## 3 Numerical Modelling and Analysis of Sound Absorption Properties
Figure 5c presents numerically predicted sound absorption spectra against resonant frequency for monomodal and bimodal bottleneck-shaped structures created at capillary radius 10 \(\upmu\)m, while Figure 5d presents results for bimodal structures (50% volume S-S additions) for varied capillary radius between 10 and 80 \(\upmu\)m. Accordingly, the modulated bimodal structures created at a 10 \(\upmu\)m capillary radius had the highest sound absorption properties, while the monosized S-S structures created at a 10 \(\upmu\)m capillary radius had the lowest. This means that the creation of bimodal structures simply increases the available pore volume and pore contacts in macroporous structures, which reduces the amount of reflected sound waves that penetrate the inlet surface of the porous body. Finally, for the monomodal and bimodal structures, optimal sound absorption spectra and properties were achieved by keeping the mean pore opening radius between 80 and 140 \(\upmu\)m, which led to a computed flow permeability between 0.60 and 1.7 \(\times\) 10\({}^{-9}\) m\({}^{2}\). As shown in Figure 5, the permeability of these macroporous structures has a direct influence on the numerical prediction of their vibroacoustic behaviour. Numerical applications of the equivalent fluid Wilson model do not directly use mean pore size or mean pore opening, but they are directly linked to the permeability of porous structures. In the inverse replication casting route of foam making, the ideal permeability values can be achieved by mechanically manipulating pore sizes (related to pore size [18]), applied infiltration pressure (related to pore openings [35]) and packing density (related to pore volume fraction [37]). Equation (1) reveals the correlation between permeability and pore structure for a deeper understanding of vibroacoustic behaviour for these kinds of structures and could be a valuable tool for modelling their behaviour. Figure 5a and Table 2 show that varying the permeability can result in a change in the absorption spectrum at either frequency maxima or minima and a change in the quarter wavelength hard-backed layer resonance peak. In addition, the dynamic tortuosity and porosity of these materials are crucial considerations. In Table 2, the manipulation of sphere-packing models to create virtual structures illustrates the difficulty in understanding which of these two properties is dominant, as both properties changed in response to changes in pore sizes or pore openings. The two properties, however, were inversely related, being lower for a bimodal structure, and their relationship could be best described using [PERSON]'s relation (\(\tau=C_{o}\varepsilon^{-n}\)) described in [17; 41]. A plot of the natural log of the dynamic tortuosity (\(\ln[\tau]\)) against the log of the porosity of the sample (\(\ln[\varepsilon]\)) shows that the relationships are linear-inverse, yielding a tortuosity factor (\(n\)) of 0.69, a shape factor (\(C_{o}\)) of 0.3, and a very good fit \(R^{2}\sim\) 0.98. There are explicit predictions of lower and upper bound values for this tortuosity factor (\(n\)) in [42; 43], ranged between 0.40 and 0.60 for most porous metallic structures. The tortuosity factor obtained in this study fell outside this range due to the reduced pore-nonuniformity and differential opening available in bimodal structures, whereas model predictions in [42; 43] were made with porous structures with monomodal pores. Earlier work described in [17] had estimated the tortuosity factor (\(n\)) for virtual macroporous structures characterised by monosized pores as 0.552, which is within the limit prescribed in [42; 43]. The enhanced bimodal structure (L-S, 50% volume at 10 m capillary radius) showed a significantly different vibroacoustic behaviour compared to the benchmark reported for bottleneck-type aluminium foam (ALF [3]), and commercial soundproofing materials like glass wool fibre (GWF [8; 9]), the Inconel 450 \(\upmu\)m-(A) and Inconel 1200 \(\upmu\)m-(B) porous metallic samples.