## 2 Theoretical Analyses  
### Impedance Model of Melamine Foam  
Following the well-known JCAL model proposed by [PERSON] et al. (20) and [PERSON] et al. (2011), the equivalent density \(\rho_{eq}(\omega)\) and modulus \(K_{eq}(\omega)\) of the porous fluid are  
\[\rho_{eq}(\omega)=\frac{\alpha_{\infty}\rho_{0}}{\phi}\Bigg{[}1-j\frac{\sigma \phi}{\omega\rho_{0}\alpha_{\infty}}\sqrt{1+j\frac{4\alpha_{\infty}{}^{2}\eta \rho_{0}\omega}{\alpha^{2}\Lambda^{2}\phi^{2}}}\Bigg{]} \tag{1}\]  
\[K_{eq}(\omega)=\frac{\gamma P_{0}/\phi}{\gamma-(\gamma-1)\Bigg{[}1-j\frac{\phi \eta}{k_{0}N_{pr}\rho_{0}\omega}\sqrt{1+j\frac{4k_{0}{}^{2}N_{pr}\rho_{0} \omega}{\eta\Lambda^{2}\phi^{2}}}\Bigg{]}^{-1}} \tag{2}\]  
where \(\omega\) is the angular frequency, \(\rho_{0}\) is the density of the air, \(\eta\) is the viscosity of the air, \(\sigma\) is the airflow resistivity, \(P_{0}\) is the mean ambient pressure, \(\phi\) is the porosity of the material considered, \(N_{pr}\) is the Prandtl number of the air, and \(\gamma=C_{p}/C_{v}\) is the specific heat ratio, in which \(C_{p}\) and \(C_{v}\) are the specific heat capacities at constant pressure and at constant volume respectively. The JCAL model involves six characteristic parameters: the static viscous permeability \(k_{0}\), the porosity \(\phi\), the tortuosity \(\alpha_{\infty}\), the viscous characteristic length \(\Lambda\), the static thermal permeability \(k_{0}{}^{\prime}\) and the thermal characteristic length \(\Lambda{}^{\prime}\).  
According to formula (1) and formula (2), the wave number \(k_{s}\) and characteristic impedance \(Z_{s}\) of the equivalent fluid medium is  
\[Z_{s}(\omega)=\sqrt{\rho_{eq}(\omega)K_{eq}(\omega)},\text{ and} \tag{3}\]  
\[k_{s}(\omega)=\omega\sqrt{\rho_{eq}(\omega)/K_{eq}(\omega)} \tag{4}\]  
For the melamine foam with a thickness of \(H\), the surface impedance at \(x=H\) of the sample backed by a rigid wall (see Figure 1) is given by  
\[Z_{PM}(\omega)=\frac{-jZ_{s}(\omega)\cot(k_{s}(\omega)H)}{\phi} \tag{5}\]  
Figure 1: Illustration of HPSET and melamine foam installed at an impedance tube. The thickness of the foam is \(H\), and the inner diameter of the HPSET is \(2R\). The diameters and the maximum length of the extended tubes are \(d_{0}\) and \(t\). All extended tubes inlets are shaped to follow the sphere curvature. The cross-sectional areas of HPSET and impedance tube are \(S\) and \(S_{0}\).  
In this paper, an experimental characterization approach, which requires direct measurements of \(\phi\) and \(k_{0}\) and an impedance tube technique [22; 23; 24], is adopted to characterize the transport parameters of melamine foam [25; 26]. Parameters of the melamine foam are listed in Table 1. The porosity \(\phi\) is measured by a porosimeter, the air-flow resistivity \(\sigma\) is directly measured by resistivimeter, and the remaining four transport parameters \((\alpha_{\infty},\Lambda,\Lambda^{\prime},k_{0}{{}^{\prime}})\) are determined by the inverse characterization techniques using a three-microphone impedance tube [24]. We use a commercially available software RokCell (v3.0, MATELYS, Lyon, France) [27] to automatically obtain the estimations of the parameters.