Acoustic cloaks for surface waves

Sebastien Guenneau
University of Liverpool

We analyse cylindrical cloaks to control surface water waves and bending waves propagating in thin-plates. The former type of waves are governed by a harmonic equation whereas the latter are solutions of a biharmonic equation which is not isomorphic to the wave equation. We discuss some extension of the theory to the full elasticity equations for coupled in plane pressure and shear waves. Finally, we present an extension of this acoustic theory to surface plasmon polaritons.

In 2006, Pendry et al. [1] and Leonhardt [2] independently showed the possibility of designing a cloak that renders any object inside it invisible to electromagnetic radiation. The experimental validation [3] of these theoretical considerations was given, a few months later, by an international team involving the former authors who used a cylindrical cloak consisting of concentric arrays of split ring resonators. This cloak makes a copper cylinder invisible to an incident plane wave at 8.5 GHz as predicted by the numerical simulations. A natural question is to check whether or not such cloaking applies to other types of waves. It turns out that the answer is positive for acoustic waves [4,5,6,7], and this has been experimentally validated for surface water waves with a micro-structured metallic cloak around 10 Hz [8].

However, when one moves to the area of elastic waves, there is no straightforward one-to-one correspondence between Navier’s and acoustic wave equations, and useful analogies with the theory of electromagnetic cloaking seem to break down [10]. In this talk, we will explain why in the special case of thin elastic plates, one can still implement the geometric transform of Pendry et al. [1]. This unexpected positive outcome comes from structural similarities between the harmonic and biharmonic equations, which were recently exploited in the analysis of stop band for periodic thin plates [10]. We will discuss both the elastic properties of the ideal cloak for bending waves [11] and its extension to fully coupled in-plane pressure and shear waves [12].

Finally, we will explain how one can extend the design of the previous cloaks to the area of surface electromagnetic waves such as surface plasmon polaritons.

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