# Constitution Method for Broadband Acoustic Metamaterials Based on the Design Theory of a One-Dimensional Distributed Transmission-Line Model

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Design Theory Based on a 1D Distributed Transmission-Line Model

_{0}and Z

_{0b}present the characteristic impedance, β and β

_{b}represent the phase constant, and l and l

_{b}(=∆d) represent the line length.

#### 2.2. Proposal Structures and Acoustic Broadband Metasurface Design

_{b}= β

_{air}, c = c

_{b}= c

_{air}). For impedance matching and feasibility, the waveguide widths are set to w = 0.8 mm and w

_{b}= w/$\sqrt{2}$ = 0.566 mm, and the unit cell length is chosen as ∆d = l

_{b}= 10 mm. The solid line in Figure 3 represents the theoretical waveguide length for each cell obtained by using Formula (2) with θ = 20 degrees and a = 20∆d = 200 mm, and these parameters are set to feasible values. This can be basically realized by selecting the one appropriate structure from #1 to #4 in Figure 2, but the acoustical length (βl) becomes smaller than the theoretical one due to the effect of the bent waveguide if we adopt the theoretical one as it is. Therefore, we need to modify the waveguide length by calculating the frequency dispersion characteristics with eigenvalue analysis.

#### 2.3. Flat Acoustic Lens Design

_{b}, and f are 200, 10, 0.8, 0.566, and 105 mm, respectively, and 1D meander acoustic waveguide structures of #1–#3 in Figure 2 and a 1D straight acoustic waveguide structure (#0) are used for the design of the lens. Given these parameters and the optimized waveguide length, we constructed the lens as shown in Figure 7. In the next section, its broadband operations are investigated by full-wave simulations and the results are presented with those of the designed acoustic metasurface in the previous subsection.

## 3. Results and Discussion

#### 3.1. Acoustic Metasurface

#### 3.2. Acoustic Flat Lens

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The concept of a broadband metasurface for refracting normal incident plane waves with an angle of θ. The size is a × Δd and the area is discretized with the 1D TL model whose size is Δd × Δd. The 2D TL model whose size is Δd × Δd is used for the acoustic metasurface and the background medium. Z

_{0b}and Z

_{0}are the characteristic impedances and β

_{b}l

_{b}and βl are the electrical lengths.

**Figure 2.**Proposed 1D meander acoustic waveguide structures for the acoustic metasurface (#1–#4) and a 2D straight acoustic waveguide structure for the background medium (#5). w and w

_{b}are waveguide widths and l and l

_{b}are waveguide lengths. Δd is the unit cell length. These waveguides are filled with the air and are formed in the rigid body.

**Figure 3.**Calculated results of the waveguide length for refracting normal incident plane waves. The solid line is the theoretical value, and the dots are the optimized values. The horizontal axis corresponds to the y coordinate in Figure 1. The refracted angle and the aperture size are chosen as θ = 20 degrees, a = 20Δd = 200 mm, respectively. The values of #1–#4 are used for the structures of #1–#4 in Figure 2. The value at y = 95 mm (#0) can be realized by a 1D straight acoustic waveguide structure (w = 0.8 mm and l = Δd = 10 mm).

**Figure 4.**Designed acoustic metasurface for refracting normal incident plane waves with the angle of 20 degrees. The size is a × ∆d = 200 × 10 mm

^{2}. A 1D straight acoustic waveguide structure (w = 0.8 mm and l = ∆d = 10 mm) is adopted to #0. The structures of #1–#4 in Figure 2 correspond to #1–#4 in this figure and the calculated waveguide lengths in Figure 3 are used for these.

**Figure 5.**The concept of a broadband acoustic flat lens. As is the case with Figure 1, the size is a × ∆d and the 1D TL model and the 2D TL model are used for the acoustic metasurface and the background medium. The position of (x, y) = (f − ∆d/2, 0) corresponds to the focal point of the lens.

**Figure 6.**Calculated results of the waveguide length for the acoustic flat lens. The solid line is the theoretical value, and the dots are the optimized values. The horizontal axis corresponds to the y coordinate in Figure 5. The focal length and the aperture size are chosen as f = 10.5∆d = 105 mm and a = 200 mm, respectively. The values of #1–#3 are used for the structures of #1–#3 in Figure 2. A 1D straight acoustic waveguide structure (w = 0.8 mm and l = ∆d = 10 mm) is used for the position at y = 95 mm, as in the case of Figure 3.

