Two-Dimensional Distributed Transmission-Line Models for Broadband Full-Tensor Anisotropic Acoustic Metamaterials Based on Transformation Acoustics

Abstract

We propose the design method for broadband acoustic metamaterials based on the concept of transformation acoustics. Two-dimensional distributed transmission-line (TL) models for full-tensor anisotropic electromagnetic metamaterials are applied to full-tensor anisotropic acoustic metamaterials and the design formulas are shown to uniquely determine the structural parameters of the unit cells. Two-dimensional acoustic waveguide unit cell structures for realizing the TL models are proposed and an acoustic carpet cloak and an acoustic illusion medium are designed according to the introduced theory. The complex sound pressure distributions in the acoustic waveguides of the unit cells are calculated by full-wave simulations to verify the validity of the proposed method, and the broadband operations of the designed carpet cloak and illusion medium are confirmed from the results.

1. Introduction

In the field of electromagnetic waves, design methods with equivalent circuit models based on the transmission-line approach [1] have been adopted to designs of electromagnetic metamaterials since the approach can abstract Maxwell’s equations and leads to physical insights and rigorous designs. In particular, these have widely been used for the designs of electromagnetic metamaterials having negative refractive index characteristics [2,3,4,5,6]. Furthermore, two-dimensional (2-D) equivalent circuit models for full-tensor anisotropic metamaterials [7,8] have been proposed and have been introduced to the design of metamaterials based on the concept of transformation electromagnetics [9] that can realize invisible cloaks [9,10,11,12,13,14,15,16,17,18,19,20,21,22], carpet cloaks [7,8,21,23,24,25,26,27,28,29,30], and illusion media [31,32]. Those lumped circuit elements independently correspond to the permittivity and the permeability tensor components and, therefore, the models can rigorously design the unit cells of the full-tensor anisotropic metamaterials without resorting to many calculations by simulators that are one of the issues in the conventional design methods. The broadband operations can also be achieved due to the intrinsic nature of non-resonant. Furthermore, 2-D distributed transmission-line (TL) models have been proposed [8]. The characteristic impedance and the line length can be determined uniquely by substituting the lumped circuit elements in the 2-D equivalent circuit models for the design formulas and, therefore, we can easily implement metamaterials based on transformation electromagnetics with planar structures. The validity and the usefulness have also been demonstrated experimentally [8,32].

Recently, the theory of the 2-D equivalent circuit models for full-tensor anisotropic electromagnetic metamaterials has been extended to acoustic metamaterials [33,34] in order to realize acoustic invisible cloaks [21,35,36,37,38,39,40,41,42,43,44,45,46,47], acoustic carpet cloaks [33,48,49,50,51,52,53,54,55,56], or acoustic illusion media [34,57,58,59,60] based on the concept of transformation acoustics [35]. As is the same in the case of the electromagnetic metamaterials, the conventional design methods have issues like many calculations for designing the unit cell structures or narrowband characteristics, but the models introduced from the electromagnetic metamaterials can solve these simultaneously. So far, the correspondences between the lumped circuit elements and acoustic material parameters (the mass density tensor and the balk modulus) have been revealed [33] and the broadband operations of an acoustic carpet cloak and an acoustic illusion medium have been confirmed by circuit simulations [33,34]; however, the 2-D distributed TL models have yet to be introduced to acoustic metamaterials.

In this paper, we introduce the 2-D distributed TL models for full-tensor anisotropic electromagnetic metamaterials to the design of full-tensor anisotropic acoustic in order to theoretically determine the structural parameters of the unit cells and realize acoustic metamaterials based on transformation acoustics. In Section 2, we recall the theory of 2-D equivalent circuit models for acoustic metamaterials, and the 2-D distributed TL models for electromagnetic metamaterials, and update the design formulas of the TL models in order to determine the structural parameters of unit cells constituting acoustic metamaterials based on transformation acoustics. We also propose 2-D anisotropic acoustic waveguide unit cell structures and determine the waveguide widths and lengths for realizing an acoustic carpet cloak and an illusion medium as examples. In Section 3, the complex sound pressure distributions in those waveguides are calculated by full-wave simulations in order to confirm the broadband operations and the validity of the proposed design theory. In Section 4, this paper is concluded.

