## Materials and Methods  
### Materials and Models  
#### 2.1.1 Polyurethane Foam and Its Theoretical Sound Absorption Model  
There were porous structures in the polyurethane foam, as shown in Figure 1, which could realize sound absorption by the sticky effect and heat conduction [15; 16; 17]. When the incident sound wave diffused in the porous structure, there existed velocity gradations among different particles, which resulted in the relative motion between the vibrant air and skeleton structures. Sticky effect of the air could generate the friction force and viscous resistance, which blocked motion of the particles. The friction and viscous effects could transform the partial sound energy to the heat energy, and the heat energy was consumed in the conduction process. By this method, the sound energy was partially reduced and the sound absorption effect was realized. Figure 1: Surface morphologies of the polyurethane foam observed by scanning electron microscope. (**a**) With the low magnification; (**b**) With the high magnification. According to the Johnson-Champoux-Allard model [15; 16; 17], calculation of sound absorption coefficient \(\alpha_{pf}\) of the polyurethane foam was obtained by the Equation (1). For the convenience of following the calculation of the sound absorption coefficients for the composite structures, the transfer matrix method was utilized in this research [21; 22; 23], and that for the polyurethane foam \(T_{pf}\) was calculated by Equation (2). In the Equation (1), Re( ) and Im( ) represented the real part and imaginary part of one complex number respectively; \(\rho_{0}\) was the density of the air with the room temperature and atmospheric pressure, 1.21 kg/m\({}^{3}\); \(c_{0}\) was the acoustic velocity in the air with the room temperature and atmospheric pressure, 340 m/s; \(T_{pf11}\) and \(T_{pf21}\) were the two components of transfer matrix \(T_{pf}\) for the polyurethane foam. In Equation (2), \(k_{pf}\) was the number of the wave in the polyurethane foam, which was calculated by Equation (3); \(d_{pf}\) was its thickness, which could be measured by the vernier caliper; \(Z_{pf}\) is its characteristic impedance, which could be derived by the Equation (4); \(j\) was the symbol of the imaginary number, \(j=\sqrt{-1}\)[15; 16; 17]. \[\alpha_{pf}=\frac{4\cdot\text{Re}\Big{(}\frac{T_{pf11}}{T_{pf21}}\cdot\frac{1} {\rho_{0}c_{0}}\Big{)}}{\Big{[}1+\text{Re}\Big{(}\frac{T_{pf11}}{T_{pf21}} \cdot\frac{1}{\rho_{0}c_{0}}\Big{)}\Big{]}^{2}+\Big{[}\text{Im}\Big{(}\frac{ T_{pf11}}{T_{pf21}}\cdot\frac{1}{\rho_{0}c_{0}}\Big{)}\Big{]}^{2}} \tag{1}\]  
\[T_{pf}=\left[ \begin{array}{cc}T_{pf11}&T_{pf12}\\ T_{pf21}&T_{pf22}\end{array} \right]=\left[ \begin{array}{cc}\cos\Big{(}k_{pf }d_{pf}\Big{)}&jZ_{pf}\sin\Big{(}k_{pf}d_{pf}\Big{)}\\ jZ_{pf}^{-1}\sin\Big{(}k_{pf}d_{pf}\Big{)}&\cos\Big{(}k_{pf}d_{pf}\Big{)} \end{array} \right] \tag{2}\]  
\[k_{pf}=\omega\sqrt{\frac{\rho(\omega)}{K(\omega)}} \tag{3}\]  
\[Z_{pf}=\sqrt{\rho(\omega)K(\omega)} \tag{4}\]  
In Equations (3) and (4), \(\omega\) was the angular frequency of the sound wave, which was gained by Equation (5), and \(f\) was sound frequency and \(\pi\) was the circular constant; \(\rho(\omega)\) and \(K(\omega)\) were complex effective density and complex effective bulk modulus of the polyurethane foam, which were obtained by Equations (6) and (7) respectively [15; 16; 17]. In Equations (6) and (7), \(\rho_{0}\) was still density of the air with the room temperature and the atmospheric pressure, 1.21 kg/m\({}^{3}\); \(\sigma\) and \(\phi\) were static flow resistivity and porosity of the polyurethane foam respectively; \(\gamma\) was the specific heat ratio of the air with the room temperature and the atmospheric pressure, 1.40; \(P_{0}\) was the static pressure of the air with the room temperature and atmospheric pressure, 1.013\(\times\)10\({}^{5}\) Pa; \(N_{u}\) was the Nusselt number, 4.36; \(P_{r}\) was the Prandtl number, 0.71; \(j\) was the still symbol of the imaginary number, \(j=\sqrt{-1}\)[15; 16; 17].