## 3 Numerical Modelling and Analysis of Sound Absorption Properties
Figure 6a illustrates how the enhanced bimodal bottleneck structures absorb sound pressure waves at lower frequencies (typically less than 2.0 kHz), comparedwith other soundproofing devices. The numerical simulations found that the sound absorption spectra of Inconel 450 \(\upmu\)m-(A) and glass wool fibre (GWF) materials flattened out at higher frequencies which may be attributed to their high pore volume fraction and reduced pore-nonuniformity. Figure 6a also illustrates how structural dilation of the internal morphology of the representative Inconel 450 \(\upmu\)m sample (A-DL1) slightly alters its sound absorption spectrum to a lower frequency but with the upside of reduced quarter wavelength hard-backed layer resonance peak (\(A_{p}\)). To create a 3D representative dual-porosity sample (A + B), selected differential stacked X-ray computerised tomography datasets of Inconel 450 \(\upmu\)m (Figure 2g) and Inconel 1200 \(\upmu\)m (Figure 2h) samples were imported into 3D imaging software for image processing, see Figure 2i. Structure dilation of this structure occurred through the addition of 20 \(\upmu\)m voxel elements to the internal topology of the microstructure, resulting in a semi-virtual structure demonstrating reduced pore volume, increased pore-nonuniformity, and high surface area. Figure 6b shows that the 1-pixel dilated structure, (A + B)-DL1, produced better sound absorption spectra compared to the original A, B, and A + B and other commercially available soundproofing materials. By contrast, the continuous addition of new pixels to the \"real\" dual-porosity sample (A + B) progressively complicates their topology, thereby reducing their pore openings and providing greater resistance to the penetration and absorption of sound pressure waves. According to Figure 6b, the direction of the pressure waves penetrating either of the surfaces (i.e., A + B or B + A) of the dual-porosity structures has very little influence on their acoustic properties, and their computed sound absorption spectra were actually better at higher frequencies than what was achieved for the enhanced bimodal bottleneck structure. Figure 6c shows the influence of changes associated with Darcian and inertial permeability values on the acoustic properties of these dual-porosity modulated materials. Table 4 clearly demonstrates that changes in fluid flow regimes result in changes in the flow permeability, and slight distortions in their viscous characteristic length are also observed. Based on the results of the study described in [38], it was found that at extremely low air velocities  
Figure 6: Plots of numerically predicted normal incidence sound absorption spectra against resonant frequency,\(f\) [Hz] for (**a**) Inconel 450 \(\upmu\)m (A), Inconel 1200 \(\upmu\)m (B) and some selected materials (**b**). Dual-porosity differential (A and B) microcellular structures (**c**) A and B samples at different fluid flowing regimes for 40 mm porous layer thickness and (**d**) plots of sound absorption properties against a dimensionless ratio of pore radii (r\({}_{\rm p}\)), pore openings (r\({}_{\rm w}\)) and sample porosity (\(\epsilon\)). (creeping flow regime), flow permeability varied depending on the pore-structure-related parameters. For high air velocity flow (typically, above 0.5 m.s\({}^{-1}\)), the presence of forced recirculation and high particle interaction within the interstitial pores of the porous matrix alters their permeability value from that obtained for low air velocity flow [38]. [PERSON] and [PERSON] [44] showed that in metal foams, fluids that move slowly are confined to their outer regions, whereas a relative velocity gradient arises for fast-moving fluids when they penetrate deeper into the pores of the porous matrix. Figure 6c shows how numerically simulated sound absorption spectra can differ depending on the key permeability values used for low or high airflow velocities--which is better for slow-moving fluid in the Darcy regime.