The obstacle is centered in the vertical direction. Figures left represent the \(k\) spectrum in the complex plane of the Laplace operator with Neumann boundary conditions obtained using Perfectly Matched Layers (PMLs). Bottom left is a zoom on top left. Trapped modes appear at the eigenfrequency \(k_0\) in red. On the other hand, we solve the scattering problem \(\Delta u+k^2u=0\) + Neumann B.C. for an incident plane wave at the frequency in green. The real part of the field is displayed above. We observe that nothing particular happens for \(k\) in a neighbourhood of \(k_0\). The blue dots correspond to the discretization of the continuous spectrum and to complex resonances. PMLs explain why the continuous spectrum is rotated.

The obstacle is slightly shifted in the vertical direction (5%). The eigenfrequency \(k_0\) which was embedded in the continuous spectrum becomes a complex resonance (in magenta) which is unveiled by the PMLs. We observe a rapid variation of the solution to the scattering problem for \(k\) in a neighbourhood of \(k_0\). This variation gets even faster as the imaginary part of the complex resonance is small.

Figures left represent the \(k\) spectrum in the complex plane of the Neumann Laplacian with two kinds of Perfectly Matched Layers (PMLs). The blue spectrum has been obtained using classical PMLs to select fields which are outgoing at infinity. The red spectrum has been obtained using conjugated PMLs to select fields which are ingoing in the left lead and outgoing in the right lead. Real eigenvalues which belong to both spectra correspond to trapped modes (eigenfunctions 1 and 3 below). Real eigenvalues which are in the spectrum with conjugated PMLs but not in the spectrum with usual PMLs correspond to reflectionless modes (eigenfunctions 2 and 4 below). For the latter eigenfrequencies, there is an incident field ingoing in the left lead whose energy is completely transmitted through the structure.

\(k=1.24\) \(k=1.45\) \(k=2.39\) \(k=2.89\)

To check the results, we computed the reflection coefficient of the solution to the scattering problem \(\Delta u+k^2u=0\) + Neumann B.C. for an incident plane wave at the frequency \(k\in(0;\pi)\) (monomode regime). The curve \(k\mapsto |R(k)|\) is displayed in orange. We observe that \(R\) vanishes for two frequencies corresponding to the ones (\(k=1.45\) and \(k=2.89\)) which have been obtained solving the eigenvalue problem with conjugated PMLs.

Below we superimpose the two spectra with usual PMLs and with conjugated PMLs for a range of \(L\), \(L\) being the width of the brick in the waveguide.

**Perfect invisibility.**We represent the total field (acoustic pressure) in the reference geometry (bottom) and in the pertubed waveguide (top). At \(x=\pm\infty\) in the perturbed waveguide, the field is the same as in the reference geometry. The reflection coefficient \(R\) and the transmission coefficient \(T\) satisfy \(R=0\), \(T=1\). More details [PDF].

Another geometry where \(R=0\) and \(T=1\). More details [PDF].

In the example below, we played with the height of the thin chimneys. More details [PDF].

**Complete reflectivity.**We represent the total field (acoustic pressure). The goemetries have been designed so that \(T=0\). All the energy is backscattered. More details [PDF].

\begin{array}{|rl} -\mbox{div}(\sigma\nabla u)=f &\mbox{ in }\Omega\\ u=0&\mbox{ on }\partial\Omega \end{array}

with a sign changing \(\sigma\).**Singular behaviour.**For certain values of \(\sigma\), very singular behaviours can appear which are not met with positive materials. Here we represent \(t\mapsto \Re e\,(u(x,y)e^{-i\omega t})\) for a given \(\omega>0\). Everything happens like if a wave was absorbed by the corner. The wave propagates to the corner but never reaches it. We talk about "black-hole phenomenon". More details [PDF], [PDF].

**Rounded corner.**The solution can be very sensitive to the geometry. Here we represent \(\delta\mapsto u^{\delta}\) as \(\delta\to0\), where \(\delta\) is the radius of the inner circle. More details [PDF], [PDF].

**Mesh refinement.**We solve numerically the above problem with a usual P1 finite element method for different meshes (we refine the mesh).

For \(\sigma_2/\sigma_1\in[-3;-1/3]\), in general the solution does not converge (below left \(\sigma_2=-1.0001\)).

For \(\sigma_2/\sigma_1\in(-\infty;0)\setminus[-3;-1/3]\), the solution does converge (below right \(\sigma_2=-3.001\)). More details [PDF].