## 3 Results
### Bandgaps for All Considered Unit Cells
Additionally, it should be noted that the widest bandgap occurs at the first acoustic bandgap. The bandgap--structures of the 150 considered normalized design combinations of Semi--Periodic unit cell designs is shown in Figure 9. Each subplot in each column represents normalized design combinations with varying \(d_{1}/C\) ratios but the same \(L/C\) ratios. Additionally, the morphology of this type of unit cell is influenced by an additional non--dimensional geometrical parameter, specifically the \(d_{2}/d_{1}\) ratio. Therefore, as analyzed in the figure, each column corresponds to subplots of normalized design combinations with different \(d/C\) ratios and \(L/C\) ratios but with the same \(d_{2}/d_{1}\) ratio. Similar to Figure 8, the \(x\)--axis shows filtered \(d_{1}/C\) ratios based on the design criterion, while the y--axis displays the normalized frequency range. After examining the subplots, it can be observed that the behavior of the Semi--Periodic unit cells' bandgap is similar to that of the Basic--Periodic unit cells. As the cylindrical holes decrease in size, there is a corresponding increase in bandgap widths. However, there is a narrow bandgap present in the first acoustic bandgap and at a lower normalized frequency range. This narrow bandgap, referred to as the first acoustic bandgap, has limited effectiveness in attenuating the target normalized frequency range for two reasons. One reason is that a small unit cell size is required for this narrow bandgap to be effective, as analyzed from the normalized frequency definition, and this would be advantageous in terms of fabrication. However, this can cause the occurrence of higher frequency bandgaps beyond the audible range, rendering them irrelevant for the desired application. The other reason is that having reliable unit cell sizes relative to the exhibited normalized bandgaps ensures significant behavior within the audible frequency range (i.e., 20 Hz to 20 kHz). In this case, the narrowness of the bandgap prevents attenuation in its corresponding frequency range. Hence, this study will focus on the normalized bandgaps beyond the first acoustic bandgap. This selection will be discussed in detail in the upcoming section, which outlines the optimal choice of a Semi--Periodic unit cell type for constructing a 1D periodic acoustic metamaterial capable of attenuating a wide range of audible frequencies (i.e., 20 Hz to 20 kHz). The bandgap--structures for the considered 150 normalized design combinations in the Tapered--Diverging unit cell and the Tapered--Converging unit cell are shown in Figures 10 and 11, respectively. These figures follow a similar organization as Figure 9, which depicted the bandgapstructure for the Semi--Periodic unit cell configuration. It is important to note that both Tapered--Diverging and Tapered--Converging designs are controlled by non--dimensional geometrical parameters that also regulate the morphology of the Semi--Periodic unit cell type. Each subplot represents different normalized design combinations with varying \(d_{1}/C\) ratios while maintaining constant \(L/C\) ratios, and each column displays subplots with varying \(d_{1}/C\) ratios alongside varying \(L/C\) ratios but with a shared \(d_{2}/d_{1}\) ratio. The \(x\)--axis reflects filtered \(d_{1}/C\) ratios based on design criteria, whereas the y--axis represents normalized frequency ranges. Upon examination of both Figures 10 and 11, it is apparent that the Tapered--Diverging unit cells and Tapered--Converging unit cells demonstrate a wider bandgap at lower \(d_{1}/C\) and \(d_{2}/d_{1}\) ratios. This behavior aligns with the bandgap characteristics exhibited by the analyzed Semi--Periodic unit cells. The first bandgap corresponds to the widest bandgap observed, which is consistent with the behavior exhibited by the Basic--Periodic unit cells.