2. Methods

2.1. Design Formulas

Firstly, we recall the theory of 2-D equivalent circuit models for full-tensor anisotropic acoustic metamaterials (see Figure 1a,b) [33] and show the relations of material parameters and circuit parameters. According to the references of [7,8,33], defining the voltage and current vectors and the 2-D equivalent circuit models of Figure 1a,b as V = [V_x, V_x+1, V_y, V_y+1] and I = [I_x, −I_x+1, I_y, −I_y+1], we can obtain those circuit equations as shown in the left column of

Table 1. Circuit equations, 2-D Maxwell’s equations, and 2-D acoustic equations for full-tensor anisotropic electromagnetic or acoustic metamaterials.

Circuit Equations	2-D Maxwell’s Equations	2-D Acoustic Equations
$-\frac{I_{x+1}-I_x}{\mathsf{\Delta }d}-\frac{I_{y+1}-I_y}{\mathsf{\Delta }d}=j\omega C^{\prime }{V}_{\mathrm{c}}$	$\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}=j\omega {\epsilon }_z{E}_z$	$-\frac{\partial v_x}{\partial x}-\frac{\partial v_y}{\partial y}=j\omega \frac{1}{K}p$
$\frac{V_{x+1}-V_x}{\mathsf{\Delta }d}=\pm j\omega M^{\prime }\frac{I_{y+1}+I_y}{2}-j\omega L_x^{\prime }\frac{I_{x+1}+I_x}{2}$	$\frac{\partial E_z}{\partial x}=j\omega {\mu }_{yx}{H}_x+j\omega {\mu }_{yy}{H}_y$	$\frac{\partial p}{\partial x}=-j\omega {\rho }_{xy}{v}_y-j\omega {\rho }_{xx}{v}_x$
$\frac{V_{y+1}-V_y}{\mathsf{\Delta }d}=-j\omega L_y^{\prime }\frac{I_{y+1}+I_y}{2}\pm j\omega M^{\prime }\frac{I_{x+1}+I_x}{2}$	$\frac{\partial E_z}{\partial y}=-j\omega {\mu }_{xx}{H}_x-j\omega {\mu }_{xy}{H}_y$	$\frac{\partial p}{\partial y}=-j\omega {\rho }_{yy}{v}_y-j\omega {\rho }_{yx}{v}_x$

from Kirchhoff’s voltage and current laws. On the other hand, Maxwell’s equations and acoustic equations in the 2-D Cartesian coordinate system with z invariance can be written as in the center and the right columns of Table 1 by assuming z-polarized TE waves or acoustic waves in a full-tensor anisotropic metamaterials and time harmonic variation, respectively. Then, considering the correspondences of voltages, currents, electric fields, magnetic fields, scalar pressures, and fluid velocities, as shown in

Table 2. Correspondences of voltages, currents, electric fields, magnetic fields, scalar pressures, and fluid velocities in the equations of Table 1.

Circuit Equations	2-D Maxwell’s Equations	2-D Acoustic Equations
$V_{\mathrm{c}}$	$E_z$	$p$
$\frac{V_{x+1}-V_x}{\mathsf{\Delta }d}$	$\frac{\partial E_z}{\partial x}$	$\frac{\partial p}{\partial x}$
$\frac{V_{y+1}-V_y}{\mathsf{\Delta }d}$	$\frac{\partial E_z}{\partial y}$	$\frac{\partial p}{\partial y}$
$\frac{I_{x+1}+I_x}{2}$	$-H_y$	$v_x$
$\frac{I_{y+1}+I_y}{2}$	$H_x$	$v_y$
$\frac{I_{x+1}-I_x}{\mathsf{\Delta }d}$	$-\frac{\partial H_y}{\partial x}$	$\frac{\partial v_x}{\partial x}$
$\frac{I_{y+1}-I_y}{\mathsf{\Delta }d}$	$\frac{\partial H_x}{\partial y}$	$\frac{\partial v_y}{\partial y}$

, we can obtain the following relations [7,8,33] between the circuit parameters per unit length (L′_x, L′_y, M′, and C′), the parameters of electromagnetic metamaterials (μ_xx, μ_xy = μ_yx, μ_yy, and ε_z), and the parameters of acoustic metamaterials (ρ_xx, ρ_xy = ρ_yx, ρ_yy, and K′):

$$\left(\begin{array}{ll}{L}_x^{\prime }& \mp M^{\prime }\\ \mp M^{\prime }& L_y^{\prime }\end{array}\right)\iff \left(\begin{array}{ll}{\rho }_{xx}& {\rho }_{xy}\\ {\rho }_{yx}& {\rho }_{yy}\end{array}\right)\iff \left(\begin{array}{ll}{\mu }_{yy}& -{\mu }_{yx}\\ -{\mu }_{xy}& {\mu }_{xx}\end{array}\right)$$

