## 2 Materials and Methods  
### Design Framework  
The initial phase of this study involved the identification of a pre--existing 1D periodic acoustic metamaterial design. Past research primarily focused on examining the influence of different lattice shapes on bandgap sensitivity through some design combinations (i.e., within the range of 9 design combinations), which was enough to prove the bandgaps' tunability through topological changes [28]. Since this current study's main objective is to find optimal designs, an extensive parametric analysis for the chosen 1D periodic acoustic metamaterial designs (i.e., including pre--existing designs proposed earlier by [PERSON] et al. [28] and the newly developed designs), will be conducted to furtherly and thoroughly investigate their sensitivity to changes in lattice geometry. It should be emphasized that the geometrical variations will specifically occur within the porosity of the acoustic metamaterial designs. In other words, modifications will be made to enhance the air geometrical part since sound waves predominantly propagate through it. For further clarifications, Figure 1 illustrates the solid part of the metamaterial (i.e., the actual shape of the metamaterial). However, Figures 2 and 3 demonstrate the air part of the metamaterial (i.e., the porous part). This concept will be further elucidated in subsequent sections detailing the methodological approach through bandgap determination using FE--[PERSON]'s theorem methodology sub--section. The present study introduces an extensive design framework by utilizing an existing 1D periodic acoustic metamaterial design as the base lattice proposed by [PERSON] et al. [28]. This base lattice is then modified geometrically to create a new class of 1D periodic acoustic metamaterials, depicted in Figure 4. The figure illustrates the various steps involved in the extensive design framework. In this study, the existing 1D periodic acoustic metamaterial air unit cell was selected as the base structure. The air unit cell for the base structure will be referred to as Basic--Periodic air unit cell in this paper. To explore different design possibilities, three distinct geometrical modifications were applied to the Basic--Periodic air unit cell. The first modification resulted in a new class of air unit cells called Semi--Periodic air unit cells. Similarly, applying the second proposed geometric modification led to a new class named  
Figure 1: Illustration of the solid part of the acoustic metamaterials made from a repeated Semi—Periodic unit cell with the corresponding cross—sectional view. (**A**) Metamaterial solid part. (**B**) Cross—sectional view. Tapered--Diverging air unit cells, and the third proposed modification resulted in another new class known as Tapered--Converging air unit cells. The proposed modifications in Figure 4 primarily involve changing the uniform cross--sectional holes to non--uniform cross--sectional holes. It is worth mentioning that starting from a pre--existing structure and then applying geometrical modifications on its components to create new structures is a common systematic practice that has been utilized in creating new designs of periodic metamaterials with enhanced mechanical bandgaps [17; 18; 19; 20; 40]. Subsequently, the bandgaps of different unit cell configurations (i.e., Basic--Periodic air unit cells, Semi--Periodic air unit cells, Tapered--Diverging air unit cells, and Tapered--Converging air unit cells) were determined using FE--[PERSON]'s theorem method. The bandgap data obtained from the bandgap determination step were utilized to filter and select optimal designs among various unit cell combinations, including Basic--Periodic air unit cells, Semi--Periodic air unit cells, Tapered--Diverging air unit cells, and Tapered--Converging air unit cells. The filtering process aimed to identify designs that cover a wide range of audible frequencies (i.e., 20 Hz to 20 kHz).