Acoustic metamaterials are artificial structures with acoustic properties not found in natural materials.^{1} They consist of periodic arrangements of subwavelength units, each unit being a designed composite made of different materials and shapes. Since the building units and their separation are small in comparison with the sound wavelength, a metamaterial behaves as a homogeneous medium with exotic properties.

A fluid or a gas can be acoustically characterized with two parameters, its mass density (*ρ*) and its bulk modulus (*B*). Although these parameters always take positive values in natural materials, mass anisotropy and negative values of both parameters are possible under dynamical conditions using acoustic metamaterials.^{1–3} These unusual properties lead to novel phenomena such as negative refraction, and interesting applications such as superlensing or acoustic cloaks.

**Figure 1. **Diagram of a simple quasi-2D sonic crystal. It consists of a periodic distribution of cylindrical cavities drilled in the upper surface of a waveguide with height *h*. The color pattern represents the sound propagation inside the waveguide. R: Cavity radius. L: Length. a: Lattice separation.

This work focuses on metamaterials intended to work with waves traveling inside two-dimensional ducts. One of the walls defining such a waveguide is drilled in order to insert a periodic arrangement of cylindrical cavities (see Figure 1). Since the cavities' length *L* extends along the third dimension, the resulting structures are named quasi-two-dimensional sonic crystals (Q2DSC). We found that they exhibit unusual properties due to the resonances embedded in the cavities. These properties appear at low frequencies, corresponding to wavelengths much larger than the cavity radius, *R*, and lattice separation, *a*.

We fabricated Q2DSC sample consisting of a cluster made of 15 rows of 9 cylinders with *R*=1cm, and covering an area of 47×25cm^{2} (see Figure 2). We characterized the sample in a 4.6×3.6m^{2} waveguide with height *h*=5cm. Transmission data indicated that the structure stops sound propagation for a frequency band around 1kHz (see Figure 2 inset).^{4} Moreover, we demonstrated that the frequency band where the sound propagation is forbidden corresponds to frequencies with negative bulk modulus, the inertial mass density remaining positive. This result suggests that this structure is a feasible device for noise reduction in ducts, where air flow should circulate continuously.

**Figure 2. **Photograph of the sample installed in a 2D waveguide. The inset shows the transmittance spectra, the shadowed zone defines the frequencies with negative bulk modulus.

In order to obtain negative mass behavior from a Q2DSC, a fluid cylinder with a density lower than that of air should be introduced inside the cavities.^{5} However, we demonstrated that a more feasible solution consists of using angularly corrugated cylindrical units, such as the one in Figure 3. These radial inclusions forbid sound propagation along the angular direction, and this induced anisotropy leads to a negative value in the effective density of the metamaterial. We performed the theoretical calculations for these structures by using the multiple scattering method^{3} in combination with the mode matching technique.^{4, 5} The regions where the metamaterial (*m*) parameters take negative values as a function of the frequency and the ratio of the cavity length to the height, *L*/*h*, are shown in Figure 4a. The narrow overlapping area between the two regions corresponds to a zone where both parameters, *ρ*_{m} and *B*_{m} are negative. Depending on the frequency, this metamaterial behaves with four possible combinations of signs in its parameters.^{6} The frequency dependence of the metamaterial parameters for the case *L*=3.5*h* is represented in Figure 4b (also the dashed line in Figure 4a). We observed that both parameters show a resonant behavior. The shadowed zones define the bands with negative values.

**Figure 3. **Diagram of the angularly corrugated cylindrical unit (left) employed in the construction of the quasi-2D metamaterial depicted on the right.

**Figure 4. **(left panel) The colored areas represent metamaterials with negative values of their mass density (ρ_{m}) and bulk modulus (*B*_{m}). (right panel) The frequency behaviors of parameters along the cut defined by the vertical dashed line in the left panel.

**Figure 5. **Sound waves traveling from the right pass through a subwavelength aperture surrounded by a metamaterial with near-zero dynamical mass density.

The metamaterial described in Figure 4b also shows an interesting feature. Its mass density crosses zero with a very small slope, in such a manner that it takes a near-zero value for selected frequencies. Under this condition, the effective sound velocity is much higher than that of air, and the acoustic energy can be transmitted through subwavelength apertures (see Figure 5). The near-zero-mass-density property also can be employed to develop many practical devices. Perfect transmission through apertures with dimensions smaller than the sound wavelength, known as tunneling, would allow the design of acoustic wires where the signal information would be transmitted without losing information, i.e., the phase of the signal at the input would be perfectly recovered at the wire output. Other applications could include non-reflecting sharp bends in waveguides, power splitters, and devices for the control of the radiation field. Control of the radiation field means that the field radiated at the end of the metamaterial surface will be conformal with the surface shape. In other words, once the sound arrives at the end of the metamaterial, all the points in the metamaterial surface emit in phase, and the emitted radiation will reproduce the shape of the metamaterial surface.^{6} The radiation patterning allowed with these structures could also be applicable to noise control.

Our future work will examine the possibility of obtaining artificial structures with similar behavior for underwater applications.

*The authors acknowledge the support by the Office of Naval Research (USA) and the Ministerio de Economía y Competitividad (Spain).*

Daniel Torrent, Victor M. García-Chocano, Rogelio Gracià-Salgado, Francisco Cervera, José Sánchez-Dehesa

Wave Phenomena Group

Polytechnic University of Valencia

Valencia, Spain

Daniel Torrent's research activities focus on the field of acoustic and electromagnetic metamaterials. His current interests include acoustic, elastic, and electromagnetic periodic systems, extraordinary absorption of waves, and classical analogues of graphene.

Víctor M. García-Chocano is a PhD student working on acoustic metamaterials. His research focuses on sound absorption and acoustic cloaks.

Rogelio Grací-Salgado is a PhD student working on acoustic metamaterials. His research explores devices for sound control inside waveguides.

Francisco Cervera is an associate professor in the department of applied physics. His current interests include acoustic metamaterials for sound absorption.

José Sánchez-Dehesa directs the Wave Phenomena Group in the electronics engineering department. His current research includes photonic and acoustic crystals, metamaterials, and acoustic barriers.

References:

1. L. Fok, M. Ambati, X. Zhang, Acoustic metamaterials, *MRS Bull.* 33, p. 931, 2008.

2. D. Torrent, J. Sánchez-Dehesa, Anisotropic mass density by radially periodic fluid structures, *Phys. Rev. Lett.* 105, p. 174301, 2010.

3. D. Torrent, J. Sánchez-Dehesa, Multiple scattering formulation of two-dimensional acoustic and electromagnetic metamaterials, *New J. Phys.* 13, p. 093018, 2011.

4. V. M. García-Chocano, R. Graciá-Salgado, D. Torrent, F. Cervera, J. Sánchez-Dehesa, Quasi-two-dimensional acoustic metamaterials with negative bulk modulus, *Phys. Rev. B* 85, p. 184102, 2012.

5. R. Graciá-Salgado, D. Torrent, J. Sánchez-Dehesa, Quasi-two-dimensional acoustic metamaterials with negative bulk modulus, *New J. Phys.* 14, p. 103052, 2012.

6. R. Graciá-Salgado, V. M. García-Chocano, D. Torrent, J. Sánchez-Dehesa, Negative mass density and ρ-near-zero quasi-two-dimensional metamaterials: design and applications, *Phys. Rev. B* 88, p. 224305, 2013.