## 2 Materials and Methods
### Bandgaps Determination through Utilization of FE--Bloch's Theorem Method  
The determination of bandgaps using [PERSON]'s theorem approach with the unit cell as a geometrical domain is a widely used methodology in the field of periodic metamaterials. This is due to [PERSON]'s theorem utilizing the unit cell as the geometrical domain that physically represents the wave propagation characteristics in periodic structures. In this methodology, the wave equation is solved for a single unit cell of the periodic structure, considering the Bloch boundary conditions, which will be referred to as Periodic Boundary Conditions (P.B.Cs) in this paper. This approach assumes that the periodic structure is infinitely periodic, allowing the use of Bloch's Theorem to calculate bandgaps. First the acoustic wave propagation equation is defined. \[\
abla.\left(-\frac{1}{\rho}\
abla P\right)-\frac{1}{c^{2}\rho}\ddot{P}=0 \tag{1}\]  
The equation governing acoustic wave propagation is a partial differential equation that describes the behavior of sound waves, where \(P\) represents the acoustic sound pressure, \(c\) is the speed of sound in the medium, and \(\rho\) is the density of the medium. Also, \(\
abla=\left(\frac{\partial}{\partial x}+\frac{d}{dy}+\frac{\partial}{\partial z}\right)\) is the Laplacian operator, which is simply the gradient. In this study, the focus is on sound wave propagation in an air medium due to the assumption of impedance mismatch between solid materials and air, leading to complete reflection of sound waves at their interface (i.e., solid part walls were assumed to be rigid boundaries). It is essential to note that impedance mismatch occurs when both mediums differ significantly in density and speed of sound. Consequently, propagation of sound waves happens dominantly in air medium. The air properties that were considered for this work were \(\rho=1.2\) kg/m\({}^{3}\) and \(c=343\) m/s. Next, it is necessary to solve the eigenvalue problem of Equation (1) in order to determine the bandgaps. As mentioned earlier in the Introduction section, bandgaps refer to frequencies at which sound waves are prevented from propagating. Relating this definition to the eigenvalue problem, bandgaps correspond to frequencies that are not covered by the eigenfrequencies of the air unit cell. \[\
abla.\left(-\frac{1}{\rho}\
abla P\right)+\frac{\omega^{2}}{c^{2}\rho}P=0 \tag{2}\]  
Equation (2) represents the eigenvalue problem of the acoustic sound wave propagation (i.e., represented in Equation (1)), where \(\omega\) denotes to the eigenfrequencies. For further elaboration, bandgaps will be the frequencies that do not satisfy the eigenvalue  
Figure 5: Design points considered in the parametric study. (**A**) 55 normalized design combinations for the Basic—Periodic unit cell. (**B**) 150 normalized design combinations considered for each of the newly developed unit cells (i.e., Semi—Periodic, Tapered—Diverging, and Tapered—Converging). equation, meaning that the corresponding eigenfrequencies are not present within the band--structure. The eigenvalue problem is solved numerically using Finite Element Method, this leads to Equation (2) to be rewritten as  
\[\omega^{2}MP+KP=0 \tag{3}\]  
Equation (3) represents the finite element equation in relation to Equation (2). In this context, \(M\) refers to the mass matrix and \(K\) represents the stiffness matrix. To determine the bandgaps through solving Equation (3) with a single unit cell of the periodic structure, the Periodic Boundary Conditions (P.B.Cs) are applied. [PERSON]'s theorem enables the accurate representation of sound waves in an infinite periodic structure by considering a single unit cell. Figure 3 illustrates the application of [PERSON]'s theorem, where a unit cell is used as a geometrical domain to represent the wave propagation in the periodic structure.