## Materials and Methods
### Experimental Validation  
Simulation of the acoustic signal propagation through a distributed Helmholtz resonator was undertaken by using a laminar flow solver, which itself is free from significant physical models. Verification of the solution methodology in the form of numerical errors associated with the quadrature scheme and discretization can be achieved by the selection of high order numerical schemes and demonstrated through a test of grid dependency.  
In order to provide a validation of the results from the computational simulations, experimental data was collected for eighteen separate plates in a dedicated experimental test rig, illustrated in Figures 7 and 8.  
Figure 5: Grid resolution in an impedance tube configured with plate No. 7 absorber: (**a**) grid resolution in the test rig and (**b**) grid resolution in the hole (Hole Diameter = 2 mm).  
Figure 6: Normalized specific acoustic impedance of perforated plate No. 7 absorber acquired by 0.1 mm and 0.2 mm resolution computational grids in the hole.  
The test rig (Figure 7) consists of an upstream duct section, a downstream duct section, and a test section installed in between them, each with an internal diameter of 165 mm. Four opposing loudspeakers were installed on the upstream duct. The test rig was designed to include anechoic terminations to minimize reflections from the change in impedance at the inlet and exit. In all test configurations, no mean flow through the duct or bias flow through the plates was applied.  
The resonant cavity, Figure 8, was constructed from a curved rectangular test plate with a length of 0.05 m and an arc length of 0.0594 m, mounted flush to the side wall of the test section duct, with an attached fan-shaped cavity of 0.051 m depth. Referring to Figure 8, the volume of the cavity is:  
\[V=\pi[(0.051+0.165/2+0.002)^{2}-(0.165/2+0.002)^{2}]\times\frac{41.25^{\circ}}{3 60^{\circ}}\times 0.05=0.002\;\mathrm{m}^{3} \tag{12}\]  
where 51 mm is the cavity depth, 165 mm is the internal diameter of the tube, 2 mm is the thickness of plate, and 41.25\({}^{\circ}\) represents the angle of the fan-shaped cavity.  
A total of 32 circular perforations, as exemplified in Figure 9, were drilled in the x and y directions, respectively. The diameters of these orifices were 2 mm, 3 mm, or 4 mm. The thickness of all experimentally tested plates was kept the same at 2 mm.  
Figure 8: The position and shape of the distributed Helmholtz resonator on the experimental test rig.  
Figure 7: Experimental Test Rig Setup.  
Detailed geometric features of the orifices are exemplified in Figure 9 and detailed in Table 2. A signal generator and amplification was employed to generate a 100 dB white noise signal within a bandwidth of 100 Hz-1000 Hz. Two microphones were installed flush with the resonator top wall and the acoustic duct wall, respectively. The resulting magnitude of the pressure response function, PRF, is defined below, based upon the Fast Fourier Transform (FFT) of the individual microphone signals:  
\[PRF=\left|\frac{FFT(P\_acoustic\_microphone\_1)}{FFT(P\_acoustic\_microphone\_2)}\right|\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case \# EXP**} & **Diameter** & \multirow{2}{*}{\begin{tabular}{c} **Plate** \\ **Circumferential** \\ **Length (mm)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Plate** \\ **Circumferential** \\ **Length (mm)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Length** \\ **Length (mm)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Distance** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Hole** \\ **Distance-x** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Hole** \\ **Distance-y** \\ **(mm)** \\ \end{tabular} } & \multirow{2