# A Perforated Plate with Stepwise Apertures for Low Frequency Sound Absorption
## 3 Results and Discuss
### Sound Absorption of PPSA Absorber Array  
Due to its Helmholtz resonance nature, PPSA absorber exhibits good sound absorption in the vicinity of the resonance. To further improve the sound absorption of PPSA absorber, only one rigid plate is inserted to separate the air cavities, thus introducing additional resonances to expand the absorption bandwidth, while the structural parameters of the PPSA remain unchanged. Without loss of simplicity, a PPSA absorber array consisting of two sub-PPSA absorbers arranged in parallel is shown in Figure 8a. According to the electroacoustic analogy method, the surface impedance of the PPSA absorber array is expressed as:  
\[Z=\left(\frac{\phi_{1}}{Z_{1}}+\frac{\phi_{2}}{Z_{2}}\right)^{-1} \tag{17}\]  
where \(\varphi_{1}\) and \(\varphi_{2}\) are the area ratios of sub-PPSA1 and sub-PPSA2 to the PPSA absorber array, \(Z_{1}\) and \(Z_{2}\) are the surface impedance of sub-PPSA1 and sub-PPSA2 absorbers, respectively. Next, we will discuss the sound absorption performance of PPSA absorber array. As an example, a PPSA absorber array with panel thickness \(t=2\) mm, air cavity \(D=100\) mm, diameters of stepped hole \(d_{1}=\)1.5 mm and \(d_{2}=4\) mm, and its relevant parameters are listed in Table 4. The sound absorption coefficients of PPSA absorber array, sub-PPSA absorber and single PPSA absorber are demonstrated in Figure 8b. It can be observed that, the two peak of the PPSA absorber array are located at 158 Hz and 208 Hz, and the peaks of sub  
Figure 8: PPSA absorber array. (**a**) Structure diagram; (**b**) Sound absorption coefficients; (**c**) The acoustic pressure field (color map) and velocity distribution (yellow arrow). PPSA1 and sub-PPSA2 absorber are positioned at 154 Hz and 213 Hz, respectively. That is, the two peaks of the PPSA absorber array are corresponding to those of sub-PPSA1 and sub-PPSA2 absorber, and the apparent frequency shift can be attributed to the interaction of sub-PPSA1 and sub-PPSA2. In addition, the PPSA absorber array has a sound absorption coefficient greater than 0.6 from 129 Hz to 257 Hz, and its sound absorption bandwidth is about 128 Hz. In contrast, the single PPSA absorber has a sound absorption coefficient greater than 0.6 from 150 Hz to 237 Hz, and its sound absorption bandwidth is about 87 Hz. Consequently, an array of PPSA absorbers can expand absorption bandwidth for low frequency under the common coupling effect of multi-local resonances. Intuitively, Figure 8c shows the distribution of the normalized sound pressure and particle velocity at two absorption peak frequencies \(f=158\) Hz and \(f=208\) Hz. As observed, at \(f=158\) Hz, the sound pressure in the air cavity of sub-PPSA1 is approximately four times higher than that of the incident sound field, and most of the particle velocity flow is distributed in sub-PPSA1 absorber. Similarly, at \(f=208\) Hz, due to strong local resonance, the maximum sound pressure and particle velocity flow are mostly concentrated in the sub-PPSA2 absorber, and even the sound pressure does not change in the Sub-PPSA1. Thus, it is confirmed that acoustic energy is mainly dissipated by the local resonances of the PPSA absorber array. For PPSA absorber array and single PPSA absorber, the measured and theoretical sound absorption of sample 1 and sample 2 are shown in Figures 9 and 10, respectively. The air cavity depth of two samples is 80 mm, and the approximate parameters for samples are given in Table 5. There are some differences between the measured and the theoretical absorption coefficients, as well as the acoustic impedance, but their sound absorption characteristics are almost identical. For PPSA absorber array1, as shown in Figure 9a, the measured sound absorption coefficient is higher than 0.5 from 141 Hz to 280 Hz, and its effective sound absorption bandwidth is about 105 Hz. While for single PPSA absorber1, the measured sound absorption coefficient is above 0.5 from 153 Hz to 247 Hz, and its effective sound absorption bandwidth is about 59 Hz.