## 2 Materials and Methods
### Transfer Matrix Method  
The transfer matrix method is a well-established method for predicting acoustic properties, including the absorption coefficients of laterally infinite multilayered assembly, considering plane wave propagation. The layers can be of different natures: solid, fluid, poroelastic, etc., and each is represented by a specific transfer matrix. The size of the matrix depends on the number of quantities that represent the acoustic field within the layer. For the poroelastic layer, the number of quantities is 6: the velocities and stresses of the porous frame (\(\
u_{1}\)\({}^{s}\), \(\
u_{3}\)\({}^{s}\), \(\sigma_{33}\)\({}^{s}\), and \(\sigma_{13}\)\({}^{s}\)) and of the fluid (\(\
u_{3}\)\({}^{f}\), and \(\sigma_{33}\)\({}^{f}\)). For the fluid layer, the transfer matrix is 2 \(\times\) 2 since the acoustic field is described with the pressure \(p\) and the component of the fluid velocity \(\
u_{3}\)\({}^{f}\). The transfer matrices relate the variables from both sides of the considered layer:  
\[\text{V(M}_{i})=[\text{T}]\text{V(M}_{i+1}) \tag{1}\]  
where \(i\) is the considered layer.  
To simulate the sound absorption phenomenon in the impedance tube, a porous layer of infinite extent was modelled in contact with a semi-infinite fluid on one side and a rigid wall on the other side. Sound plane waves are incident normally on the porous layer, as shown in Figure 4a. On the right side, the condition of rigid wall end was introduced, and the fluid-porous continuum condition was applied on the left side.  
Figure 4: (**a**) TMM model of porous layer with rigid backing; (**b**) TMM model with porous layer and included air gap between the wall and the porous material.  
Figure 3: Impedance tube for measuring the normal sound absorption coefficients at the acoustic laboratory of the University of Campania “[PERSON]”, Department of Architecture and Industrial Design.  
In order to investigate the effect of introducing an air layer between the wall and the porous material, as shown in Figure 4b, the fluid transfer matrix was introduced into the global transfer matrix using a suitable coupling transfer matrix and a hard wall termination was applied to the fluid layer.  
The surface impedance can be determined from the assembled transfer matrix; hence, the reflection coefficient and absorption coefficient. Details on the transfer matrix representation and the calculation of the acoustic indicators can be found in ref. [26].  
For the bulk modulus of the air saturating the porous medium, the Champoux-Allard model was adopted:  
\[K=\frac{P0}{1-\frac{\gamma-1}{\gamma\alpha^{\prime}(\omega)}} \tag{2}\]  
where \(\alpha^{\prime}(\omega)\) is  
\[\alpha^{\prime}(\omega)=1+\frac{8\eta}{j\omega B^{2}\Lambda^{\prime 2}\rho_{0} }\Bigg{[}1+j\frac{\omega B^{2}\rho_{0}}{\eta}\left(\frac{\Lambda^{\prime}}{4} \right)^{2}\Bigg{]}^{1/2} \tag{3}\]  
The model by [PERSON] et al. was used for the coefficient \(G(\omega)\), which is required to calculate the modified Biot's densities to obtain the complex wave numbers of the waves that propagate in a porous material [24]:  
\[G(\omega)=\left[1+j\omega\eta\rho_{0}\left(\frac{2\alpha_{\infty}}{\phi \Lambda\sigma}\right)^{2}\right]^{1/2} \tag{4}\]  
where P0 is the barometric pressure, \(\omega\) is the angular frequency, and \(\gamma\), \(\eta\), \(B^{2}\), \(\rho_{0}\) are specific heat ratio, dynamic viscosity, Prandtl number, and density of the saturating air, respectfully.  
Since the Young's modulus, \(E\), of the foams was not obtained, the rigid limit was considered, i.e., \(E\) was set sufficiently large so that the porous layer behaved as a rigid.