## 5 Conclusions  
This work commenced by proposing an extensive design framework for designing 1D periodic acoustic metamaterials for attenuating noises passing within the audible frequency range of 20 Hz to 20 kHz. The focus is on creating and finding optimal designs and evaluating their effectiveness in noise reduction by analyzing the acoustic bandgaps and acoustic pressure response in the audible frequency range (i.e., 20 Hz to 20 kHz). The geometrical configurations are controlled using non--dimensional parameters as outlined earlier in the methodology section. A total of 505 normalized design combinations are considered during the parametric analysis. The noise attenuation for each design combination is evaluated through determining its corresponding bandgaps using the FE--[PERSON]'s theorem unit cell--dependent approach. In all the design configurations that include the four--unit cell geometries, it is shown that further reduction in the cavity hole diameter results in an increase in the widths of the bandgaps. Furthermore, the widest bandgap for all the periodic designs was found to be the first bandgap; however, the first acoustic bandgap for the Semi--Periodic designs was found to be too narrow to operate in for effective noise attenuation. Two key features (i.e., the normalized size of the widest bandgap and the normalized centroid of the widest bandgap; see Supplementary Materials) are introduced to analyze and evaluate the effectiveness of the noise attenuation in various designs. Analyzing the normalized Size of the Widest Bandgap in relation to non--dimensional geometrical parameters across all designs showed a consistent trend: as the holes' diameters decrease, the normalized Size of the Widest Bandgap increases. Additionally, analyzing the normalized Centroid of the Widest Bandgap relationship to non--dimensional geometrical parameters led to two main conclusions. The first indicated that designs with lower \(L/C\) values saw an decrease in the normalized Centroid of the Widest Bandgap as \(d/C\) decreased in the Basic--Periodic designs and \(d_{1}/C\) decreased in the Semi--Periodic, Tapered--Diverging, and Tapered--Converging designs with a fixed \(d_{2}/d_{1}\) value, while designs with higher \(L/C\) values observed a slight decrease or constant centroid with increasing \(d/C\) in Basic--Periodic designs and increasing \(d_{1}/C\) in Semi--Periodic, Tapered--Diverging, and Tapered--Converging designs while maintaining a fixed value for \(d_{2}/d_{1}\). It is expected that the optimal designs must have both a larger size for their widest bandgap as well as a lower centroid value for this bandgap to help reduction for the lower frequency range (i.e., 0-800 Hz), which is more vital for noise reduction. These characteristics are generally found in unit cells with smaller ratios of the hole diameter to the unit cell size (\(C\)) (\(\frac{d_{1}}{C}\), \(\frac{d_{2}}{C}\), \(\frac{d_{3}}{d_{1}}\)). This leads to hole sizes that may be too close to the visco--thermal boundary layer thickness. Designs that met this criterion were not taken into consideration by introducing a filtering--out rule in the optimal design filtering step. To evaluate the frequency ranges of the bandgaps and calculate their coverage, a comprehensive efficient filtering approach was employed. Various unit cell sizes ranging from 10 mm to 50 mm were considered and led to 202,505 designs, which were systematically examined, considering the established filtering--out rule and the limitations due the visco--thermal layers. The optimal unit cells in each configuration were identified based on their highest bandgap coverage. The results demonstrate that the optimal Semi--Periodic unit cell with \(C=42.9\) mm, \(\frac{L}{C}=0.3\), \(\frac{d_{1}}{C}=0.1\), \(\frac{d_{2}}{d_{1}}=0.5\) exhibits the highest bandgap coverage.