**Figure 7.**Designed acoustic flat lens. The size is a × ∆d = 200 × 10 mm

^{2}. A 1D straight acoustic waveguide structure (w = 0.8 mm and l = ∆d = 10 mm) is adopted for #0. The structures of #1–#3 in Figure 2 correspond to #1–#3 in this figure and calculated waveguide lengths of Figure 6 are used for these.

**Figure 8.**Configuration for full-wave simulations of the designed acoustic metasurface. The acoustic metasurface of 200 × 10 mm

^{2}in Figure 4 is sandwiched between two background media whose sizes are 200 × 40 and 200 × 150 mm

^{2}. The background media are composed of the structure of #5 in Figure 2. Twenty input ports are set to the left-hand side boundaries of acoustic waveguides, and other boundaries are chosen as absorption boundaries.

**Figure 9.**Complex sound pressure distributions for the acoustic metasurface: (

**a**) 3.063 kHz (λ = 8∆d = 80 mm); (

**b**) 3.500 kHz (λ = 7∆d = 70 mm); (

**c**) 4.084 kHz (λ = 6∆d = 60 mm); (

**d**) 4.900 kHz (λ = 5∆d = 50 mm); (

**e**) 6.125 kHz (λ = 4∆d = 40 mm); (

**f**) 8.167 kHz (λ = 3∆d = 30 mm). The left and right figures present the amplitude and phase, respectively. Incident plane waves illuminate the acoustic metasurface perpendicularly from the left-hand side of the analysis area and are refracted by that. The broken lines represent the theoretical propagation direction of incident and refracted waves.

**Figure 10.**Configuration for full-wave simulations of the designed acoustic flat lens. The acoustic metasurface of 200 × 10 mm

^{2}in Figure 7 is sandwiched between two background media whose sizes are 200 × 40 and 200 × 150 mm

^{2}. The background media are composed of the structure of #5 in Figure 2. As is the case with Figure 8, twenty input ports are set to the left-hand side boundaries of acoustic waveguides and other boundaries are chosen as absorption boundaries.

**Figure 11.**Complex sound pressure distributions for the acoustic flat lens: (

**a**) 3.063 kHz (λ = 8∆d = 80 mm); (

**b**) 3.500 kHz (λ = 7∆d = 70 mm); (

**c**) 4.084 kHz (λ = 6∆d = 60 mm); (

**d**) 4.900 kHz (λ = 5∆d = 50 mm); (

**e**) 6.125 kHz (λ = 4∆d = 40 mm); (

**f**) 8.167 kHz (λ = 3∆d = 30 mm). The left and right figures present the amplitude and phase, respectively. Incident plane waves illuminate the lens perpendicularly from the left-hand side of the analysis area and a focus is formed by these. The broken lines represent the theoretical propagation direction of incident and refracted waves, and the intersection is the theoretical position of the focus.

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**MDPI and ACS Style**

Nakagawa, T.; Nagayama, T.; Fukushima, S.; Watanabe, T.
Constitution Method for Broadband Acoustic Metamaterials Based on the Design Theory of a One-Dimensional Distributed Transmission-Line Model. *Crystals* **2022**, *12*, 1528.
https://doi.org/10.3390/cryst12111528

**AMA Style**

Nakagawa T, Nagayama T, Fukushima S, Watanabe T.
Constitution Method for Broadband Acoustic Metamaterials Based on the Design Theory of a One-Dimensional Distributed Transmission-Line Model. *Crystals*. 2022; 12(11):1528.
https://doi.org/10.3390/cryst12111528

**Chicago/Turabian Style**

Nakagawa, Tomoya, Tsutomu Nagayama, Seiji Fukushima, and Toshio Watanabe.
2022. "Constitution Method for Broadband Acoustic Metamaterials Based on the Design Theory of a One-Dimensional Distributed Transmission-Line Model" *Crystals* 12, no. 11: 1528.
https://doi.org/10.3390/cryst12111528