# A Perforated Plate with Stepwise Apertures for Low Frequency Sound Absorption
## 2 Models and Methods
### Experiment Measurements  
To further verify the feasibility of the theoretical and numerical models, the sound absorption coefficients of PPSA absorber at normal incidence are measured by SW422 impedance tube with diameter of 100 mm, as shown in Figure 4a, and PPSA samples are made of resin materials by 3D printing using SLA (light curing molding) equipment. The outer diameter of the specimen is about 99 mm, so the sample is placed in the specimen tube with sealing strips or tape to ensure the tightness.  
The sound absorption measurement in impedance tube follows the two-microphone transfer function method [30], and the principle is described below. The sound pressure \(p_{1}\) and \(p_{2}\) of two microphones located at position 1 and position 2 are expressed as:  
\[p_{1}=P_{I}e^{jk_{0}x_{1}}+P_{R}e^{-jk_{0}x_{1}},p_{2}=P_{I}e^{jk_{0}x_{2}}+P_{ R}e^{-jk_{0}x_{2}} \tag{13}\]  
The transfer function of the incident wave and the reflected wave are written as:  
\[H_{I}=\frac{p_{2I}}{p_{1I}}=e^{-jk_{0}(x_{1}-x_{2})}=e^{-jk_{0}s},H_{R}=\frac{p _{2R}}{p_{1R}}=e^{jk_{0}(x_{1}-x_{2})}=e^{jk_{0}s} \tag{14}\]  
According to Equation (13), the transfer function of the total sound field is denoted as:  
\[H_{12}=\frac{p_{2}}{p_{1}}=\frac{e^{jk_{0}x_{2}}+re^{-jk_{0}x_{2}}}{e^{jk_{0}x _{1}}+re^{-jk_{0}x_{1}}} \tag{15}\]  
Substituting Equation (14) into Equation (15), the reflection coefficient is expressed as:  
\[r=\frac{H_{12}-H_{I}}{H_{R}-H_{12}}e^{jk_{0}x_{1}} \tag{16}\]  
where \(x_{1}\) and \(x_{2}\) are the distances between microphone position 1 or position 2 and the front surface of the test sample, respectively, and \(s\) is the distance between two microphone position 1 and position 2. Once the reflection coefficient is determined, the sound absorption coefficient of the test sample can be calculated.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **PPSA1** & **PPSA2** & **PPSA3** \\ \hline \(d(d_{1}/d_{2})\) (mm) & \(1.5/4\) & \(1.5/4\) & \(1.5/4\) \\ \(t(t_{1}/t_{2})\) (mm) & \(1/1\) & \(1/1\) & \(1/1\) \\ \(\sigma(\sigma_{1}/\sigma_{2})\) (\%) & \(0.27/1.92\) & \(0.23/1.60\) & \(0.18/1.28\) \\ \(D\) (mm) & \(25\) & \(50\) & \(100\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for three PPSA absorbers in the theoretical and simulated comparison.  
Figure 3: The theoretical and simulated results for PPSA absorbers. (**a**) Sound absorption coefficients; (**b**) Specific acoustic impedance.  
The sound absorption coefficients of a PPSA absorber in the frequency range of 100-300 Hz from theoretical, numerical and experimental results are plotted in Figure 4b. The parameters of the PPSA are \(d_{1}\) =1.5 mm, \(d_{1}\) =3 mm, \(t_{1}\) = 1 mm, \(t_{2}\) = 1 mm, \(p_{1}\) = 0.24%, \(p_{2}\) = 1.08% and the air cavity depth is \(D\) = 80 mm. The peak frequencies of three absorption coefficient curves are almost at 190 Hz, and the peaks are 0.999, 0.999 and 0.992, and the frequency ranges of absorption coefficients greater than 0.6 are 152-244 Hz, 153-242 Hz and 154-239 Hz, respectively. The results showed that there are tiny errors between the measured and predicted results. These unavoidable deviations may be due to factors such as machining accuracy and inaccurate mounting. In conclusion, the proposed theoretical and numerical models are feasible for predicting the sound absorption of the PPSA absorber. Meanwhile, it implies that the expected sound absorption is also relied on the accuracy of the sample processing.