## Materials and Methods
### Materials and Models
#### 2.1.2 Composite Sound Absorbing Structure and Its Theoretical Sound Absorption Model  
It was well known that the microperforated panel could achieve an outstanding low-frequency sound performance [18; 19; 20]. Thus, in order to improve low-frequency sound absorption properties of the polyurethane foam, a prepositive microperforated polymethyl methacrylate panel was fixed in front, which formed the composite sound-absorbing structure, and its sound absorption coefficient could be calculated through the transfer matrix method [21; 22; 23].  
The transfer matrix \(T_{mpma}\) of the prepositive microperforated polymethyl methacrylate panel was derived according to [PERSON]'s theory [18; 19; 20], as shown in the Equation (8). Here, \(Z_{s}\) was acoustic impedance, as shown in Equation (9), and its real part and imaginary part were symboled as \(R\) and \(X\) respectively; \(j\) is the still symbol of the imaginary number, \(j=\sqrt{-1}\). Moreover, \(R\) and \(X\) could be calculated by Equations (10) and (11) respectively [18; 19; 20].  
\[T_{mpma}=\left[\begin{array}{cc}1&Z_{s}\\ 0&1\end{array}\right] \tag{8}\]  
\[Z_{s}=R+jX \tag{9}\]  
\[R=\frac{32\mu\rho_{0}}{\varepsilon}\frac{t}{d^{2}}k_{r} \tag{10}\]  
\[X=\frac{t\omega\rho_{0}}{\varepsilon}k_{m} \tag{11}\]  
In Equations (10) and (11), \(\mu\) was the viscosity coefficient of the air with normal atmospheric temperature, \(1.506\cdot 10^{-5}\) m\({}^{2}\)/s; \(\rho_{0}\) was still density of the air, \(1.21\) Kg/m\({}^{3}\); \(\varepsilon\) was the perforating rate, which could be calculated by the Equation (12); \(k_{r}\) was the acoustic resistance constant, which was obtained by Equation (13); \(\omega\) was angular frequency of the sound wave; \(k_{m}\) was the acoustic mass constant, which was achieved by Equation (14) [18; 19; 20]. Moreover, in the Equations (13) and (14), \(k\) was the perforated panel constant, which was derived by Equation (15). Furthermore, in the Equations (10)-(15), these symbols \(t\), \(d\), and \(b\) represented thickness of the microperforated polymethyl methacrylate panel, diameter of the microholes, and distance between the neighboring holes, respectively [18; 19; 20]. From Equations (8)-(15), it could be found that the transfer matrix of the microperforated polymethyl methacrylate panel was determined by its structural parameters of \(t\), \(d\), and \(b\) for a given frequency range \([f_{\min},f_{\max}]\).  
\[\varepsilon=\frac{\pi}{4}\bigg{(}\frac{d}{b}\bigg{)}^{2} \tag{12}\]  
\[k_{r}=\sqrt{1+\frac{k^{2}}{32}}+\frac{\sqrt{2}}{8}k\frac{d}{t} \tag{13}\]  
\[k_{m}=1+\frac{1}{\sqrt{\left(9+\frac{k^{2}}{2}\right)}}+0.85\frac{d}{t} \tag{14}\]  
\[k=\sqrt{\frac{\omega}{\mu}}\frac{d}{2} \tag{15}\]According to the derived transfer matrix \(T_{pf}\) of the polyurethane foam in Equation (2) and the calculated transfer matrix \(T_{mpma}\) of the microperforated polymethyl methacrylate panel in Equation (8), total transfer matrix \(T_{cs}\) of the composite structure was obtained based on the transfer matrix method [21; 22; 23], as shown in Equation (16). Moreover, similar to the calculation of sound absorption coefficient \(\alpha_{pf}\) of the polyurethane foam through Equation (1), theoretical sound absorption coefficient \(\alpha_{cs}\) of the composite structure could be calculated by Equation (17).  
\[T_{cs}=\left[\begin{array}{cc}T_{cs11}&T_{cs12}\\ T_{cs21}&T_{cs22}\end{array}\right]=T_{mpma}\cdot T_{pf} \tag{16}\]  
\[\alpha_{cs}=\frac{4\cdot\text{Re}\big{(}\frac{T_{cs11}}{T_{cs21}}\cdot\frac{1} {\rho\circ 0}\big{)}}{\left[1+\text{Re}\big{(}\frac{T_{cs11}}{T_{cs21}}\cdot\frac{1} {\rho\circ 0}\big{)}\right]^{2}+\left[\text{Im}\big{(}\frac{T_{cs11}}{T_{cs21}} \cdot\frac{1}{\rho\circ 0}\big{)}\right]^{2}} \tag{17}\]