## Materials and Methods
### Numerical Simulation
The normalized specific acoustic impedance of the plate absorber is defined by [PERSON] [4] as:  
\[z=r+ix=(1+R)/(1-R) \tag{9}\]  
where \(r\) is the normalized specific acoustic resistance, which is a measure of the resistance that the perforated plate presents to the acoustic flow, \(x\) is the normalized specific acoustic reactance, which describes the phase difference between the driving pressure difference and the resultant orifice velocity. \(R\) represents the reflection coefficient. For a PPA installed at the end of an impedance tube, the acoustic  
Figure 3: An impedance tube configured with two microphones. energy that is not reflected by the absorber is absorbed by the PPA and the acoustic energy absorption coefficient, therefore, becomes [4]:  
\[\alpha=1-\left|R\right|^{2}=\frac{4r}{\left(1+r\right)^{2}+x^{2}} \tag{10}\]  
As can be seen from Equation (10), the optimum absorption effect of a PPA takes place at the frequency where acoustic reactance is zero, which also coincides with the resonance frequency of a PPA [4]. The range of geometric configurations considered is summarized in Table 1. All plates have the same porosity of 0.0123 and the same thickness of 2.5 mm. Figure 4 illustrates a selection of the geometric configurations considered. The ratios of hole separation distance to hole diameter (x/d, y/d) vary from 1.2, to 1.5, 2, 2.5, 3, 4, 5, where x is the hole pitch in the horizontal direction, y is the hole pitch in the vertical direction, and d is the diameter of the hole. The size of the perforations varies from 2 mm to 3 mm and 4 mm. Those plates with 2 mm, 3 mm, and 4 mm perforations are installed in impedance tubes with diameters of 90 mm, 135 mm, and 180 mm, respectively. The length of all orifics is kept consistent at 2.5 mm. All plates are backed by a 25 mm deep cylindrical air cavity, which has an identical cross-section to upstream impedance tubes. Real world perforated plates are sometimes more densely perforated in one direction than in other directions; therefore, perforated plates (plate 23, plate 24, and plate 25) with different aspect ratios were taken into consideration as well. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case No. **} & \multicolumn{2}{c}{**Diameter of the**} & \multirow{2}{*}{**x/d**} & \multirow{2}{*}{**y/d**} & \multirow{2}{*}{**Effective Porosity, \(\sigma_{\textit{eff}}\)**} \\  & **Impedance Tubes and** & & & & \\  & **Perforated Plates (mm)** & & & & \\ \hline
1 & 90 & 2 & 1.2 & 1.2 & 0.545 \\
2 & 90 & 2 & 1.5 & 1.5 & 0.349 \\
3 & 90 & 2 & 2 & 2 & 0.196 \\
4 & 90 & 2 & 2.5 & 2.5 & 0.126 \\
5 & 90 & 2 & 3 &  
The porosity of a perforated plate is defined as the ratio of the total opening area to the whole surface area of the perforated plate, \(\sigma=nA_{0}/A_{p}=A/A_{p}\), where \(n\) is the number of holes, which equals to 25 for the PPA simulated in this work and 32 for those experimentally tested. In the above definition, \(A_{0}\) is the cross-section area of a single hole, \(A\) represents the total opening area of all holes, and \(A_{p}\) stands for the total surface area of the perforated plate backed by a cavity. In this paper, a new parameter, effective porosity, is introduced, and for a parallel distributed plate, as shown in Figure 4, it is defined to be:  
Figure 2:0  
where \(d\) is diameter of the hole, \(x\) and \(y\) are hole separation distance in two directions, as shown in Figure 4, \(\pi d^{2}/4\) is the opening area in a rectangular perforation region whose area is \(xy\). Effective porosity depicts the uniformity of hole distribution in a perforated plate backed by a cavity. It is equal to classical porosity if all holes are distributed evenly on a perforated surface. It will be larger than the classical porosity if some or all holes are more densely distributed in a local area of a noise damping surface. The phenomena of acoustic radiation takes place in close proximity to the individual holes in the perforated plate.