## 2 Materials  
A 5 mm thick transparent Methyl Methacrylate sheet was used to manufacture the perforated panels (PP) in this study. The surface mass of the panel was 6 kg/m\({}^{2}\). This material has high optical transmittance when compared with the conventional glass, and great strength and good toughness. The panels were laser cut by a computer controlled process. Figure 1 shows the design of the perforated panels used for the preliminary investigation. Four different perforated panels, named A, B, C, and D were made, with their hole diameter and hole spacing, as given in Table 1. Two perforation rates were calculated. The first was the actual rate that was obtained by dividing the hole area by the tube diameter (only considering the 10 cm diameter, corresponding to the \"low frequency\" tube, see Section 3.1), while the second was calculated assuming the hole pattern to be reproduced on an ideally infinite surface, and it was obviously higher than the first one. The micro-perforated membrane (MPM) absorbers that were studied in this paper were made while using a polyester transparent film, having a thickness of 0.09 mm and a surface density of 137 g/m\({}^{2}\). The holes in MPM were made by micro-driller that was mounted on a three-axis numerically controlled system. A proper drill bit was chosen, as the application of theoretical models suggested that the best performance could be obtained by using a hole diameter of 0.30 mm, so that the actual resulting diameter was as close as possible to the desired value. However, microscopic analysis (Figure 2) showed unavoidable fluctuations in actual hole diameters that spanned from 0.20 to 0.35 mm. Therefore, taking advantage of a segmentation algorithm that was applied to microscopic images, the actual hole area was calculated, and the \"effective\" circular hole diameter was found. According to the actual frequency distribution of the effective diameters, the average value of 0.267 mm was considered in the subsequent calculations. Three MPM, named M4, M5, and M6, were prepared with the hole spacing being set accordingly at 4 mm, 5 mm, and 6 mm. Their perforation rates were 0.36%, 0.23%, and 0.16%, respectively. No significant difference appeared between actual and \"infinite-extension\" perforation rate, given the hole dimension. Sample M6 resulted from the optimization process and was, consequently, only used in the preparation of the triple-layer panel. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Panel ID** & **Hole Diam.** & **Spacing** & **Hole Number** & **Perf. Rate** & **Perf. Rate\({}_{\infty}\)** \\ \hline  & **(mm)** & **(mm)** & & **(\%)** & **(\%)** \\ \hline A & 3.0 & 20 & 16 & 1.44 & 1.77 \\ B & 4.0 & 20 & 16 & 2.56 & 3.14 \\ C & 1.5 & 10 & 69 & 1.55 & 1.77 \\ D & 4.0 & 10 & 69 & 11.04 & 12.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the characteristics of different panels under investigation. Figure 1: Different layout of the four perforated panels: (**a**) panel A; (**b**) panel B; (**c**) panel C; and, (**d**) panel D.  
Laying the selected specimens and spacers in series formed the multi-layer structures that were studied in this paper. The different layers could be tightly combined together through fixing screws at the panel corners. The spacers were made of the same material as the perforated panels mentioned above, which were adapted to form an air cavity of desired thickness. The spacers had different diameters, so that they fitted the tube diameter, in order to ensure that air volume behind each hole corresponded to the expected value. Plasticine and adhesive tape was used to seal every visible gap to avoid that measurements could be affected by sound escaping through air gaps between different parts. It should be noted that the last air cavity, the one right in front of the rigid surface, was formed while using the measuring tube termination, so that its thickness could be conveniently controlled. One sample for each configuration was made, but several measurements were carried out.