## 3 Numerical Modelling and Analysis of Sound Absorption Properties  
The sound absorption spectra and properties of these modulated soundproofing structures were determined using numerical simulations of propagating pressure waves across their interstices and analysis of the resulting normal incidence sound absorption spectra for frequencies between 0.1 kHz and 6.5 kHz. The numerical modelling and simulation procedures used here are related to those reported in [9, 13, 16] and were performed in COMSOL MultiphysicsTM using the acoustic module. A two-dimensional (2D) geometry (Figure 4b) was created to mimic an AFD 1200-AcoustiTubeTM measurement set up [8] with air as the fluid domain (FD) of the geometry. Essentially, this geometry consists of a 40 mm-thick hardback porous layer (PSL), a 40 mm-thick air-dominated zone (FD), and a 20 mm-thick perfectly matched layer (PML). It was set up so that, as near as possible, the simulations would be compared with the data generated by a three-dimensional model (Figure 4a) and experimental results reported in [3, 6, 7, 13, 14, 15, 16] as shown in Figure 1. In the FD and PSL domains, triangular mesh structures with minimum grid size, maximum grid size and growth rates of 21 um, 52 um and 1.3 um, respectively, were applied. The PML domain was confined to a perfect match layer. The result was a mesh density of ~26,000 cells for the 2D simulations and ~950,000 cells for 3D simulations, both of which are within the capacity of the computational resources required for this exercise. Figure 4: _Cont._  
Helmholtz linear acoustic pressure (Equation (2)) involving total acoustic pressure (\(p\)), fluid density (\(\rho_{o}\)), propagation coefficient (\(k\)), monopoles (\(Q\)) and dipoles (\(q\)) was resolved on the PML and FD domains. In the PML domain, the equivalent fluid Wilson [28] and semi-phenomenological JCA [24] models were interchanged to estimate their real specific surface acoustic impedance (\(Z\)) and normal incidence sound absorption coefficients (\(A_{\rm C}\)), Equation (3), for the bottleneck-shaped and tetrakaidekahedron-shaped macroporous structures, respectively. \[\
abla.\left(-\frac{1}{\rho_{o}}(\
abla p-q)\right)-\frac{w^{2}}{\rho_{o}c_{o} ^{2}}p=Q \tag{2}\]  
\[Z=\left[\frac{{\rm i}Z_{c}\cot(kL)}{\varepsilon\rho_{o}c_{o}}\right]\text{ and }A_{\rm C}=\left[1-\left(\frac{Z-Z_{o}}{Z+Z_{o}}\right)^{2}\right] \tag{3}\]  
where \(Z_{c}\) is the characteristic impedance of the porous sample, \(Z_{o}\) is the specific surface acoustic impedance of air, \(\varepsilon\) is the sample porosity, \(\rho_{o}\) is the air density and \(c_{o}\) is the speed of sound in air. A background pressure field boundary condition was applied to the FD zone and a perfectly matched layer to the PML domain, while a periodic Floquet boundary condition was applied to the lateral faces (sides). In the selected Helmholtz linear acoustic model, the inlet and outlet sections of the PML and PSL are conditioned to sound hard boundary walls at a 20 \(\upmu\)Pa reference pressure. The simulation results from the 2D and 3D created geometries overlap each other with little or no difference (R\({}^{2}\)\(\sim\)0.99), and the computational time for the 3D geometries is 10\(\times\) higher than that of the 2D geometry (\(\sim\)1 min) according to the specified frequency range between 0.1 and 6.5 kHz. In addition to the normal incidence absorption spectra, a number of key properties were determined to allow for a quantitative assessment and comparison of the acoustic performance of these modulated structures. These properties are the noise reduction coefficient (NRC), sound absorption average (SAA), quarter wavelength resonance peak in absorption (\(A_{p}\)) and frequency of peak absorption (\(f_{p}\)). The arithmetic mean of the normal incidence sound absorption coefficient for the quarter wavelength layer resonance frequencies (0.25, 0.5, 1.0 and 2.0 kHz) was calculated as NRC, while the average sound absorption coefficient for the twelve third-octave bands (0.20-2.5 kHz) was determined as SAA [17; 29].