(1)

$$C^{\prime }\iff \frac{1}{K^{\prime }}\iff {\epsilon }_{z^{\prime }}$$

(2)

where the upper signs correspond to the case of Figure 1a (ρ_xy = ρ_yx < 0) and the lower signs correspond to the case of Figure 1b (ρ_xy = ρ_yx > 0). Moreover, according to the concept of transformation acoustics [35], the mass density tensor (ρ′) and the bulk modulus (K′) for mimicking transformed coordinate system can be calculated from the following formulas:

$${\rho }^{\prime }=(\det \mathit{A}){({\mathit{A}}^T)}^{-1}\rho {\mathit{A}}^{-1}$$

(3)

$$K^{\prime }=(\det \mathit{A})K$$

(4)

where A is the Jacobian matrix, and ρ and K are the mass density and the bulk modulus for mimicking the original coordinate system, respectively.

Secondly, in order to theoretically determine the unit cell structures and realize broadband acoustic metamaterials based on transformation acoustics, we recall the design formulas of 2-D distributed TL models for full-tensor anisotropic electromagnetic metamaterials (see Figure 2a,b) [8], and extend the theory to acoustic metamaterials. The design formulas of the TL models for the case with electromagnetic metamaterials can be written as

$$L_x^{\prime }-M^{\prime }=\frac{2Z_{0x}}{\omega \mathsf{\Delta }d}\tan \frac{\beta l}{2}$$

(5)

$$L_y^{\prime }-M^{\prime }=\frac{2Z_{0y}}{\omega \mathsf{\Delta }d}\tan \frac{\beta l}{2}$$

(6)

$$M^{\prime }=\frac{2}{\omega \mathsf{\Delta }d}\frac{Z_{0\mathrm{M}}^2(Y_{0x}+Y_{0y})\tan \frac{\beta l}{2}+Z_{0\mathrm{M}}(\mathrm{cosec}{\beta }_{\mathrm{M}}{l}_{\mathrm{M}}-\cot {\beta }_{\mathrm{M}}{l}_{\mathrm{M}})}{{\cos }^2\frac{\beta l}{2}+Z_{0\mathrm{M}}(Y_{0x}+Y_{0y})\sin \beta l\cot {\beta }_{\mathrm{M}}{l}_{\mathrm{M}}-Z_{0\mathrm{M}}^2{(Y_{0x}+Y_{0y})}^2{\sin }^2\frac{\beta l}{2}}$$

(7)

$$C^{\prime }=\frac{1}{\omega \mathsf{\Delta }d}\left\{\left(Y_{0x}+Y_{0y}\right)\sin \beta l\cos {\beta }_{\mathrm{M}}{l}_{\mathrm{M}}+Y_{0\mathrm{M}}{\cos }^2\frac{\beta l}{2}\sin {\beta }_{\mathrm{M}}{l}_{\mathrm{M}}-Z_{0\mathrm{M}}{(Y_{0x}+Y_{0y})}^2{\sin }^2\frac{\beta l}{2}\right\}$$

(8)

and according to the reference of [8], these can be obtained by comparing Z-parameters of the equivalent circuit models with those of the TL models. βl = β_xl_x = β_yl_y is also assumed to solve these formulas simultaneously. Then, substituting (1) and (2) for (5)–(8), we can obtain the following design formulas for acoustic metamaterials that are extended from those for electromagnetic metamaterials:

$${\rho }_{xx}\pm {\rho }_{xy}=\frac{2Z_{0x}}{\omega \mathsf{\Delta }d}\tan \frac{\beta l}{2}$$

(9)

$${\rho }_{yy}\pm {\rho }_{xy}=\frac{2Z_{0y}}{\omega \mathsf{\Delta }d}\tan \frac{\beta l}{2}$$

(10)

$$\mp {\rho }_{xy}=\mp {\rho }_{yx}=\frac{2}{\omega \mathsf{\Delta }d}\frac{Z_{0\mathrm{M}}^2(Y_{0x}+Y_{0y})\tan \frac{\beta l}{2}+Z_{0\mathrm{M}}(\mathrm{cosec}{\beta }_{\mathrm{M}}{l}_{\mathrm{M}}-\cot {\beta }_{\mathrm{M}}{l}_{\mathrm{M}})}{{\cos }^2\frac{\beta l}{2}+Z_{0\mathrm{M}}(Y_{0x}+Y_{0y})\sin \beta l\cot {\beta }_{\mathrm{M}}{l}_{\mathrm{M}}-Z_{0\mathrm{M}}^2{(Y_{0x}+Y_{0y})}^2{\sin }^2\frac{\beta l}{2}}$$

