## 3 Numerical Modelling and Analysis of Sound Absorption Properties
Figure 5 plots the numerical predictions of the normal incidence sound absorption coefficient against resonant frequency for bottleneck-type structures, as well as for glass  
Figure 4: (**a**) Three-dimensional and (**b**) two-dimensional half-tube model representations of total acoustic pressure field (Pa) computed for 40 mm thick hard-backed porous layer at a resonant frequency of 6551.3 Hz. -wool fibre (GWF) and monomodal aluminium foam (ALF), which are also shown in Figure 1. Table 2 contains the pore-structure related input parameters, which were obtained via microscale simulation, as described earlier. These plots illustrate the numerical applicability of the Wilson equivalent fluid model to predict the ability to enhance the vibroacoustic behaviour of bottleneck-type structures by assessing the effect of key pore-structure-related parameters. To perform these simulations, property parameters such as dynamic tortuosity, porosity, and permeability (largely dictated by pore openings and sizes) must be defined and varied within the range realistically possible during foam processing. Figure 5a,b show the influence of different capillary radius (\(r_{w}\)) on the normal incidence sound absorption spectra for bottleneck-type structures with monomodal pores, typically the L-L and S-S structure types. As shown in Table 2, increasing the capillary radius between the connecting pores in the sphere-packing model increases the pore openings, the permeability, and a slight increase in porosity. With increasing porosity, there is less structural content per unit volume in the material, which results in a consistent decrease of their sound absorption properties and a shift in their sound absorption spectrum to frequency maxima as capillary radius increases. Table 2 shows that the highest characteristics of sound absorption were observed for L-L structures created at a 10 \(\upmu\)m capillary radius characterised by a mean pore opening radius of 141.6 \(\upmu\)m, a porosity of 63.4% and a permeability of 1.03 \(\times\) 10\({}^{-9}\) m\({}^{2}\), respectively. Figure 5b, however, illustrates that the sound absorption property of the S-S structure created at a 10 \(\upmu\)m capillary radius was observably poor, a result of the very small pore opening radius (74.3 \(\upmu\)m\(\sim\)created by small pores), which results in greater reflections at audible frequencies. Table 2 and Figure 5b indicate that the S-S structures' sound absorption properties are improved by increasing their capillary radius, thereby extending the mean opening radius (\(r_{w}\)) to a range between 110 and 130 \(\upmu\)m.  
Figure 5: Plots of numerically predicted normal incidence sound absorption spectra for virtually-generated macroporous structures characterised by: (**a**) monomodal large-sized pores (2.0-2.5 mm/L-L) for capillary radius between 10 and 80 \(\upmu\)m, (**b**) monomodal small-sized pores (0.5–1.0 mm/S-S) for capillary radius between 10 and 80 \(\upmu\)m, (**c**) monomodal and bimodally distributed pores generated at 10 \(\upmu\)m capillary radius and (**d**) bimodal (50% volume S-S) for capillary radius between 10 and 80 \(\upmu\)m, against resonant frequency,\(f\) [Hz] at 40 mm porous layer thickness. As illustrated in Figure 5a,b, the use of these selected pore-structure-related parameters improves only their quarter wavelength hard-backed layer resonance peak in absorption (\(A_{p}\)) and shifts their absorption spectra toward frequency minima, but at the downside of dips in the characterisation of sound absorption spectra for frequencies above 1.8 kHz. Therefore, the creation of bimodal structures increases the number of pore openings (coordination number), the available pore space (\(\varepsilon\)), and reduces the pore-nonuniformity (tortuosity) of the structure. As a result, the absorption bandwidth of the materials flattens out and the sound absorption properties improve at these high frequencies, as shown in Figure 5c,d.