## 3 Results
### Acquisition of Model
\[l_{op}=C\cdot 0.96\sqrt{A} \tag{14}\]  
where \(A_{0}\) stands for area of an individual perforation and A is the overall opening area of all orifices. A correcting parameter C is introduced to consider the extent to which the perforation region is acting  
Figure 13: Acoustic velocity magnitude contour near an orifice with different hole separation distance: (a) plate No.7 absorber: x/d = 5, y/d = 5, (b) plate No.2 absorber: x/d = 1.5, y/d = 1.5.  
like a single large hole. C is large when holes are close to each other and it is zero when holes are far away from each other. A simple model is proposed in this paper by assuming that acoustic end correction lengths caused by these effects are additive. \[l_{ec}=l_{sh}+l_{op}=0.96\sqrt{A_{0}}+C\cdot 0.96\sqrt{A} \tag{15}\]  
where \(\sigma_{eff}\) represents the effective porosity of the perforation area; \(l_{sh}\) is the end correction length due to acoustic radiation effect near an individual hole and \(l_{op}\) stands for the end correction length due to the acoustic radiation effect due to the overall perforation area. C is a factor for the overall perforation area radiation effect. The factor C for all twenty-five numerically investigated PPAs was determined from Equations (1) and (15), as listed in Table 3, and then plotted against effective porosities, as presented in Figure 14.  
A model relating the overall perforation area radiation effect to the effective porosity was obtained by regression analysis, yielding:  
Figure 12.0  
The total end correction length considering both acoustic radiation effects then becomes:  
Figure 12.1  
The resonant frequency of a distributed Helmholtz resonator  
Figure 12.2 \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case No. CFD** & \(\sigma_{\mathit{eff}}\) & \(f_{r}\) **(Hz)** & \(l_{ec}\) **(mm)** & **C** \\ \hline \(1\) & 0.545 & 429 & 5.562 & 0.654 \\ \(2\) & 0.349 & 457 & 4.275 & 0.503 \\ \(3\) & 0.196 & 492 & 2.970 & 0.349 \\ \(4\) & 0.126 & 517 & 2.194 & 0.258 \\ \(5\) & 0.087 & 535 & 1.696 & 0.199 \\ \(6\) & 0.049 & 561 & 1.062 & 0.125 \\ \(7\) & 0.031 & 576 & 0.721 & 0.085 \\ \(8\) & 0.545 & 373 & 8.141 & 0.639 \\ \(9\) & 0.444 & 401 & 6.452 & 0.506 \\ \(10\) & 0.250 & 435 & 4.512 & 0.354 \\ \(11\) & 0.160 & 461 & 3.320 & 0.260 \\ \(12\) & 0.111 & 481 & 2.529 & 0.198 \\ \(13\) & 0.063 & 510 & 1.544 & 0.121 \\ \(14\) & 0.040 & 530 & 0.955 & 0.075 \\ \(15\) & 0.545 & 331 & 11.000 & 0.647 \\ \(16\) & 0.444 & 357 & 8.764 & 0.516 \\ \(17\) & 0.250 & 392 & 6.037 & 0.355 \\ \(18\) & 0.160 & 417 & 4.471 & 0.263 \\ \(19\) & 0.111 & 437 & 3.411 & 0.201 \\ \(20\) & 0.063 & 470 & 1.988 & 0.117 \\ \(21\) & 0.040 & 492 & 1.179 & 0.069 \\ \(22\) & 0.105 & 473 & 2.721 & 0.2234 \\ \(23\) & 0.079 & 490 & 2.123 & 0.1665 \\ \(24\) & 0.052 & 508 & 1.522 & 0.1194 \\ \(25\) & 0.039 & 521 & 1.136 & 0.0891 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical results of resonance frequencies and end correction lengths for all simulated perforated plates. where \(l_{ec}\) represents the total end correction length of a PPA, \(l_{sh}\) represents end correction length as a result of acoustic radiation effect due to individual perforation, and \(l_{op}\) represents end correction length as a result of acoustic radiation effect due to the overall perforation region. The above model was fitted to the available data within an effective porosity range of 3.14-55% and an aspect ratio range of 0.4:1-1:1. The model provides an overall assessment for the impact of hole interaction effect on acoustic radiation effect by introducing a simple additive term \"overall perforation area radiation effect\", under the assumption that acoustic radiation effect from the overall perforation area does not change the end correction length 0.96\(\sqrt{A_{0}}\) caused by acoustic radiation from each individual hole.