## Abstract

We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Ω of ℝ^{3} and equal to unity outside Ω. This region is perforated with an extremely large number of small holes D_{m}'s of maximum radius a, a << 1, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order M:=O(a^{β − 2}), the minimum distance between the holes is d ∼ a^{t}, and the surface impedance functions of the form λ_{m}∼λ_{m,0}a^{−β} with β > 0 and λ_{m,0} being constants and eventually complex numbers. Under some natural conditions on the parameters β,t, and λ_{m,0}, we characterize the equivalent medium generating approximately the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one, respectively, as a→0. As applications of these results, we discuss the following findings: If we choose negative-valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction ñ(x)=−n(x),x∈Ω and ñ(x)=1,x∈ℝ^{3}\∖Ω. This means that with this process, we can switch the sign of the index of the refraction from positive to negative values. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction ñ is ñ(x)=1,x∈ℝ3. This means that the region Ω modeled by the original index of refraction n is approximately cloaked.

Original language | British English |
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Pages (from-to) | 3607-3622 |

Number of pages | 16 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 39 |

Issue number | 13 |

DOIs | |

State | Published - 15 Sep 2016 |

## Keywords

- 35Q40
- 35Q61
- 45Q05
- acoustic scattering
- multiple scattering
- refraction index
- small inclusions
- subclass 35J08