## 1 Introduction  
Helmholtz resonators are commonly employed to reduce sound pressure levels across a broad range of applications, including the built environment, industrial installations, and propulsion devices, such as gas turbines [1; 2; 3]. Combustion instability represents a significant problem in the application of low NOx emission LPC (lean premixed combustion) gas turbines. Helmholtz resonators are commonly applied to achieve attenuation effect near the instability frequencies. However, attenuation bandwidth due to Helmholtz resonance effect is often very narrow. In comparison, attenuation of noise by distributed Helmholtz resonators takes place within a much wider frequency range thanks to the stronger acoustic resistance of those smaller perforations [1; 2; 3]. The distributed Helmholtz resonator can be thought of as a special type of perforated plate absorber (PPA). The noise attenuation effect of a PPA is typically characterized by its peak damping frequency and absorption bandwidth. The peak damping frequency often coincides with the resonance frequency of a PPA [4; 5; 6; 7]:  
\[f_{r}=\frac{c}{2\pi}\sqrt{\frac{A}{Vl_{e}}} \tag{1}\]  
\[l_{e}=l+l_{ec}=l+0.96\sqrt{A_{0}} \tag{2}\]  
where \(f_{r}\) stands for resonance frequency, c represents the speed of sound, \(V\) represents the volume of the resonator cavity, \(A\) is the total opening area of all perforates, \(l_{e}\) is effective length of the opening, which includes the physical plate thickness \(l\) and an acoustic radiation \"end effect\" length \(l_{ec}\). \(A_{0}\) is the cross section area of an individual perforation. A distributed Helmholtz resonator with multiple small holes is able to significantly improve the noise damping bandwidth compared to a single neck Helmholtz absorber. On the other hand, when multiple holes are employed, additional acoustic interaction occurs between neighboring holes, thereby changing the resonance frequency of the resonator. A number of researchers, including [PERSON] [5], [PERSON] [8], and [PERSON] [9], have demonstrated that the end correction length is affected by the interaction effect between neighboring perforations. [PERSON] [8] tested an orifice in the center of a partition across a tube and proposed that to take into account the effect of hole interaction, the end correction length for the orifice should be corrected by a \"Fok's function\", \(\psi\)[8]:  
\[\psi=\left(1-1.4092\xi+0.33818\xi^{3}+0.06793\xi^{5}-0.02287\xi^{6}+0.03015\xi^ {7}-0.01641\xi^{8}\right)^{-1} \tag{3}\]  
where \(\xi\) is the hole diameter (d) divided by hole separation distance (b). [PERSON]'s function is later cited by other authors to consider the effect of hole interaction on acoustic radiation strength [10; 11; 12; 13; 14]. The acoustic effective length of the perforated plate then becomes:  
\[l_{e}=l+l_{ec}=l+\frac{0.96\sqrt{A_{0}}}{\psi(\xi)} \tag{4}\]  
Substituting for [PERSON]'s function gives,  
\[l_{e}=l+0.96\sqrt{A_{0}}\,\left(1-1.4092\xi+0.33818\xi^{3}+0.06793\xi^{5}-0.02 287\xi^{6}+0.03015\xi^{7}-0.01641\xi^{8}\right)\]  
As shown in Figure 1, where Fok's function is plotted against hole diameter (d) divided by hole separation distance (b), Fok's function is always greater than 1 and increases with increasing d/b. Fok's function appears in the denominator in Equation (4), therefore, the acoustic end correction length \(0.96\sqrt{A_{0}}\) gradually diminishes with decreasing hole separation distance due to hole interaction effect. [PERSON] [10] provided an explanation for the diminishing end correction by a simple diagram, as illustrated in Figure 2. He described the end correction effect as being the result of an \"attached air mass\" in the neighborhood of the perforation. [PERSON] [10] claimed that a portion of \"attached masses\" for a perforation will overlap with the neighboring perforations if they are sufficiently close to each other. The overlapped portions of attached masses merge into one portion of attached mass, and as a result the average attached mass for each perforation is reduced.