## Analogue Transformational Acoustics: An alternative theoretical framework for acoustic metamaterials

In this project, we proposed a new formulation of the transformational approach to acoustic metamaterials based on the idea of analogue gravity. The aim was to obtain a more natural extension of the powerful transformational techniques devised for optics.

### Introduction

The transformational paradigm has been very successful in the design of optical metamaterials. Since its first proposal [1] (and its extension [2]), this idea has been exploited in a variety of different ways [3-14]. The intention of this study was to fully extend this approach to the field of acoustics. Contrary to electromagnetism, the theoretical framework of acoustics does not immediately generalize to a relativistic theory. It is not easy, therefore, to apply directly the techniques developed in transformational optics. A first attempt to construct a design technique for acoustic metamaterials inspired by the approach of [1] was given in [15] and [16] in two and three dimensions respectively. Specifically, it has been shown (numerically) that it is possible to use transformational techniques to build acoustic cloaks. The aim of this proposal was to construct a new designing technique, which is more elegant than those perviously used [15-16] and more powerful in terms of predicting power. This was done by using tools borrowed from General Relativity, more specifically, the so-called Analogue Gravity paradigm [17].

### Analogue Gravity and acoustics.

The concept of Analogue Gravity is based on the development of analogies (typically based on condensed matter physics) to probe aspects of the physics of curved spacetime. The best-known of these analogies is the use of sound waves in a moving fluid as an analogue for light waves in a curved spacetime. In particular, the following theorem is proven in [17]: If a fluid is barotropic and inviscid, and the flow is irrotational (though possibly time dependent) then the equation of motion for the velocity potential describing an acoustic disturbance is identical to the d'Alembertian equation of motion for a minimally coupled massless scalar field propagating in a (3+1)-dimensional Lorentzian geometry. Under these conditions, the propagation of sound is governed by an effectve acoustic metric, which describes a (3+1)-dimensional Lorentzian (pseudo-Riemannian) geometry. The metric depends algebraically on the density, velocity of flow, and local speed of sound in the fluid. This theorem allows us to rewrite the sound equation as a relativistic equation for a scalar field. Traditionally, it provides (at least in principle) a concrete laboratory model for curved space quantum field theory in a realm that is technologically accessible to experiment. In terms of transformational acoustics, this theorem implies that we can apply in a straightforward way the line of reasoning of [2] for electromagnetism.

In order to achieve this goal, however, one will need to redefine the acoustic metric in such a way that the d'Alembertian equation of motion presents explicitly the conductivity tensor and the unperturbed density, but this does not pose any real problem. It is important to keep in mind that analogy is not identity and, as consequence, that the analogue model reflects only a certain number of important features of general or special relativity. This, in our case, will translate in some limit when applying analogue gravity to transformational acoustic. We will evaluate such limits during the first stage of this study. The possibility of designing metamaterials with non-trivial acoustic properties could lead to the development of interesting new technologies in the space sector. One example would be better shielding for payloads but also acoustic (perfect) lenses could have an important role in the geological analysis of asteroids and planetary surfaces for scientific and mining purposes.

### Study Objectives.

This study was focused on the development and testing of a transformational acoustic based on Analogue Gravity. The proposed structure of the study was:

1. The construction of the actual formalism. Using equations inspired from [17], a set of transformations that would allow the design of acoustic metamaterials, as in [2], are constructed,
hence, further developing a “Geometric Acoustic” suitable for our purposes.

2. Using the new equations, the study designs the acoustic cloaks described in [15] and [16]. This serves as a checkpoint for the correctness of the calculations.

3. Using the new equation to design exotic acoustic metamaterials, in particular, the study investigates the existence of metamaterials able to generate dumb holes or the acoustic equivalent of trapped surfaces with the
aid of numerical simulations.

### Study Conclusion.

The study was concluded at the beginning of September 2012. The research activity was developed along two different branches. The first one, developed by the Imperial College, contains a detailed analysis of the structure of elasticity and some general proposals on how to construct a transfromational approach to acoustics. The second one, developed by the University of Valencia, succceeded in using the Analogue gravity paradigm to devise a new approach to transformation acoustics [18].

### References

[1] J. Pendry, D. Schurig, and D. Smith, "Controlling Electromagnetic Fields", Science 312, 1780 (2006).[2] Bergamin, L., "Generalized transformation optics from triple spacetime metamaterials", Phys. Rev. A 78, 043825 (2008), arXiv:0807.0186

[3] U. Leonhardt and T. G. Philbin, "General relativity in electrical engineering", New J. Ph. 8, 247(2006), arXiv:cond-mat/0607418 .

[4] D. Schurig, J. Mock, B. Justice, S. Cummer, J. Pendry, A. Starr, and D. Smith, Science 314, 977 (2006).

[5] D. Schurig, J. Pendry, and D. Smith, Opt. Express 15,14772 (2007).

[6] D. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 (2003).

[7] D. Smith, P. Kolilnko, and D. Schurig, J. Opt. Soc. Am. B 21, 1032 (2004).

[8] Z. Jacob, L. Alekseyev, and E. Narimanov, Opt. Express14, 8247 (2006).

[9] U. Leonhardt and T. Philbin, Prog. Opt. 53, 69-152 (2009), arXiv:0805.4778 [physics.optics].

[10] S. Tretyakov and I. Nefedov, in Metamaterials’, edited by F. Bilotti and L. Vegni (Rome, 2007)

[11] S. Tretyakov, I. Nefedov, and P. Alitalo, New Journal of Physics, vol. 10, p. 115028, 2008 arXiv:0806.0489[physics.optics].

[12] J. B. Pendry, Phys. Rev. Lett. vol. 85, p. 3966, 2000.

[13] U. Leonhardt, Science 312, 1777 (2006).

[14] D. Smith, P. Kolilnko, and D. Schurig, J. Opt. Soc. Am. B vol. 21, p.1032, 2004.

[15] S. A. Cummer and D. Schurig New J. Phys. 9 45 2007

[16] H. Chen and C. T. Chan, Appl. Phys. Lett. 91, 183518 (2007); ibid 2010 J. Phys. D: Appl. Phys. 43 113001 (2008)

[17] C. Barcelo, S. Liberati and M. Visser, “Analogue Gravity”, Living Rev. Relativity, 8, (2005), 12. [Online Article]: cited [13/04/2011], https://www.livingreviews.org/lrr-2005-12

[18] C. Carcí, S. Carloni, C. Barcel&0acute;, G. Jannes, J. Sanchez-Dehesa and A. Martíez, “Analogue Transformation Acoustics”, Ariadna Final Report, ID 11-1301, 2012