A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Ω of 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,0a^?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:

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