## 3 Methods  
### Impedance Tube Method  
The two-microphones impedance tube method was applied in this paper to measure the acoustic absorption of resonant absorbers, according to the process that is detailed in ISO 10534-2:1998 standard [40]. In this method, the plane sound wave is generated by a loudspeaker at one end of the tube and is reflected by the specimen at the other end. Two microphones that are set at different positions on the internal surface of the tube are used to record the sound energy decay. The reflection coefficient R of the specimen is calculated from the corrected complex acoustic transfer function. Subsequently, the normal incidence sound absorption coefficient \(\alpha\) is calculated as \(\alpha=1-|\text{R}|^{2}\). The frequency range in which the measurements are reliable depends on the length of the tube for the minimum frequency and on the tube diameter and the microphone spacing for the maximum frequency. Two impedance tubes with different internal diameters (10 cm and 4 cm) are needed in order to cover the whole spectrum range in which the specimens are expected to absorb. These tubes are made of transparent Methyl Methacrylate, 5 mm thick and 85 cm long. At the sending end of the larger tube, a 11 cm loudspeaker is sealed into a wooden case and properly isolated from the tube structure by means of an elastic pad, while a 5 cm loudspeaker is used for the smaller tube (Figure 4). On the other end, there is a shorter tube with a movable rigid termination, which is suitable to form the last air cavity of the test structure. All of the gaps between the interfaces are carefully sealed with plasticine. In the measurements, two microphones (Core Sound) with a flat frequency response from 20 Hz to 20 kHz are used. In the large tube, two different spacings are available, one at 6 cm and one at 20 cm, which correspond to a low frequency limit, respectively, of 283 Hz and 85 Hz, and a high frequency limit of 2 kHz and 770 Hz, respectively. Thus, the large tube covers one-third octave bands from 100 Hz to 1.6 kHz. In the smaller tube a 3 cm spacing between microphones is used, resulting in a frequency range from 566 Hz to 4.9 kHz, allowing for measurements up to the 4 kHz one-third octave band.  
Figure 4: The 10 cm standing wave tube used to measure sound absorption coefficients in the frequency range from 100 Hz to 1.6 kHz. One of the triple layer samples is mounted in place.  
Figure 3: Schematic configuration of the different samples under investigation: (**a**) double leaf perforated panels (PP); (**b**) double leaf one perforated panel and a micro-perforated membrane (PP-MPM); (**c**) double leaf double micro-perforated membranes (MPM-MPM); (**d**) triple leaf optimized triple-layer absorber (PP-MPM-MPM). Black box represents the rigid surface.  
The whole measurement system is controlled by a MATLAB (2018, Mathworks, Natick, MA, USA) graphic user interface, which generated and played a 5 s linear sweep from 70 Hz to 3 kHz, used in the combination with the larger impedance tube, and from 500 Hz to 5 kHz used in combination with the smaller one. In the measurements, the temperature and the relative humidity were monitored with a precision electronic thermo-hygrometer and were subsequently applied in the calculation process.