## 3 Results
### Validation of the Model  
The model, as described earlier in Equations (17) and (18), was determined based on numerical results alone. In this section, the model is validated by comparison with experimental results for eighteen distributed Helmholtz resonators. Effective porosities of these experimentally tested perforated plates range from 0.0436 to 0.4987, porosities of these plates ranges from 0.0338 to 0.135, and perforation aspect ratios range from 1:1 to 1:3. Figures 15-17 display the magnitude of pressure response functions between microphone 1 in the resonator and microphone 2 in the acoustic propagation duct. The experimental results suggest that resonance frequencies of PPAs shift to lower frequencies as the holes approach each other. This finding is in agreement with CFD investigations within a wide range of effective porosities from approximately 0.04 to 0.54. This range of 0.04-0.54 covers the effective porosity of most practical PPAs that exhibit a hole-hole acoustic interaction effect. An even lower porosity than 0.04 does not exhibit a noticeable hole-hole interaction effect. A higher effective porosity than 0.54 is not investigated because a PPA with higher effective porosity is nearly a single hole in terms of acoustic radiation effect (for example, for a plate with an effective porosity of 0.6, the ratio of hole diameter (d) to hole center distance (b) is 0.874, which means the orifice is so close that only a distance of 0.126d exists in between their edges). Figure 14: Proposed model for end correction factor as a function of effective porosity. Figure 16: Experimental pressure response functions for plate 7 to plate 12. Figure 17: Experimental pressure response functions for plate 13 to plate 18. Figure 15: Experimental pressure response functions for plate 1 to plate 6. Resonance frequencies of all eighteen PPAs are determined from Figures 15-17 and then listed in Table 4. Then, the experimental end correction length was calculated by employing Equation (1). End correction lengths acquired by the proposed model (Equation (17)) are similarly provided in Table 4.  
The experimental end correction lengths for all eighteen distributed Helmholtz resonators are plotted against the model results, as shown Figure 18. The figure shows that the proposed model resulting from the regression of the CFD simulation data successfully provides a satisfactory agreement with the experimental results. The relative errors with respect to the experimental results are provided in Table 4 as well. Both experimental results and model results predict an increase of end correction length with closer hole separation distance due to the hole-hole interaction effect. By contrast, as listed in Table 4, predictions according to Fok's function (Equation 4) always generate end correction lengths less than \(0.96\sqrt{A_{0}}\), which are much lower than experimental results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case \#** & \(\sigma_{\textit{eff}}\) & \(f_{\textbf{r}}\)** & \(\textbf{Hz}\)**\(-\)**EXP** & \(f_{\textbf{r}}\)** & \(\textbf{Hz}\)**\(-\)**Model** & \(I_{\textit{ac}}\)**\(-\)**Fok (mm)** & \(I_{\textit{ac}}\)**\(-\)**EXP (mm)** & \(I_{\textit{ac}}\)**\(-\)**Model (mm)** &
\begin{tabular}{c} **Model Relative** \\ **Error (\%)** \\ \end{tabular} \\ \hline  
21 \\ \hline \hline \end{tabular} \(\sigma_{\textit{eff}}\): Effective porosity; \(f_{\textbf{r}}\) (Hz)-EXP: resonant frequency obtained by experimental method; \(f_{\textbf{r}}\) (Hz)-Model: resonant frequency obtained by the proposed model as given in Equation (18); \(I_{\textit{ac}}\)_Fok:_ acoustic end correction length obtained by Fok’s function as given in Equation (4); \(I_{\textit{ac}}\)-EXP: acoustic end correction length from experiment; \(I_{\textit{ac}}\)-Model: predicted acoustic end correction length by using the proposed model as given in Equation (17).  
\end{table} 
Table 4: Comparison of experimental and fitted model results for end correction lengths.