## 4 Discussion
### Acoustic Pressure Response of 1D Acoustic Metamaterials Made from Repeated Optimal Unit Cell  
The acoustic pressure response of a 1D periodic acoustic metamaterial composed of tessellated optimal unit cells, which were determined earlier in the filtered step mentioned in the preceding sub--section, is examined. Computing the acoustic pressure response of 1D metamaterials made from these optimal unit cells serves two purposes. Firstly, it serves as proof--of--concept for determining bandgaps using FE--Bloch's Theorem (i.e., one of the components in the extensive design framework), and secondly, evaluating the acoustic sound pressure levels of these periodic structures. One objective of this study is to examine the acoustic sound pressure levels in four different 1D periodic acoustic metamaterials. Each of these metamaterials has a unique unit cell size, resulting in variations in dimensions. To ensure a consistent actuation scenario across all four metamaterials, a building rule was implemented. This involved using a box with dimensions of 500 mm by 45 mm by 45 mm, and filling it with optimal unit cells to create each respective 1D periodic acoustic metamaterial. The number of unit cells required for each metamaterial was determined by dividing the length of the box by the size of the unit cell. This approach allows for the examination of the acoustic pressure response of the 1D periodic acoustic metamaterials and enables comparisons between different metamaterials with varying unit cell sizes. Figure 14A showcases the actuation scenario, where there is a designated actuation location point at one end of the box length and five sensing location points. One sensing location point is positioned at the other end of the box length (i.e., denoted as X), while two are located in the middle and at opposite ends of the tessellation axes (i.e., denoted as Y and Z). Additionally, there are two sensing locations close to one end of the box length, aligned with opposite axes to tessellation (i.e., denoted as Y\({}^{\prime}\) and Z\({}^{\prime}\)). The external sound source placed at the actuation location was in the form of an impulse signal with \(P=1\) Pa, \(SPL=94\) dB applied across the calculated frequency range (i.e., 20 Hz-20 kHz), which is depicted in Figure 14B. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{Metamaterial} & \multicolumn{4}{c}{Design Combination} & \multirow{2}{*}{Bandgaps Coverage} \\ \cline{2-2} \cline{4-6}  & \(C\) (mm) & \(L/C\) & \(d_{1}/C\) & \(d_{2}/d_{1}\) & \\ \hline Semi—Periodic & 42.9 & 0.3 & 0.1 & 0.5 & 0.955 \\ Tapered—Diverging & 38.7 & 0.3 & 0.1 & 0.5 & 0.944 \\ Tapered—Converging & 21.1 & 0.1 & 0.1 & 0.9 & 0.936 \\ \hline \multicolumn{6}{c}{Design Combination} & \multirow{2}{*}{Bandgaps Coverage} \\ \cline{2-2} \cline{4-6}  & \(C\) (mm) & \(L/C\) & \(d/C\) & \\ \hline Basic—Periodic & 21.1 & 0.1 & 0.1 & 0.935 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bandgaps coverage for the optimal designs. The acoustic pressure responses of the optimal 1D periodic acoustic metamaterials are shown in Figures 15-18 and S3-S6. These metamaterials consist of tessellated optimal unit cells that follow the introduced building rule and are actuated according to the actuation scenario. To assess the effectiveness of these optimal metamaterials, the acoustic sound pressure levels were compared between five sensing locations within the metamaterials and a block of air. Panels B in Figures 15-18 and Panels A to D in Figures S3-S6 illustrate this comparison. Table 2 shows the average sound pressure level in each sensing location (i.e., denoted **X**, **Y**, **Z**, **Y\({}^{\prime}\)**, **Z\({}^{\prime}\)** in the table), which was determined using the centroid of the area under the curve of the acoustic pressure responses (i.e., represented in Panels B-G in Figures 15-18 and Panels A to D in Figures S3-S6), as well as the overall average sound pressure level in all the sensing locations of the four optimal metamaterials.