Homogenization scheme for acoustic metamaterials

We present a homogenization scheme for acoustic metamaterials that is based on reproducing the lowest orders of scattering amplitudes from a finite volume of metamaterials. This approach is noted to differ significantly from that of coherent potential approximation, which is based on adjusting the effective-medium parameters to minimize scatterings in the long-wavelength limit. With the aid of metamaterials’ eigenstates, the effective parameters, such as mass density and elastic modulus can be obtained by matching the surface responses of a metamaterial's structural unit cell with a piece of homogenized material. From the Green's theorem applied to the exterior domain problem, matching the surface responses is noted to be the same as reproducing the scattering amplitudes. We verify our scheme by applying it to three different examples: a layered lattice, a two-dimensional hexagonal lattice, and a decorated-membrane system. It is shown that the predicted characteristics and wave fields agree almost exactly with numerical simulations and experiments and the scheme's validity is constrained by the number of dominant surface multipoles instead of the usual long-wavelength assumption. In particular, the validity extends to the full band in one dimension and to regimes near the boundaries of the Brillouin zone in two dimensions.

Figure 1

(a) Schematic of the unit cells and their boundaries for a typical metamaterial. The inner heterogeneous scatterers are denoted by gray disks, the external boundaries are marked by black solid lines, and the internal unit boundaries are delineated by dashed lines. (b) The relevant SU (marked $\Omega$) of the metamaterial. Its boundary is denoted by the red lines. The incident waves are illustrated as circular wave fronts in the medium $M$, external to the SU.

Figure 2

(a) Schematic of the layered lattice. The thick black lines represent the stiff plates, whereas, the elastic layer is shaded in gray. Wave propagation is perpendicular to the plates. (b) and (c) Displacement fields for the first two nontrivial eigenstates of the relevant SU.

Figure 3

The effective parameters of the layered lattice. The black (red) curves represent the real (imaginary) parts of the effective parameters in (a), (c), and (d). In (b), the black curve denotes the amplitude, and the red curve denotes the phase. The band gaps are shaded in gray, whereas, the blank region denotes the pass bands. In the region $[{\tilde{\omega }}_{1_{+}},{\omega }_{2_{+}}]$, the wave vector $\overline{k}$ is negative in value. However, it is plotted in the positive domain because of the usual convention.

Figure 4

Intensity transmission coefficients $T$ for the layered lattice samples comprising (a) one layer, (b) five layers, and (c) ten layers of the SU. The solid curves are the results predicted from the effective parameters in Fig. 3, and the open circles are the numerically simulated results. The band gaps are shaded in gray, whereas, the pass bands are shown as blank regions. Excellent agreement is seen between the homogenization predictions and the numerical simulations.

Figure 5

(a)–(c) The displacement fields for the layered lattice with five layers of the SU at three representative frequencies. The sample is shaded in gray. The black curves denote the effective fields obtained from homogenization prediction, whereas, the red lines represent the fields obtained by simulations. The black dots mark the displacements of the plates. It is seen that the homogenization theory cannot reproduce the internal displacement patterns of the metamaterial at higher frequencies, even though its prediction of the scattered field remains accurate. (d) Schematic of the incident, reflected, and transmitted waves for the system whose displacement field patterns are shown in (a)–(c).

Figure 6

The unit cell and eigenstates of the hexagonal lattice. (a) The unit cell of the hexagonal lattice. Heavy components are colored gray, whereas, the light components are left blank. The numberings are for ease of identifying the different components within the unit cell. (b)–(d) The first three nontrivial eigenstates for the SU of the hexagonal lattice. Here, the SU is taken to be a circular region, whereas, the unit cell is hexagonal in shape. Such a difference is immaterial at low frequencies but can become noticeable at high frequencies, especially near the BZ boundaries.

Figure 7

The effective parameters of the hexagonal lattice as deduced from the homogenization theory. The black (red) curves represent real (imaginary) parts of the effective parameters in (a), (c), and (d). In (b), the black curve denotes the amplitude, and the red curve denotes the phase of impedance. The band gaps are shaded in gray, whereas, the pass bands are left blank. For comparison, the results obtained from the CPA are plotted as dashed lines. It is seen that, at the limit of $\overline{k}a\ll 1$, the agreement with our scheme is very good. At higher frequencies, the present homogenization scheme is capable of reproducing the scattered fields rather accurately. It is seen that the CPA has failed to predict the imaginary parts in (c) and (d) as well as the resonant feature for the effective modulus at ${\tilde{\omega }}_{1_{+}}$.

Figure 8

(a)–(c) The scattered fields for the hexagonal lattice with 21 units at three representative frequencies. The upper semicircular regions are the effective fields predicted from our homogenization scheme, and the lower semicircular regions are the results of numerically simulating the actual sample with all of its structural complexities. It is seen that, except for the inner regions of the metamaterial, the agreement is exceptionally good. (d) Schematic illustrating the relevant scattering process whose wave fields are shown in (a)–(c).

Figure 9

The band structures of the hexagonal lattice. The solid lines are predicted from our homogenization scheme, whereas, the open circles are the results of numerical simulations on the original heterogeneous composite. It is seen that the prediction of the homogenization scheme misses the band around $\omega =0.8$. This is the deaf mode that cannot be excited by propagating waves. The wave mode at the Dirac cone (marked by the arrow) is shown in the right panel.

Figure 10

(a) Schematic illustrating the decorated-membrane system. The black line represents the membrane, whereas, the central flat rectangle denotes the attached rigid platelet. The two gray rectangles on the sides denote the aluminum frame. (b) and (c) The first and second eigenmodes of the decorated membrane. Circular symmetry is assumed.

Figure 11

The effective parameters of the decorated-membrane structure. (a) The effective wave vector, (b) the effective impedance $\overline{Z}$, and (c) the effective mass density $\overline{\rho }$. The units for the material parameters are normalized to those of rubber: $Z_{\mathrm{r}}=1.9\times {10}^5\mathrm{N}\mathrm{s}{\mathrm{m}}^{-3}$ and ${\rho }_{\mathrm{r}}=1.3\times {10}^3{\mathrm{kg}/\mathrm{m}}^3$, respectively. The black (red) curves stand for real (imaginary) parts in (a) and (c). In (b), the black curve denotes the amplitude, whereas, the red curve denotes the phase. The band gaps are shaded in gray, whereas, the pass bands are left blank.

Figure 12

(a) and (b) The reflection coefficient $R$ and (c) and (d) the transmission coefficient $T$ of the decorated-membrane structure for the airborne sound. The open circles are the experimental results obtained using the impedance tube measurements, whereas, the solid curves are calculated from the effective parameters shown in Fig. 11. The band gaps are shaded in gray, and the pass bands are without shading. Notice that the area of one unit cell is 50.3 cm${}^2$. Notice that the frequency here is the angular frequency $\omega =2\pi \times f$. Excellent agreement is seen between the homogenization theory and the experimental results.