide always produces some non exponentially decaying scattered field at $+\infty$ for wavenumbers smaller than a constant that we explicit. As a consequence, for such wavenumbers, the flies cannot be made completely invisible to an observer equipped with a measurement device located at $+\infty$

Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

We investigate the eigenvalue problem $-\text{div}(\sigma \nabla u) = \lambda u\ (\mathscr{P})$ in a 2D domain $\Omega$ divided into two regions $\Omega_{\pm}$. We are interested in situations where $\sigma$ takes positive values on $\Omega_{+}$ and negative ones on $\Omega_{-}$. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with $(\mathscr{P})$: for certain configurations, when the interface between the subdomains $\Omega_{\pm}$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem $(\mathscr{P})$. We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.Comment: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), 09/12/2016. arXiv admin note: text overlap with arXiv:1304.478

Pathology of essential spectra of elliptic problems in periodic family of beads threaded by a spoke thinning at infinity

We construct "almost periodic'' unbounded domains, where a large class of elliptic spectral problems have essential spectra possessing peculiar structure: they consist of monotone, non-negative sequences of isolated points and thus have infinitely many gaps.Peer reviewe

On the breathing of spectral bands in periodic quantum waveguides with inflating resonators

We are interested in the lower part of the spectrum of the Dirichlet Laplacian $A^\varepsilon$ in a thin waveguide $\Pi^\varepsilon$ obtained by repeating periodically a pattern, itself constructed by scaling an inner field geometry $\Omega$ by a small factor $\varepsilon>0$. The Floquet-Bloch theory ensures that the spectrum of $A^\varepsilon$ has a band-gap structure. Due to the Dirichlet boundary conditions, these bands all move to $+\infty$ as $O(\varepsilon^{-2})$ when $\varepsilon\to0^+$. Concerning their widths, applying techniques of dimension reduction, we show that the results depend on the dimension of the so-called space of almost standing waves in $\Omega$ that we denote by $\mathrm{X}_\dagger$. Generically, i.e. for most $\Omega$, there holds $\mathrm{X}_\dagger=\{0\}$ and the lower part of the spectrum of $A^\varepsilon$ is very sparse, made of bands of length at most $O(\varepsilon)$ as $\varepsilon\to0^+$. For certain $\Omega$ however, we have $\mathrm{dim}\,\mathrm{X}_\dagger=1$ and then there are bands of length $O(1)$ which allow for wave propagation in $\Pi^\varepsilon$. The main originality of this work lies in the study of the behaviour of the spectral bands when perturbing $\Omega$ around a particular $\Omega_\star$ where $\mathrm{dim}\,\mathrm{X}_\dagger=1$. We show a breathing phenomenon for the spectrum of $A^\varepsilon$: when inflating $\Omega$ around $\Omega_\star$, the spectral bands rapidly expand before shrinking. In the process, a band dives below the normalized threshold $\pi^2/\varepsilon^2$, stops breathing and becomes extremely short as $\Omega$ continues to inflate

Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools

We consider the linear water-wave problem in a periodic channel pi(h)subset of R-3, which is shallow except for a periodic array of deep potholes in it. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which includes the spectral Steklov boundary condition posed on the free water surface. We apply methods of asymptotic analysis, where the most involved step is the construction and analysis of an appropriate boundary layer in a neighborhood of the joint of the potholes with the thin part of the channel. Consequently, the existence of a spectral gap for small enough h is proven.Peer reviewe

Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions

We investigate a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We consider cases where an observer can produce incident plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coefficients of the inclusion, we show that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a second step, for a given real wavenumber and a given domain D, we present a constructive technique to prove that there exist inclusions supported in D for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm which is described at the end of the paper and which allows to provide examples of (approximated) invisible inclusions.Comment: 20 pages, 7 figure

THE BAND-GAP STRUCTURE OF THE SPECTRUM IN A PERIODIC MEDIUM OF MASONRY TYPE

We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane R-2. The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane R-2 is decomposed into an infinite union of the translates of the rectangular periodicity cell Omega(0), and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of Omega(0) consist of a neighborhood of the boundary of the cell of the width h and thus has an area comparable to h, where h > 0 is a small parameter. Using the methods of asymptotic analysis we study the position of the spectral bands as h -> 0 and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided h is small enough.Peer reviewe