One path to acoustic cloaking

A complete analysis of coordinate transformations in elastic media by Milton et al has shown that, in general, the equations of motion are not form invariant and thus do not admit transformation-type solutions of the type discovered by Pendry et al for electromagnetics. However, in a two-dimensional (2D) geometry, the acoustic equations in a fluid are identical in form to the single polarization Maxwell equations via a variable exchange that also preserves boundary conditions. We confirm the existence of transformation-type solutions for the 2D acoustic equations with anisotropic mass via time harmonic simulations of acoustic cloaking. We discuss the possibilities of experimentally demonstrating acoustic cloaking and analyse why this special equivalence of acoustics and electromagnetics occurs only in 2D. The transformation-based solutions to the Maxwell equations reported by Pendry et al [1] yield a general method for rendering arbitrarily sized and shaped objects electromagnetically invisible. Other approaches for almost eliminating electromagnetic scattering in all directions from certain classes of objects of limited size have been described [2]–[4], and Leonhardt [5, 6] has given a recipe that can cloak an object in the short wavelength limit that thus applies to different classes of waves, including electromagnetic and acoustic. Similar transformation-based approaches have been analysed for the related problem of electrical impedance tomography [7, 8]. The solution found by Pendry et al stands out for its theoretically perfect performance regardless of wavelength and object size and shape. It is undoubtedly of interest whether the transformation-based electromagnetic solutions, which were simulated in [9] and experimentally confirmed in [10], can be extended to waves in other systems. From one perspective, it seems that the answer could be yes. The concepts of wavevector, wave impedance, and power flow are universal, and the manner in which permittivity 1 Author to whom any correspondence should be addressed. New Journal of Physics 9 (2007) 45 PII: S1367-2630(07)37675-1 1367-2630/07/010045+8$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2 DEUTSCHE PHYSIKALISCHE GESELLSCHAFT and permeability control these in electromagnetics is closely analogous to that by material properties in other systems. On the other hand, the Maxwell equations possess a special symmetry that is an important element of the equivalence between coordinate transformations and material properties for electromagnetic waves. The field equations that describe waves in most other systems do not have this same symmetry. Milton et al [11] analysed the full equations of motion for a general elastic medium under coordinate transformations and found that, in general, the form of the equations of motion are not invariant to transformations. The possibility of engineering artificial elastic media that obeyed these modified equations of motion was discussed, but no simple scheme for realizing a transformation-based application, such as acoustic cloaking, was reported. There is at least one special case, however, in which the equivalence between electromagnetics and elastodynamics is complete. Moreover, this case is practically useful and can potentially be demonstrated experimentally. It has long been known that in two dimensions (2D) acoustics and electromagnetics in isotropic media are exactly equivalent [e.g., 12]. We show below that this isomorphism holds for anistropic media as well. For an inviscid fluid with zero shear modulus, the linearized equations of state for small amplitude perturbations from conservation of momentum, conservation of mass, and linear relationship between pressure and density are [13]