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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Design Formulas

_{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 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, 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′):

_{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:

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

_{x}l

_{x}= β

_{y}l

_{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:

_{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

_{M}l

_{M}) for acoustic metamaterials.

#### 2.2. Proposed Acoustic Waveguide Unit Cell Structures

_{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:

_{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

_{M}are values of electrical lengths normalized by k

_{b}Δd

_{b}(k

_{b}is the wavenumber for the background medium) and represent the solutions of βl/k

_{b}Δd

_{b}and β

_{M}l

_{M}/k

_{b}Δd

_{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

_{b}= 10 mm, in the following, for simplicity.

_{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

#### 3.2. Acoustic Illusion Medium

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**2-D equivalent circuit models for full-tensor anisotropic acoustic metamaterials. (

**a**) ρ

_{xx}< 0 case; (

**b**) ρ

_{xx}> 0 case.

**Figure 2.**2-D distributed TL models for determining the structural parameters of unit cells of full-tensor anisotropic acoustic metamaterials to be introduced from electromagnetic metamaterials. (

**a**) ρ

_{xx}< 0 case; (

**b**) ρ

_{xx}> 0 case. In the case of acoustic metamaterials, the phase constants (β and β

_{M}) of TLs correspond to those of acoustic waves.

**Figure 3.**2-D acoustic waveguide unit cell structures for realizing full-tensor anisotropic acoustic metamaterials. (

**a**) ρ

_{xx}< 0 case; (

**b**) ρ

_{xx}> 0 case. The waveguides are formed in the rigid body and are filled with the air.

**Figure 6.**Concept of an acoustic carpet cloak. (

**a**) Original coordinate system on a flat surface and the reflection from the flat surface; (

**b**) transformed coordinate system on a bump and the reflection from an acoustic carpet cloak covering the bump. The same reflected wave as that from the flat surface is generated by controlling the trajectory of the incident acoustic wave, with the carpet cloak mimicking the transformed coordinate system and it looks as if there is not the bump on the flat surface.

**Figure 7.**Concept of an illusion medium. (

**a**) Original coordinate system on a groove and scatterings of the incident acoustic wave by the groove; (

**b**) transformed coordinate system on a flat surface and scatterings of the incident acoustic wave from an illusion medium on the flat surface. The same scattered waves as that from the groove are generated by controlling the trajectory of the incident acoustic wave with the illusion medium and it looks like as if there is the groove on the flat surface.

**Figure 8.**Calculated results of the waveguide widths and lengths for the acoustic carpet cloak (w

_{M}= 0.943 mm). (

**a**) w

_{x}; (

**b**) w

_{y}; (

**c**) l; (

**d**) l

_{M}.

**Figure 9.**Calculated results of the waveguide widths and lengths for the acoustic illusion medium (w

_{M}= 0.667 mm). (

**a**) w

_{x}; (

**b**) w

_{y}; (

**c**) l; (

**d**) l

_{M}.

**Figure 10.**Designed acoustic carpet cloak and the additional designed structures for comparison. (

**a**) Carpet cloak; (

**b**) flat surface; (

**c**) bump. These sizes are 2p × h = 200 mm × 100 mm (2p × h) and those are discretized by using unit cells of Figure 3 (carpet cloak case) and Figure 4 (flat surface and bump cases) whose sizes are Δd × Δd = Δd

_{b}× Δd

_{b}= 10 mm × 10 mm. The structural parameters of Figure 8 are adopted to the unit cells of the carpet cloak, and those for the flat surface and the bump are w

_{b}= 1.0 mm and l

_{b}= 11 mm.

**Figure 11.**Designed acoustic illusion medium and the additional designed structure for comparison. (

**a**) Illusion medium; (

**b**) groove. These sizes are 2p × h = 200 mm × 100 mm (2p × h) and those are discretized by using unit cells of Figure 3 (illusion medium case) and Figure 4 (groove case) whose sizes are Δd × Δd = Δd

_{b}× Δd

_{b}= 10 mm × 10 mm. Structural parameters of Figure 9 are adopted to the unit cells of the carpet cloak, and those for the groove are w

_{b}= 1.0 mm and l

_{b}= 11 mm.

**Figure 12.**Full-wave simulation setup. (

**a**) Acoustic carpet cloak; (

**b**) flat surface; (

**c**) bump. The designed structures in Figure 10a-c are placed at the center of the bottom of the background medium area (400 mm × 200 mm). The bottom boundaries are set as the rigid body walls.

**Figure 13.**Complex sound pressure distributions for the acoustic carpet cloak, the flat surface, and the bump. (

**a**) 1.0 kHz (λ/∆d = 22.3); (

**b**) 1.5 kHz (λ/∆d = 14.8); (

**c**) 2.0 kHz (λ/∆d = 11.1); (

**d**) 2.5 kHz (λ/∆d = 8.91); (

**e**) 3.0 kHz (λ/∆d = 7.42); (

**f**) 3.5 kHz (λ/∆d = 6.36). Left: Amplitude. Right: Phase.

**Figure 14.**Full-wave simulation setup. (

**a**) Acoustic illusion medium; (

**b**) groove. The designed structures in Figure 11a,b are placed at the center of the bottom of the background medium area (400 mm × 200 mm). In the case of (

**b**), the groove is on the lower position than the flat surface level. The bottom boundaries are set as the rigid body walls.

