Acoustic passive cloaking using thin outer resonators

Abstract

We consider the propagation of acoustic waves in a 2D waveguide unbounded in one direction and containing a compact obstacle. The wavenumber is fixed so that only one mode can propagate. The goal of this work is to propose a method to cloak the obstacle. More precisely, we add to the geometry thin outer resonators of width $\varepsilon$ and we explain how to choose their positions as well as their lengths to get a transmission coefficient approximately equal to one as if there were no obstacle. In the process, we also investigate several related problems. In particular, we explain how to get zero transmission and how to design phase shifters. The approach is based on asymptotic analysis in presence of thin resonators. An essential point is that we work around resonance lengths of the resonators. This allows us to obtain effects of order one with geometrical perturbations of width $\varepsilon$. Various numerical experiments illustrate the theory.

Appendix: auxiliary results

Proof of Proposition 2.1. We reproduce the material of [4, Proposition 3.4]. Looking at the behavior of $\mathbb {S}\overline{W}-W$ for $|x|>d$ and using that $\mathbb {S}$ is unitary (see relations (4)), one finds that $\mathbb {S}\overline{W}-W$ is a vector of functions which solve the homogeneous problem (1) and which are exponentially decaying at infinity. In other words, $\mathbb {S}\overline{W}-W$ is a vector of trapped modes. But since by definition the $W^{\pm }$ are orthogonal to trapped modes for the $\mathrm {L}^2(\Omega )$ inner product, we deduce that $\mathbb {S}\overline{W}=W$.

Lemma 6.1

The constant $C_{\Xi }$ appearing in the decomposition (14) of the function $Y^1$ is real.

Proof

Since there holds $\Delta Y^1=0$ in $\Xi$, for all $\kappa >0$, we have

$$\begin{aligned} 0=\int \limits _{\Xi ^\kappa } (Y^1-\overline{Y^1})\Delta Y^1-\Delta (Y^1-\overline{Y^1})Y^1\,\mathrm{d}\xi _x\mathrm{d}\xi _y \end{aligned}$$

with $\Xi ^\kappa :=\{(\xi _x,\xi _y)\in \Xi ,\,\xi _y<0 \text{ and } |\xi |<\kappa \}\cup \{(\xi _x,\xi _y)\in (-1/2;1/2)\times [0;\kappa )\}$. Integrating by parts and taking the limit $\kappa \rightarrow +\infty$, we get $C_\Xi -\overline{C_\Xi }=0$. This shows that $C_\Xi$ is real.$\square$

Lemma 6.2

The constant $\Gamma$ corresponding to the constant behavior of $\gamma$ at A (see (21)) is such that

$$\begin{aligned} \Im m\,(\omega \Gamma )=(|W^+(A)|^2+|W^-(A)|^2)/4. \end{aligned}$$

Proof

Since the function $\gamma$ is outgoing, we have the expansion $\gamma =s_{\pm }\mathrm {w}^\pm +\tilde{\gamma }$ for $\pm x >d$ where $s_{\pm }\in \mathbb {C}$ and where $\tilde{\gamma }$ is exponentially decaying at infinity. Integrating by parts in

$$\begin{aligned} 0=\int \limits _{\Omega ^\kappa }(\Delta \gamma +\omega ^2\gamma )W^{\pm }-\gamma \,(\Delta W^{\pm } +\omega ^2W^{\pm })\,\mathrm{d}z, \end{aligned}$$

and taking the limit $\kappa \rightarrow +\infty$ as in (17), we obtain

$$\begin{aligned} s_{\pm }=iW^{\mp }(A)/(2\omega ). \end{aligned}$$

(63)

On the other hand, integrating by parts in

$$\begin{aligned} 0=\int \limits _{\Omega ^\kappa }(\Delta \gamma +\omega ^2\gamma )\overline{\gamma }-\gamma \,(\Delta \overline{\gamma } +\omega ^2\overline{\gamma })\,\mathrm{d}z, \end{aligned}$$

and taking again the limit $\kappa \rightarrow +\infty$, we obtain $2\omega (|s_{-}|^2+|s_{+}|^2)-2\Im m\,\Gamma =0$. From (63), this yields the desired result.$\square$

Lemma 6.3

Let $\gamma _j$, $j=1,2$, be the functions introduced in (51). We have $\gamma _1(A_2)=\gamma _2(A_1)$.

Proof

Integrating by parts in

$$\begin{aligned} 0=\int \limits _{\Omega ^\kappa }(\Delta \gamma _1 +\omega ^2\gamma _1)\gamma _2-\gamma _1\,(\Delta \gamma _2 +\omega ^2\gamma _2)\,\mathrm{d}z, \end{aligned}$$

using that for $j=1,2$, there holds $\partial _\nu \gamma _j=\delta _{A_j}$ on $\partial \Omega$, and taking the limit $\kappa \rightarrow +\infty$, we find $\gamma _1(A_2)=\gamma _2(A_1)$.$\square$