## 5 Conclusions
To further validate the results from the bandgap analysis using [PERSON]--[PERSON]'s theorem and filtering step, the acoustic pressure responses for all the optimal designs with the suggested actuation scenario were examined. The attenuated frequency regions observed in the acoustic pressure response were in perfect agreement with their respective bandgap frequency ranges from the [PERSON]--[PERSON]'s theorem, further proving the effectiveness of the optimal designs for acoustics attenuation of the desired frequency ranges. The results were further ranked for their effectiveness in sound pressure level reduction; this aligned with rankings determined from the optimal design filtering step, which relied on unit cell--dependent bandgap data. The present study has provided significant insight into the use of metamaterials for acoustics reduction within the audible frequency range. Through systematic exploration of numerous unit cell configurations, optimal metamaterial designs for noise attenuation have been identified. These designs can be effectively employed in the fabrication of acoustics filters for various applications such as the automotive and aviation industries. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14167272/s1](https://www.mdpi.com/article/10.3390/app14167272/s1)]([https://www.mdpi.com/article/10.3390/app14167272/s1](https://www.mdpi.com/article/10.3390/app14167272/s1)), Figure S1: The normalized Size of the Widest Bandgap (S.O.W.B) and the normalized Centroid of the Widest Bandgap (C.O.W.B) for the considered designs combinations sharing Basic--Periodic unit cell type; Figure S2: The normalized Size of the Widest Bandgap (S.O.W.B) and the normalized Centroid of the Widest Bandgap (C.O.W.B) for the considered designs combinations in the geometrically modified unit cell types. A) Semi--Periodic. B) Tapered--Diverging. C) Tapered--Converging; Figure S3: The acoustic pressure of the other considered sensing locations (i.e., Y, Z, Y\({}^{\prime}\), Z\({}^{\prime}\)) for the 1D acoustic metamaterial made from repeated optimal Basic--Periodic unit cell; Figure S4: The acoustic pressure of the other considered sensing locations (i.e., Y, Z, Y\({}^{\prime}\), Z\({}^{\prime}\)) for the 1D acoustic metamaterial made from repeated optimal Semi--Periodic unit cell; Figure S5: The acoustic pressure of the other considered sensing locations (i.e., Y, Z, Y\({}^{\prime}\), Z\({}^{\prime}\)) for the 1D acoustic metamaterial made from repeated optimal Tapered--Diverging unit cell; Figure S6: The acoustic pressure of the other considered sensing locations (i.e., Y, Z, Y\({}^{\prime}\), Z\({}^{\prime}\)) for the 1D acoustic metamaterial made from repeated optimal Tapered--Converging unit cell; Table S1: Unit cell size to wavelength ratio for the considered designs in this comparative study. Conceptualization, M.S. and A.S.; methodology, M.S., M.O. and A.S.; software, M.S. ; validation, M.S. ; formal analysis, M.S. ; investigation, M.S. ; resources, A.S.; data curation, M.S. ; writing--original, M.S. ; writing--review, A.S., M.O. and N.M.; visualization, M.S. and A.S.; supervision, A.S. and N.M.; project administration, A.S.; funding acquisition, A.S. All authors have read and agreed to the published version of the manuscript. This research received funding from both Natural Sciences and Engineering Research Council of Canada (CRDF) 524018-18 and RGPIN-2022-03338) as well as the Government of Canada through the Federal Economic Development Agency for Southern Ontario (FedDev Ontario) in the Waterloo Institute for Sustainable Aeronautics at the University of Waterloo (WISA 53072-10071). Not applicable. Data Availability Statement: Data will be made available on request. The authors declare no conflicts of interest.