## 3 Results
### Bandgaps for All Considered Unit Cells  
Based on the previously demonstrated and discussed [PERSON]'s theorem, the benchmarked methodology dependent on unit cell analysis was employed to determine the bandgaps for different types of unit cells. These included the Basic--Periodic unit cells, Semi--Periodic unit cells, Tapered--Diverging unit cells, and Tapered--Converging unit cells. The benchmarked method was used to evaluate 505 normalized design combinations in order to assess their respective bandgaps as both an initial consideration and core component of this parametric study. It is important to emphasize that this study will examine bandgaps in the normalized format. Computing the normalized bandgaps requires understanding of the normalized eigen--value problem of Equation (2). The normalization of the eigen--value problem is done within its spatial domain. For instance, consider an eigenvalue problem of a one--dimensional acoustic sound wave propagation partial differential equation and normalize its single spatial domain relative to the unit cell size (\(C\)). Equation (2) can be rewritten as  
\[-\frac{1}{\rho}\frac{\partial^{2}P}{\partial x^{2}}+\frac{4\pi^{2}f^{2}}{\rho c ^{2}}P=0 \tag{5}\]  
Upon normalization, we obtain  
\[-\frac{1}{\rho}\frac{\partial^{2}P}{\partial x^{*2}}+\frac{4\pi^{2}}{\rho}n^{ 2}P=0 \tag{6}\]  
where  
\[n=\frac{fC}{c} \tag{7}\]  
Equation (5) represents the eigenvalue problem of the one--dimensional acoustic wave propagation partial differential equation. Equation (6) represents the spatial normalized eigen--value problem of a one--dimensional acoustic sound wave propagation, where \(x^{*}=\frac{x}{C}\) is the normalized spatial domain and \(n\) is the normalized frequency. Equation (7) presents the definition of normalized frequency, where \(f\) represents the actual frequency, \(C\) signifies the size of the unit cell, and \(c\) denotes the speed of sound in the medium through which sound wave propagation occurs (i.e., air medium). The normalized frequency is  
Figure 7: Benchmarking the bandgaps’ unit cell-dependent methodology through re-generating the results for the documented 1D acoustic metamaterial made up from Basic-Periodic unit cell with the geometrical parameters of \(C=30\) mm, \(\frac{L}{C}=0.2\), and \(\frac{C}{C}=0.5\) investigated by earlier efforts [28]. (**A**) Bandgap-structure. (**B**) The input signal. (**C**) Acoustic pressure response. the same for unit cells with the same non--dimensional geometrical parameters but with different unit cell sizes. Further elaboration on the reason behind storing the normalized frequency will be provided in the subsequent section. Looking at Figures 8-11, these figures demonstrate the bandgap--structures of the considered normalized design combinations in each considered unit cell type (i.e., Basic--Periodic unit cells, Semi--Periodic unit cells, Tapered--Diverging unit cells, and Tapered--Converging unit cells), where each figure corresponds to one unit cell type. Figure 11: Bandgaps for the 150 considered Tapered—Converging unit cell normalized design combinations. **(A)**\(\frac{d_{2}}{d_{1}}=0.5\). **(B)**\(\frac{d_{2}}{d_{1}}=0.7\). **(C)**\(\frac{d_{2}}{d_{1}}=0.9\). Figure 8 illustrates the bandgap--structures of the 55 different normalized design combinations of Basic--Periodic unit cell designs. Each subplot represents normalized design combinations with varying \(d/C\) ratios but the same \(L/C\) ratio. Therefore, this plot consists of five subplots corresponding to different \(L/C\) ratios, ranging from 0.1 to 0.3 in increments of 0.05, as described in this study's methodology section. The x--axis in each subplot represents the filtered \(d/C\) ratio based on the design criterion explained earlier. Meanwhile, the y--axis corresponds to the normalized frequencies. Upon examining the subplots, it becomes apparent that there is a noteworthy increase in bandgap widths as the diameter of the cylindrical hole decreases. This observation aligns with findings presented in previous research [28].