(11)

$$\frac{1}{K^{\prime }}=\frac{1}{\omega \mathsf{\Delta }d}\left\{\left(Y_{0x}+Y_{0y}\right)\sin \beta l\cos {\beta }_{\mathrm{M}}{l}_{\mathrm{M}}+Y_{0\mathrm{M}}{\cos }^2\frac{\beta l}{2}\sin {\beta }_{\mathrm{M}}{l}_{\mathrm{M}}-Z_{0\mathrm{M}}{(Y_{0x}+Y_{0y})}^2{\sin }^2\frac{\beta l}{2}\right\}$$

(12)

The upper signs of (9)–(11) correspond to the case of Figure 2a (ρ_xy = ρ_yx < 0) and the lower signs correspond to the case of Figure 2b (ρ_xy = ρ_yx > 0). It is seen from these formulas that we can uniquely determine the characteristic impedances (Z_0x, Z_0y, Z_0M) and the electrical lengths (βl and β_Ml_M) for acoustic metamaterials.

2.2. Proposed Acoustic Waveguide Unit Cell Structures

Figure 3a,b shows the proposed 2-D acoustic waveguide unit structures for full-tensor anisotropic acoustic metamaterials and Figure 4a,b is for the background medium to be connected to Figure 3a,b, respectively. Furthermore, these waveguides are formed in the rigid body and are filled with the air. w_x, w_y, w_M, and w_b are the waveguide widths, l (=l_x = l_y), l_M, and l_b are the waveguide lengths, and Δd and Δd_b (=Δd) are the waveguide unit cell lengths. The waveguide widths can be determined as the following formulas:

$$w_x=\frac{Z_{0b}}{Z_{0x}}{w}_b,w_y=\frac{Z_{0b}}{Z_{0y}}{w}_b,w_{\mathrm{M}}=\frac{Z_{0b}}{Z_{0\mathrm{M}}}{w}_b$$

(13)

where Z_0b is the characteristic impedance of the 2-D distributed TLs for the background medium shown in Figure 5. The ratio of the characteristic impedances can be obtained from (9)–(12). On the other hand, the waveguide lengths can be calculated with

$$l=\sqrt{2}\varphi l_b,l_{\mathrm{M}}=\sqrt{2}{\varphi }_{\mathrm{M}}{l}_b$$

(14)

ϕ and ϕ_M are values of electrical lengths normalized by k_bΔd_b (k_b is the wavenumber for the background medium) and represent the solutions of βl/k_bΔd_b and β_Ml_M/k_bΔd_b obtained from (9)–(12), respectively. $\sqrt{2}$ denotes the 2-D effect of the unit cell structures [1,8].

2.3. Design of an Acoustic Carpet Cloak and an Illusion Medium

Here, we design an acoustic carpet cloak for hiding a bump on a flat surface of Figure 6 [33] and an acoustic illusion medium for mimicking a groove on a flat surface of Figure 7 [34] to show the design examples with the proposed unit cell structures in the previous subsection. Incidentally, materials of the flat surface, the bump, and the groove are assumed to the rigid body in the following design.

We first consider defining the original coordinate systems, such as Figure 6a and Figure 7a, and transforming those to Figure 6b and Figure 7b, respectively. These transformation formulas are chosen as [33,34]

$$x^{\prime }=x,y^{\prime }=y+A\left(1-\frac{y}{h}\right){\left\{1-{\left(\frac{x}{p}\right)}^2\right\}}^2$$

(15)

$$x^{\prime }=x,y^{\prime }=\frac{y+A{\left\{1-{\left(\frac{x}{p}\right)}^2\right\}}^2}{h+A{\left\{1-{\left(\frac{x}{p}\right)}^2\right\}}^2}h,$$

(16)

respectively, and A, h, and p are selected as 30 mm, 100 mm, and 100 mm with Δd = Δd_b = 10 mm, in the following, for simplicity.

Next, we obtained Z_0b/Z_0x, Z_0b/Z_0y, ϕ, and ϕ_M by using (3), (4), (9)–(12), (15), and (16) with Z_0M/Z_0b = 1.061 (carpet cloak case) or 1.500 (illusion medium case), and calculated w_x, w_y, l, and l_M from (13) and (14) with w_b = 1.0 mm and l_b = 11 mm. Figure 8 and Figure 9 show the results for the acoustic carpet cloak case (w_M = 0.943 mm) and the acoustic illusion medium case (w_M = 0.667 mm), respectively, and these are determined by considering the feasibility. Furthermore, the designed acoustic carpet cloak and illusion medium are illustrated in Figure 10a and Figure 11a. We carry out the full-wave simulations and compare those results with the cases of the flat surface (see Figure 10b), the bump (see Figure 10c), and the groove (see Figure 11b) in the next section.

