## 2 Theoretical Modeling of the Combination Structure
\[\omega=2\pi f \tag{9}\]  
\[\rho(\omega)=\rho\bigg{[}1+\bigg{(}3^{2}+\frac{4\omega\rho}{\sigma\phi}\bigg{)} ^{-0.5}-j\frac{\sigma\phi}{\omega\rho}\bigg{(}1+\frac{\omega\rho}{4\sigma\phi} \bigg{)}^{0.5}\bigg{]} \tag{10}\]  
\[K(\omega)=\gamma P_{0}\bigg{[}\gamma-(\gamma-1)\bigg{(}1-N_{u}\bigg{(}j\frac{ 8\omega\rho P_{r}}{\sigma\phi}+N_{u}\bigg{)}^{-1}\bigg{)}\bigg{]}^{-1} \tag{11}\]  
In Equation (9), \(f\) is the sound frequency. In Equations (10) and (11), \(\gamma\) is the specific heat ratio of the air, 1.40; \(P_{0}\) is the static pressure of the air, 1.013*105 Pa; \(N_{u}\) is the Nusselt number, 4.36; \(j\) is symbol of the imaginary number; \(P_{r}\) is the Prandtl number, 0.71 [7; 13; 22; 28]. Meanwhile, in Equations (10) and (11), the static flow resistivity \(\sigma\) and porosity \(\phi\) of the porous metal used in this study were 9.2*103 Pa-s/m and 0.9, respectively, which were obtained by the measurement based on their definitions. The transfer matrix \(M\) of the microperforated panel can be obtained by Equation (12). Here, \(Z_{s}\) is the acoustic impedance rate of the microperforated panel, which consists of the real part \(R\) and the imaginary part \(X\), as shown in Equation (13). The \(R\) and \(X\) can be calculated through Equations (14) and (15), respectively. \[[M]=\left[ \begin{array}{cc}1&Z_{s}\\ 0&1\end{array} \right] \tag{12}\]  
\[Z_{s}=R+jX \tag{13}\]  
\[R=\frac{32(\mu+\upsilon)\rho}{\varepsilon}\frac{t}{d^{2}}k_{r} \tag{14}\]  
\[X=\frac{t\omega\rho}{\varepsilon}k_{m} \tag{15}\]  
In Equations (14) and (15), \(\mu\) is the viscosity coefficient of the air, \(1.506\times 10^{-5}\) m\({}^{2}\)/s; \(\upsilon\) is the temperature conduction coefficient of the metal panel, \(2.0\times 10^{-5}\) m\({}^{2}\)/s; \(\varepsilon\) is the microperforating rate, which can be calculated by Equation (16); \(k_{r}\) is the acoustic resistance constant, which can be obtained by Equation (17); \(k_{m}\) is the acoustic mass constant, which can be derived by Equation (18) [8; 16; 23]. In Equations (17) and (18), \(k\) is the perforated panel constant, which can be calculated through Equation (19). \[\varepsilon=\frac{\pi}{4}\bigg{(}\frac{d}{b}\bigg{)}^{2} \tag{16}\]  
\[k_{r}=\sqrt{1+\frac{k^{2}}{32}}+\frac{\sqrt{2}}{8}k\frac{d}{t} \tag{17}\]  
\[k_{m}=1+\bigg{(}9+\frac{k^{2}}{2}\bigg{)}^{-0.5}+0.85\frac{d}{t} \tag{18}\]  
\[k=\sqrt{\frac{\omega}{\mu+\upsilon}}\frac{d}{2} \tag{19}\]  
The transfer matrices \(S_{1}\) of the cavity backing the porous metal and \(S_{2}\) of the cavity backing the microperforated panel can be calculated by Equations (20) and (21), respectively. \[[S_{1}]=\left[ \begin{array}{cc}\cos\!\left(\omega c^{-1}D_{1}\right)&jpc \sin\!\left(\omega c^{-1}D_{1}\right)\\ \frac{j}{\rho c}\sin\!\left(\omega c^{-1}D_{1}\right)&\cos\!\left(\omega c^{- 1}D_{1}\right)\end{array} \right] \tag{20}\]  
\[[S_{2}]=\left[ \begin{array}{cc}\cos\!\left(\omega c^{-1}D_{2}\right)&jpc \sin\!\left(\omega c^{-1}D_{2}\right)\\ \frac{j}{\rho c}\sin\!\left(\omega c^{-1}D_{2}\right)&\cos\!\left(\omega c^{- 1}D_{2}\right)\end{array} \right] \tag{21}\]  
In this way, the theoretical sound absorption coefficients of the investigated sound absorbers in Figure 1 could be obtained, forming the theoretical basis for further parameter optimization, finite element simulation, and experimental validation.