# A Novel Acoustic Ventilation Barrier Based on the Block-Shaped Space-Coiling Metamaterial
## 2 The Design and Working Principle of the Proposed Acoustic  
### Ventilation Barrier  
This section introduces the design and working principles of the acoustic ventilation barrier proposed in this article. As shown in Figure 1, the acoustic ventilation barrier is made of the block-shaped metamaterial in which the meta-unit is a rectangular helix with a square hole for ventilation, and the front face is also a square. The detailed structures of this meta-unit are shown in Figure 1a. The outer and inner side lengths are denoted by \(L\) and \(l\), respectively, and the thickness is denoted by \(t\). The meta-unit is designed to have a total of six acoustic channels arranged in a symmetrical manner in which the spacings of the three layers on one side are denoted by \(d_{1}\), \(d_{2}\), and \(d_{3}\), respectively. The thicknesses of the internal walls are identical, which is denoted by \(w\). The width of opening \(d_{0}\) was set as \((d_{1}+d_{2}+d_{3})/3\), and \(a\) denotes the length of the resonant room. Evidently, \(t\) is computed by \(t=2(d_{1}+d_{2}+d_{3})+7w\), and \(a\) is computed by \(a=L-d_{0}\). Table 1 gives the design parameters of a benchmark meta-unit. On the one hand, the spatially coiled acoustic channels form Fano-like resonances and therefore hamper the propagation of sound waves belonging to certain frequency ranges. On the other hand, the square hole in the center allows air to pass through, obtaining good ventilation performance. Additionally, the proposed acoustic ventilation barrier can be easily achieved by putting the block-shaped meta-units together in which two adjacent square holes are separated by two space-coiling blocks (as shown in Figure 2b). The Fano-like resonance and the consistently quasi-balanced monopolar and dipolar response strength in a broadband endow the proposed meta-unit with a broad working frequency range. Evidently, compared to other acoustic ventilation  
Figure 1: A comparison of recently proposed acoustic ventilation barriers: (**a**) the cylindrical space-coiling meta-unit [20]; (**b**) the modified cylindrical space-coiling meta-unit [24]; (**c**) the customizable cylindrical space-coiling meta-unit [25]; (**d**) the rectangular column-shaped acoustic barrier [26]; and (**e**) the block-shaped meta-unit [27]. barriers made of cylinder-shaped meta-units, the proposed acoustic ventilation barrier is much easier to install. One can easily build such acoustic ventilation barriers by stacking the proposed meta-units together. To study the working principle of the proposed metamaterial, a benchmark meta-unit was designed, and its design parameters are shown in Table 1. The sound attenuation effect of the metamaterial made of the benchmark meta-units was numerically studied using the pressure acoustic module of COMSOL Multiphysics (v.6.3) in which a frequency domain study is performed to analyze the sound transmission through the meta-unit. The proposed meta-unit was placed at the center of a rectangular waveguide, whose size is 1055 \(\times\) 80 \(\times\) 80 mm (shown in Figure 3). And both the meta-unit and the waveguide were modeled as acoustically rigid, where both ends of the waveguide were set as plane-wave radiation boundaries, while the incident wave was only excited at one side. The narrow region acoustics model was employed to consider the losses due to viscosity and thermal dissipation. The maximum element size was set to 50 mm, which satisfies the Nyquist-Shannon sampling theorem (the allowable element size is computed by \(L_{m}=V_{a}/(4f_{u})=70.8\) mm, where \(V_{a}=340\),000 mm/s is the sound speed, and \(f_{u}=1200\) Hz is the cut-off frequency). Figure 4 shows the simulated TL of the metamaterial made of the benchmark meta-units, where the TL is defined as  
\[\mathrm{TL}=10\log_{10}\left|\frac{W_{in}}{W_{out}}\right| \tag{1}\]  
where \(W_{in}\) and \(W_{out}\) are the input and output power of incident wave. As shown in Figure 4, a Fano-like resonance effect is achieved, forming two resonance frequencies, namely 536 Hz and 934 Hz.