**Figure 15.**Complex sound pressure distributions for the acoustic illusion medium and the groove. (

**a**) 1.0 kHz (λ/∆d = 22.3); (

**b**) 1.5 kHz (λ/∆d = 14.8); (

**c**) 2.0 kHz (λ/∆d = 11.1); (

**d**) 2.5 kHz (λ/∆d = 8.91); (

**e**) 3.0 kHz (λ/∆d = 7.42); (

**f**) 3.5 kHz (λ/∆d = 6.36). Left: Amplitude. Right: Phase.

**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 |
---|---|---|

$-\frac{{I}_{x+1}-{I}_{x}}{\mathsf{\Delta}d}-\frac{{I}_{y+1}-{I}_{y}}{\mathsf{\Delta}d}=j\omega {C}^{\prime}{V}_{\mathrm{c}}$ | $\frac{\partial {H}_{y}}{\partial x}-\frac{\partial {H}_{x}}{\partial y}=j\omega {\epsilon}_{z}{E}_{z}$ | $-\frac{\partial {v}_{x}}{\partial x}-\frac{\partial {v}_{y}}{\partial y}=j\omega \frac{1}{K}p$ |

$\frac{{V}_{x+1}-{V}_{x}}{\mathsf{\Delta}d}=\pm j\omega {M}^{\prime}\frac{{I}_{y+1}+{I}_{y}}{2}-j\omega {L}_{x}^{\prime}\frac{{I}_{x+1}+{I}_{x}}{2}$ | $\frac{\partial {E}_{z}}{\partial x}=j\omega {\mu}_{yx}{H}_{x}+j\omega {\mu}_{yy}{H}_{y}$ | $\frac{\partial p}{\partial x}=-j\omega {\rho}_{xy}{v}_{y}-j\omega {\rho}_{xx}{v}_{x}$ |

$\frac{{V}_{y+1}-{V}_{y}}{\mathsf{\Delta}d}=-j\omega {L}_{y}^{\prime}\frac{{I}_{y+1}+{I}_{y}}{2}\pm j\omega {M}^{\prime}\frac{{I}_{x+1}+{I}_{x}}{2}$ | $\frac{\partial {E}_{z}}{\partial y}=-j\omega {\mu}_{xx}{H}_{x}-j\omega {\mu}_{xy}{H}_{y}$ | $\frac{\partial p}{\partial y}=-j\omega {\rho}_{yy}{v}_{y}-j\omega {\rho}_{yx}{v}_{x}$ |

**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 |
---|---|---|

${V}_{\mathrm{c}}$ | ${E}_{z}$ | $p$ |

$\frac{{V}_{x+1}-{V}_{x}}{\mathsf{\Delta}d}$ | $\frac{\partial {E}_{z}}{\partial x}$ | $\frac{\partial p}{\partial x}$ |

$\frac{{V}_{y+1}-{V}_{y}}{\mathsf{\Delta}d}$ | $\frac{\partial {E}_{z}}{\partial y}$ | $\frac{\partial p}{\partial y}$ |

$\frac{{I}_{x+1}+{I}_{x}}{2}$ | $-{H}_{y}$ | ${v}_{x}$ |

$\frac{{I}_{y+1}+{I}_{y}}{2}$ | ${H}_{x}$ | ${v}_{y}$ |

$\frac{{I}_{x+1}-{I}_{x}}{\mathsf{\Delta}d}$ | $-\frac{\partial {H}_{y}}{\partial x}$ | $\frac{\partial {v}_{x}}{\partial x}$ |

$\frac{{I}_{y+1}-{I}_{y}}{\mathsf{\Delta}d}$ | $\frac{\partial {H}_{x}}{\partial y}$ | $\frac{\partial {v}_{y}}{\partial y}$ |

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**MDPI and ACS Style**

Nagayama, T.; Toshima, A.; Fukushima, S.; Watanabe, T.
Two-Dimensional Distributed Transmission-Line Models for Broadband Full-Tensor Anisotropic Acoustic Metamaterials Based on Transformation Acoustics. *Crystals* **2022**, *12*, 1557.
https://doi.org/10.3390/cryst12111557

**AMA Style**

Nagayama T, Toshima A, Fukushima S, Watanabe T.
Two-Dimensional Distributed Transmission-Line Models for Broadband Full-Tensor Anisotropic Acoustic Metamaterials Based on Transformation Acoustics. *Crystals*. 2022; 12(11):1557.
https://doi.org/10.3390/cryst12111557

**Chicago/Turabian Style**

Nagayama, Tsutomu, Akihiro Toshima, Seiji Fukushima, and Toshio Watanabe.
2022. "Two-Dimensional Distributed Transmission-Line Models for Broadband Full-Tensor Anisotropic Acoustic Metamaterials Based on Transformation Acoustics" *Crystals* 12, no. 11: 1557.
https://doi.org/10.3390/cryst12111557