## 2 Materials and Methods
### NEAM Model  
Inside a porous material, the sound propagation can be described, in time domain, with Equations (1) and (2). Likewise, in frequency domain, with \(e^{i\omega t}\) notation, the equations convert to the following:  
\[i\omega\rho_{ef}\hat{V}_{m} =-\
abla\hat{P}_{m} \tag{9}\] \[i\omega C_{ef}\hat{P}_{m} =-\
abla\hat{V}_{m} \tag{10}\]  
Being \(i=\sqrt{-1}\), \(\rho_{ef}\) is the effective density \(\rho_{f}\alpha(\omega)\), \(C_{ef}\) is the effective compressibility \(\frac{\beta(\omega)}{k_{f}}\), and \(\hat{V}_{m}\) and \(\hat{P}_{m}\) are the frequency-dependent particle velocity and pressure, respectively. Following the work of [PERSON] et al. [21], the use of partial fraction expansion in the effective density and the effective compressibility takes the form of the following:  
\[\rho_{ef}(\omega)\approx\rho_{\infty}+\sum_{k=1}^{Np}\frac{A}{i\omega+a_{k}} \tag{11}\]  
\[C_{ef}(\omega)\approx C_{\infty}+\sum_{k=1}^{Nc}\frac{B}{i\omega+b_{k}} \tag{12}\]  
where \(\rho_{\infty}\) and \(C_{\infty}\) are the asymptotic values of density and compressibility when the frequency tends to infinity, \(A\) and \(B\) are real function constants, and \(a_{k}\) and \(b_{k}\) are the function poles. As stated by [PERSON] et al. [26], poles must be positive to get stable solutions. Substituting (11-12) into (9-10), with the addition of the auxiliary functions \(\varphi_{m}\) and \(\Phi_{m}\), the inverse Fourier transform yields [27],  
\[\rho_{\infty}\frac{\partial\mathbf{u_{m}}}{\partial t}+\sum_{k=1}^{Np}A\mathbf{u_{m}} -\sum_{k=1}^{Np}Aa_{k}\varphi_{m}=-\
abla p_{m} \tag{13}\]  
\[\frac{1}{\rho_{\infty}c_{\infty}^{2}}\frac{\partial p_{m}}{\partial t}+\sum_ {k=1}^{Nc}Bp_{m}-\sum_{k=1}^{Nc}Bb_{k}\Phi_{m}=-\
abla\mathbf{u_{m}} \tag{14}\]  
where \(c_{\infty}=1/\sqrt{\rho_{\infty}C_{\infty}}\) is the asymptotic sound speed, and \(\varphi_{m}\) and \(\Phi_{m}\) are related to the convolution integral of the particle velocity and the pressure as, i.e., along \(X\)-axis,  
\[\varphi_{m}(x,t)=\int_{0}^{t}\mathbf{u_{m}}\left(x,t^{{}^{\prime}}\right)e^{-a_{k }(t-t^{{}^{\prime}})}dt^{\prime} \tag{15}\]  
The convolution terms in Equations (5), (6), (13) and (14) demonstrate that the material response is not instantaneous and accounts for the dispersion effects. Under the hypothesis of an immediate action-response material behavior, the integral contribution may be neglected, so (13) and (14) reduce to  
\[\rho_{\infty}\frac{\partial\mathbf{u_{m}}}{\partial t}+\sum_{k=1}^{Np}A\mathbf{u_{m}} =-\
abla p_{m} \tag{16}\]  
\[\frac{1}{\rho_{\infty}c_{\infty}^{2}}\frac{\partial p_{m}}{\partial t}+\sum_ {k=1}^{Nc}Bp_{m}=-\
abla\mathbf{u_{m}} \tag{17}\]  
Either way, the discrete summation in Equations (16) and (17) should give a numerical real value, and thus, they may be replaced for a constant, frequency independent variable. In the same way, the asymptotic values of density and compressibility should have a real value that can be replaced with another numerical variable. Taking this hypothesis into account, the concepts of equivalent effective density, \(\Omega\), and equivalent effective compressibility, \(\Psi\), can be introduced, as in Equations (1) and (2). The proposed Numerical Equivalent Acoustic Material (NEAM) characterizes porous materials through real values of numerical tortuosity, \(\Omega_{A}\), and numerical viscosity, \(\Omega_{B}\), both describing the viscous and inertial interactions, with the density being a value that depends solely on the propagation medium. Thereby, no asymptotic values must be used. Understanding that fluid-structure thermal exchanges may exist, due to frame elasticity and therefore compression-expansion changes, and there should also be some variables that can describe such behavior. Accordingly, the numerical compressibility, \(\Psi_{A}\), and the numerical thermolable, \(\Psi_{B}\), could be introduced. As well, the static compressibility would just depend on the propagation medium.