# A Perforated Plate with Stepwise Apertures for Low Frequency Sound Absorption
## 3 Results and Discuss  
### Sound Absorption of PPSA and Single Perforated Palle(PP) Absorber  
To explain the feasibility of the proposed PPSA structure for low frequency sound absorption, comparisons of the sound absorption of PPSA absorber with that of single PP absorber are given here, with a depth of 80 mm for each air cavity. Figure 5 displays the theoretical sound absorption coefficients and acoustic impedance of PPSA and single PP absorbers with the same perforation ratio. The relevant parameters of PPSA and PP are listed in Table 2. In Figure 5a, the resonance peaks of PPSA, PP1 and PP2 absorber are located at 168 Hz, 195 Hz and 219 Hz, and the bandwidth of the sound absorption coefficient well above 0.6 are 76 Hz, 40 Hz and 110 Hz, respectively. In Figure 5b, PPSA has a relatively matched acoustic resistance from 1 to 1.41 in the range of 100-300 Hz. In contrast, the acoustic resistance of PP is extremely inadequate, especially the maximum acoustic resistance of PP1 with a large aperture is below 0.5. In addition, compared to the PP absorbers, the zero acoustic reactance of the PPSA absorber is shifted to a lower frequency. Even though, PP2 has a relatively large absorption bandwidth, its resonant frequency occurs at higher frequency, and a large number of perforations will increase the manufacturing costs. Thus, PPSA is more suitable for low-frequency sound absorption.  
Figure 4: Experimental setup and results. (**a**) Impedance tube system and test sample; (**b**) Absorption coefficients of PPSA absorber from three models: theoretical, FEM and experimental.  
Figure 6 illustrated the theoretical sound absorption coefficients and acoustic impedance of PPSA and single PP absorbers with the same hole spacing. The parameters of PPSA and PP are given in Table 3. First, comparing PPSA and PP1, their resonant frequencies are close to 153 Hz and their sound reactance curves are almost overlapping as shown in Figure 6b. For PPSA with \(d_{1}\) = 2 mm and \(d_{2}\) = 4 mm, its thickness is \(t\) = 2 mm, while for PP1 with \(d\) = 4 mm, its thickness is 6.5 mm, thus the panel thickness of PPSA is about 3.25 times that of PP1. That is, for the same cavity depth and hole spacing, when PPSA and PP absorbers are resonating at the same frequency, PP with the large hole diameter requires a larger thickness. Meanwhile, the resonance peak of PPSA is close to 1, and that of PP1 is 0.88. The sound absorption coefficient of PPSA is greater than 0.6 from 135 Hz-196 Hz, and that of PP1 is greater than 0.6 from 143 Hz-182 Hz. Obviously, the sound absorption of PPSA is better than that of PP1. Furthermore, for PP2 of the thickness \(t\) = 2 mm, its resonance occurs at a higher frequency of 226 Hz with a peak of only 0.6. This is due to its inadequate acoustic resistance and high acoustic reactance. Therefore, the introduction of stepped holes not only effectively increase the acoustic resistance, but also shift the zero acoustic reactance of the PPSA absorber to lower frequencies.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **PPSA** & **Single PP1** & **Single PP2** \\ \hline \(d(d_{1}/d_{2})\) (mm) & 1.5/3 & 3 & 1.5 \\ \(t(t_{1}/t_{2})\) (mm) & 1/1 & 2 & 2 \\ \(\sigma(\sigma_{1}/\sigma_{2})\) (\%) & 0.21/0.81 & 1.02 & 1.02 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for PPSA and single PP absorbers with the same perforation ratio.  
Figure 5: Sound absorption of PPSA and single PP absorbers with the same perforation ratio. (**a**) Sound absorption coefficients; (**b**) Specific acoustic impedance.  
Figure 6: Sound absorption of PPSA and single PP absorbers with the same hole spacing. (**a**) Sound absorption coefficients; (**b**) Specific acoustic impedance.