Abstract
We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Omega of R-3 and equal to unity outside Omega. 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(beta-2)), the minimum distance between the holes is d similar to a(t), and the surface impedance functions of the form lambda(m) similar to lambda(m,0)a(-beta) with beta > 0 and lambda(m,0) being constants and eventually complex numbers. Under some natural conditions on the parameters beta, t, and lambda(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:
1. 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 (n) over tilde (x) = -n(x), x is an element of Omega and (n) over tilde (x) = 1, x is an element of R-3\(Omega) over bar. This means that with this process, we can switch the sign of the index of the refraction from positive to negative values.
2. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction (n) over tilde is (n) over tilde (x) = 1, x is an element of R-3. This means that the region Omega modeled by the original index of refraction n is approximately cloaked. Copyright (C) 2015 John Wiley & Sons, Ltd.