## 4 Discussion
Upon examining this relationship, it becomes evident that for an air medium with fixed kinematic viscosity \(\mu\) and density \(\rho\), the vorticity boundary layer is contingent on the propagation frequency. Furthermore, there exists an inversely proportional correlation between these two variables--as the propagation frequency decreases, an increase in vorticity boundary layer thickness is observed. \[l_{ent}=\frac{l_{vor}}{\sqrt{P_{r}}} \tag{9}\]  
The thickness of the entropy boundary layer, introduced by [PERSON] [42], is represented by Equation (9). This boundary layer depends on the vorticity boundary layer thickness (\(l_{vor}\)) and the Prandtl number of air (\(P_{r}\)). The relationship between the entropy boundary layer thickness and the propagation frequency is indirectly determined through its dependence on the corresponding vorticity boundary layer thickness. It should be noted that this work focuses solely on accounting for the entropy boundary layer thickness during the filtering process, as it exceeds its corresponding vorticity counterpart in size. The propagation frequency (\(\omega\)) used to determine the thickness of the entropy boundary layer (\(l_{ent}\)) in a unit cell with bandgaps pertains to the lower bound of the first acoustic bandgap (i.e., the widest bandgap). The rationale behind choosing this lower limit as the propagation frequency (\(\omega\)) stems from the understanding that there exists an inverse relationship between propagation frequency and entropy boundary layer thickness. This implies that operating at the lower bound frequency results (i.e., the lower limit frequency in which the metamaterial will be operated) in the maximum thickness for the entropy boundary layer could be attained within metamaterial structures. \[f_{l}=\frac{c}{C}C.O.W.B\left[1-\left(\frac{S.O.W.B}{2C.O.W.B}\right)\right] \tag{10}\]  
Equation (10) represents the frequency that corresponds to the lower bound of the first acoustic bandgap in Hz. This equation was written in such a way that relates the lower bound of the first acoustic bandgap (i.e., the widest bandgap in all the 505 normalized design combinations) to its corresponding bandgaps' key features (i.e., the normalized size of the widest bandgap (\(S.O.W.B\)) and the normalized centroid of the widest bandgap  
Figure 12: The proposed filtering approach to determine the optimal unit cell design combination for attenuating most of the audible frequency range (i.e., 20 Hz to 20 kHz). (\(C.O.W.B\)); see Supplementary Materials). After formulating this equation, the entropy boundary layer (\(l_{ent}\)) equation was re--written in a way that incorporates Equation (10). \[l_{ent}=\sqrt{\frac{a}{f_{l}}}\;;\;a=\frac{\mu}{\pi\rho P_{r}} \tag{11}\]  
Equation (11) presents the modified form of the entropy boundary layer, incorporating the minimum frequency in Hz for the first acoustic bandgap. The constant variable \(a\) represents properties specific to air medium. A filtering criterion is then applied to exclude unit cells with design combinations where either the diameter of a uniform cross--sectional hole in Basic--Periodic unit cell configurations (\(d\)) or the smallest diameter of a non--uniform cross--sectional hole in Semi--Periodic, Tapered--Diverging, and Tapered--Converging unit cell configurations (\(d_{2}\)) is less than 20 times its associated entropy boundary layer thickness (\(l_{ent}\)). Fourth, the data corresponding to the filtered bandgaps for unit cells that satisfy the predetermined design criteria were analyzed in order to assess their bandgap coverage. Subsequently, an optimal unit cell design was found for each configuration (i.e., Basic--Periodic, Semi--Periodic, Tapered--Diverging, and Tapered--Converging) based on the evaluation of bandgap coverage. The optimal combinations for unit cell design in each configuration were determined using the suggested filtering approach.