3. Results and Discussion

3.1. Acoustic Carpet Cloak

Figure 12a shows the setup for the full-wave simulations of the acoustic carpet cloak and the size of the analysis area is set to 400 mm × 200 mm. The designed carpet cloak in Figure 10a is arranged at the center of the bottom in that area, and the other area corresponds to the background medium constituted by using the structures of Figure 4. Forty input ports are connected to the waveguides of the upper boundary and a Gaussian beam with a beam waist of 100 mm is injected from there. Furthermore, the rigid body walls are set as the bottom boundary and other boundaries are selected as the absorption boundary. We calculated the complex sound pressure distributions in the waveguides by the Acoustics Module of COMSOL and compared the results with those for the cases with the flat floor in Figure 10b (see Figure 12b) and the bump in Figure 10c (see Figure 12c). Incidentally, the number of meshes in these simulations was approximately 1.4 million and the calculation time was approximately one hour.

The calculated results of the amplitude and phase distributions of the sound pressure are shown in Figure 13a–f, and the frequencies are set to 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 kHz, respectively. The values of the wavelength per unit cell length (λ/Δd) are 22.3, 14.8, 11.1, 8.91, 7.42, and 6.36. It is seen from the results of Figure 13a–d that the carpet cloak can sufficiently suppress the scattered waves by the bump and mimics the flat surface. On the other hand, the level of the scattered wave becomes large in the cases of Figure 13e,f. This reason is that the influence by the discretization error becomes large due to the shorter wavelength than in the other cases. Therefore, we can conclude from the results above that the designed acoustic carpet cloak can operate at least up to 2.5 kHz (λ/Δd = 8.91).

3.2. Acoustic Illusion Medium

The setup for the full-wave simulations of the acoustic illusion medium is shown in Figure 14a. The size of the analysis area, the input ports, and the boundary conditions are set to the same as those for the acoustic carpet cloak, and the designed illusion medium in Figure 11a is placed at the center of the bottom in that area. As is the case in Figure 13, we calculated the complex sound pressure distributions in the waveguides by the Acoustics Module of COMSOL and compared the results with those for the case with the groove in Figure 11b (see Figure 14b). Incidentally, the number of meshes and the calculation time were almost the same as in the acoustic carpet cloak case.

The calculated results of the amplitude and phase distributions of the sound pressure are shown in Figure 15a–f, and the frequencies are set to the same values as in the cases of Figure 13a–f, respectively. Differences of distributions in the background medium region between the acoustic illusion medium case and the groove case are slightly large in the cases of the higher frequencies from these results. However, it can be seen that the scattered waves from the groove are sufficiently mimicked by the acoustic illusion medium and the performance is better than that of the acoustic carpet cloak. The reason for this is considered to be that the area discretized by the unit cells is larger than that in the case of the acoustic carpet cloak and the influence of the discretization error is small. Therefore, it can be concluded from the results, and those of the acoustic carpet cloak, that the broadband operations and the validity of the proposed design method can be shown.

4. Conclusions

Firstly, the theory of 2-D distributed TL models for the full-tensor anisotropic electromagnetic metamaterials was introduced to acoustic metamaterials in order to theoretically design unit cell structures constituting broadband acoustic metamaterials based on the concept of transformation acoustics. The formulas of 2-D equivalent circuit models for full-tensor anisotropic acoustic metamaterials were recalled, and the design formulas of the TL models for electromagnetic metamaterials were updated to those for determining the structural parameters of unit cells of acoustic metamaterials.

Secondly, 2-D acoustic waveguide unit cell structures were proposed and the design formulas were shown to determine the waveguide widths and lengths according to the updated theory of the 2-D distributed TL models. To show the design examples and the usefulness, an acoustic carpet cloak and an illusion medium were designed with the proposed structures and those structural parameters were determined by using the design formulas theoretically.

Finally, full-wave simulations were carried out with the Acoustics Module of COMSOL and complex pressure distributions were calculated. The results show the broadband operations of the designed acoustic carpet cloak and the illusion medium, and it is concluded that the validity of the proposed method introduced from electromagnetic metamaterials has been demonstrated.