## 3 Results  
### Benchmarking the Bandgaps through Using [PERSON]'s Theorem Single Unit Cell--Dependent Methodology  
The main emphasis of this study lies in the examination of bandgaps, which serve as indicators for frequency ranges where 1D acoustic metamaterials effectively attenuate sound. In order to achieve accurate and reliable results, it is crucial to benchmark a methodology that determines bandgaps in the considered 1D acoustic metamaterials. Specifically, we employ [PERSON]'s theorem single unit cell--dependent methodology (i.e., one of the components in the extensive design framework).  
This study replicates the bandgap findings for a 1D acoustic metamaterial composed of a Basic--Periodic unit cell with geometrical parameters of \(C=30\) mm, \(\frac{L}{C}=0.2\), and \(\frac{d}{C}=0.5\), as previously investigated [28]. Figure 7A showcases the replicated bandgap--structure results for this documented unit cell. The bandgaps in this study are represented using a series of plots known as bandgap--structures. These structures display the black regions indicating the presence of bandgaps. Traditionally, band--structure plots were used to depict eigenfrequencies, with frequencies that did not fall within these bands being referred to as bandgaps. However, this research proposes using the bandgap--structure plots instead of the conventional band--structure because attenuation goals for vibrations and acoustics rely on identifying and understanding these essential gaps [17]. Additionally, Figure 7B displays the input signal of the external sound source applied, while Figure 7C displays the acoustic pressure response for a 1D acoustic metamaterial made from seven corresponding repeated Basic--Periodic unit cells within a frequency range of 20 Hz to 12 kHz, while applying rigid boundary conditions on the faces opposite to the tessellation axis (i.e., the \(x\) axis). The acoustic pressure response was calculated using modal analysis according to the selected methodology in this study (see Supplementary Materials). The external sound source (i.e., impulse signal sound source with \(P=1\) Pa, \(SPL=94\) dB applied across the calculated frequency range (20 Hz to 12 kHz), and represented in Figure 7B) was applied at one end of the metamaterial and the acoustic sound pressure level was determined at the other end of the metamaterial. By employing [PERSON]'s theorem single unit-- cell--dependent methodology, these re--generated results were obtained with an aim to validate and align with those originally generated by [PERSON] et al. [28], confirming their accuracy and consistency as depicted in Figure 7A. Figure 7C demonstrates the presence of bandgap frequency regions by exhibiting significant attenuation within these specific ranges. It is important to acknowledge that by applying rigid boundary conditions or radiation boundary conditions to the faces opposite of the tessellation axis, similar attenuation frequency regions (i.e., bandgaps) can be achieved. This phenomenon has been studied and documented previously [28]. The attenuation regions of these metamaterials depend on their periodicity. Moreover, the bandgaps are influenced by the Periodic Boundary Conditions (PBCs), which are consistent for both a 1D periodic acoustic metamaterial with reflection boundary conditions and one with radiation boundary conditions applied to its opposing tessellated axis faces.  
The bandgaps in various 1D acoustic metamaterials, including the Basic--Periodic unit cells, Semi--Periodic unit cells, Tapered Diverging unit cells, and Tapered Converging unit cells, will be determined using the benchmarked [PERSON]'s theorem methodology. A total of 505 normalized design combinations will be analyzed.