## 2 Methodology
### Mathematical Model of the MPP+PAM  
The acoustic absorption mechanism of a thin-plate acoustic metamaterial is primarily based on the principle of local resonance, serving to achieve the total absorption of a certain frequency of acoustic waves. In this study, the thin-plate acoustic metamaterial was composed of a rectangular aluminum thin plate and a small cylindrical iron block, whose individual metamaterial structure is depicted in Figure 5. The following derivation of the absorption coefficient of this composite structure is based on the modal superposition method. The transfer matrix of the PAM is:  
\[N= \begin{bmatrix}1&Z_{PAM}\\ 0&1\end{bmatrix}  \tag{15}\]  
where \(Z_{\text{PAM}}\) is the impedance rate of the PAM. A mass block with the dimensional diameter \(R_{\text{mass}}\) is attached to the thin plate. In the Cartesian coordinate system, the coordinates of the center of the mass block circle are assumed to be \((x_{0},y_{0})\), and the equation of the vibration motion of the thin plate with the mass block attached is:  
\[\rho_{s}\frac{\partial^{2}w}{\partial t^{2}}+\rho_{mass}H\frac{\partial^{2}w}{ \partial t^{2}}+D\
abla^{4}w=2p_{in}e^{i\omega t}-2\rho_{0}c_{0}\frac{\partial w }{\partial t} \tag{16}\]  
where \(\rho_{mass}\) is the surface density of the mass block, \(p_{in}\) is the amplitude of the incident acoustic wave, \(H\) is the characteristic impedance of air, and \(\rho_{0}c_{0}\) is the combination of four step functions characterizing the impact of the mass block equation. For the position of the mass block to take 1, and the other positions of the thin plate to take 0, the expression is:  
\[H=\left[H(x-x_{0})-H(x-x_{0}-R/2)\right]\cdot\left[H(y-y_{0})-H(y-y_{0}-R/2)\right] \tag{17}\]  
The vibration displacement of a thin plate can be expressed as:  
\[w(x,y,t)=\sum_{m=1}^{M}\phi_{m}(x,y)q_{m}(t) \tag{18}\]  
Figure 5: Schematic of the thin-plate acoustic metamaterial. where \(m\) is the modal number, \(\phi_{m}\) (\(x\), \(y\)) is the modal type function of each order for a thin plate under fixed constraints, and \(q_{m}\) (\(t\)) is the generalized coordinate corresponding to the modal type function of each order. Here,  
\[\phi_{m}(x,y)=2\sin(k_{x}x)\sin(k_{y}y) \tag{19}\]  
\[q_{m}(t)=\widetilde{q}_{m}e^{j\omega t} \tag{20}\]  
Substituting Equation (20) into Equation (18), multiplying by the orthogonal modal function \(\phi_{n}\) (\(x\), \(y\)), and integrating over the surface of the thin plate, we obtain:  
\[-\omega^{2}M_{m}\widetilde{q}_{m}-\omega^{2}\sum\limits_{n=1}^{N}Q_{m,n} \widetilde{q}_{m}+K_{m}\widetilde{q}_{m}+E_{m}\widetilde{q}_{m}=2p_{in}H_{m}- j\omega D_{m}\widetilde{q}_{m} \tag{21}\]  
Equation (21) can be expressed as a matrix using the expression:  
\[-\omega^{2}\big{[}M_{p}\big{]}\widetilde{q}-\omega^{2}[Q]\widetilde{q}+[K] \widetilde{q}+[E]\widetilde{q}+j\omega[D]\widetilde{q}=2p_{in}L \tag{22}\]  
where \([M_{p}]\) is the sheet mass matrix, \([Q]\) is the additional mass matrix, \([E]\) is the damping matrix with a damping factor of 0.15, \([K]\) is the sheet stiffness matrix, \([D]\) is the flexural stiffness matrix, and \(L\) is the nodal displacement. The modal migration matrix \([Y]\) can be expressed as:  
\[Y=\frac{j\omega}{-\omega^{2}\big{[}M_{p}\big{]}-\omega^{2}[Q]+[K]+[D]+j\omega [E]} \tag{23}\]  
The amplitude of the structural vibration velocity of the PAM at the vertical incidence of the acoustic wave is:  
\[V=Y\times(2p_{in}L_{m}) \tag{24}\]  
The average structural vibration velocity of the PAM is:  
\[H=\left[H(x-x_{0})-H(x-x_{0}-R/2)\right]\cdot\left[H(y-y_{0})-H(y-y_{0}-R/2)\right] \tag{17}\]0  
The impedance of the PAM is the ratio of the sound pressure difference between the two sides of the structure to the average vibration velocity of the structure, which can be expressed as:  
\[H=\left[H(x-x_{0})-H(x-x_{0}-R/2)\right]\cdot\left[H(y-y_{0})-H(y-y_{0}-R/2)\right] \tag{17}\]1  
The transfer matrix of MPP+PAM is:  
\[\Pi_{c}=MC_{1}NC_{2}= \begin{bmatrix}\Pi_{11}&\Pi_{12}\\ \Pi_{21}&\Pi_{22}\end{bmatrix}