## 2 Theoretical Modeling of the Combination Structure  
Two combination structures made of porous metal and a microperforated panel were investigated, as shown in Figure 1a,b. One consisted of a porous metal + cavity + microperforated panel + cavity, and the other consisted of a microperforated panel + cavity + porous metal + cavity. Two traditional sound absorbers consisting of a porous metal + cavity and a microperforated panel + cavity, as shown in Figure 1c,d, were also investigated as a contrast. The structural parameters of the porous metal were its thickness \(D\) and the corresponding cavity length \(D_{1}\), and those of the microperforated panel were the thickness of the panel \(t\), the diameter of the microhole \(d\), the distance of the neighboring microholes \(b\) (distribution of the microholes was in a square arrangement), and the corresponding cavity length \(D_{2}\), which were consistent with the labels of these structural parameters in Figure 1.  
Theoretical models of the sound absorption coefficients of these sound absorbing structures were constructed through the transfer matrix method [21], as shown in Equation (1). Here, \(\alpha\) is the sound absorption coefficient; \(TT_{11}\) and \(TT_{21}\) are two components of the total transfer matrix of the sound absorber, which can be calculated through Equations (2)-(5) for the corresponding sound absorbing  
Figure 1: Schematic diagrams of the investigated sound absorbing structures. (**a**) Porous metal + cavity + microperforated panel + cavity; (**b**) Microperforated panel + cavity + porous metal + cavity; (**c**) Porous metal + cavity; (**d**) Microperforated panel + cavity. structures in Figure 1a-d; \(\rho\) is the density of the air, 1.21 kg/m\({}^{3}\); \(c\) is the acoustic velocity in air, 340 m/s; Re( ) and Im( ) represent the real part and imaginary part of one complex number, respectively. \[\alpha=4\text{Re}\bigg{(}\frac{TT_{11}}{TT_{21}}.\frac{1}{\rho c}\bigg{)} \bigg{\{}\bigg{[}1+\text{Re}\bigg{(}\frac{TT_{11}}{TT_{21}}.\frac{1}{\rho c} \bigg{)}\bigg{]}^{2}+\bigg{[}\text{Im}\bigg{(}\frac{TT_{11}}{TT_{21}}.\frac{1} {\rho c}\bigg{)}\bigg{]}^{2}\bigg{\}}^{-1} \tag{1}\]  
\[TT_{a}=\left[ \begin{array}{cc}TT_{11}&TT_{12}\\ TT_{21}&TT_{22}\end{array} \right]=[P][S_{1}][M][S_{2}] \tag{2}\]  
\[TT_{b}=\left[ \begin{array}{cc}TT_{11}&TT_{12}\\ TT_{21}&TT_{22}\end{array} \right]=[M][S_{2}][P][S_{1}] \tag{3}\]  
\[TT_{c}=\left[ \begin{array}{cc}TT_{11}&TT_{12}\\ TT_{21}&TT_{22}\end{array} \right]=[P][S_{1}] \tag{4}\]  
\[TT_{d}=\left[ \begin{array}{cc}TT_{11}&TT_{12}\\ TT_{21}&TT_{22}\end{array} \right]=[M][S_{2}] \tag{5}\]  
In Equations (2)-(5), \(P\), \(S_{1}\), \(M\), and \(S_{2}\) represent the transfer matrix of the porous metal, of the cavity backing the porous metal, of the microperforated panel, and of the cavity backing the microperforated panel, respectively. Transfer matrix \(P\) of the porous metal can be calculated by Equation (6) based on the Johnson--Champoux--Allard model [22]. Here, \(k_{p}\) is the number of waves in the porous material, which can be obtained by Equation (7); \(Z_{p}\) is the characteristic impedance of the porous material, which can be obtained by Equation (8). \[[P]=\left[ \begin{array}{cc}\cos\Big{(}k_{p}D\Big{)}&jZ_{p}\sin\Big{(}k_{p} D\Big{)}\\ jZ_{p}^{-1}\sin\Big{(}k_{p}D\Big{)}&\cos\Big{(}k_{p}D\Big{)}\end{array} \right] \tag{6}\]  
\[k_{p}=\omega\sqrt{\frac{\rho(\omega)}{K(\omega)}} \tag{7}\]  
\[Z_{p}=\sqrt{\rho(\omega)K(\omega)} \tag{8}\]  
In Equations (7) and (8), \(\omega\) is the angular frequency, which can be obtained by Equation (9); \(\rho(\omega)\) is the complex effective density, which can be obtained through Equation (10); \(K(\omega)\) is the complex effective bulk modulus, which can be calculated with Equation (11).