## 1 Introduction
Fluid parts affected by these exchanges can be estimated by the pore size ratio to the viscous skin depth thickness, \(\delta=\left(2\eta/\omega\rho_{0}\right)^{1/2}\), where \(\eta\) is the fluid shear viscosity and \(\omega\) the angular frequency. The skin depth, \(\delta\), also controls the limit for low- and high-frequency approximations [11]. In addition, the limit is given when the skin depth is much bigger than the pore radius. For frequencies under the low frequency limit, viscous losses are important all over the fluid, and the generalized governing equations, in time domain, are [5],  
\[\rho_{f}\alpha_{0}\frac{\partial\mathbf{u}_{\mathbf{m}}}{\partial t}+\frac{\eta\phi}{ k_{0}}\mathbf{u}_{\mathbf{m}}=-\
abla p_{m} \tag{3}\]  
\[\frac{\gamma}{k_{f}}\frac{\partial p_{m}}{\partial t}=-\
abla\mathbf{u}_{\mathbf{m}} \tag{4}\]  
where \(\alpha_{0}\) is the low frequency tortuosity approximation, \(\phi\) is the material porosity, \(k_{0}\) is the static permeability, and \(\gamma\) is the specific heats ratio \(\left(C_{p}/C_{v}\right)\). The propagation velocity can be calculated as \(c_{m}=\left(\rho_{m}\alpha_{0}\gamma/k_{f}\right)^{1/2}\). When frequency increases, the skin depth becomes small and the viscous effects only concentrate near the pore boundaries, and so they can be neglected [5]. For this case, the governing equations arise to the high-frequency approximation [6],  
\[\rho_{m}\alpha_{\infty}\frac{\partial\mathbf{u_{m}}}{\partial t}+2\frac{\rho_{m} \alpha_{\infty}}{\Lambda}\sqrt{\frac{\eta}{\pi\rho_{m}}}\int_{-\infty}^{t} \frac{\partial\mathbf{u_{m}}/\partial t^{\prime}}{\sqrt{t-t^{\prime}}}dt^{\prime}= -\
abla p_{m} \tag{5}\]  
\[\frac{1}{K_{f}}\frac{\partial p}{\partial t}+2\frac{\gamma-1}{K_{f}\Lambda^{ \prime}}\sqrt{\frac{\eta}{\pi Pr\rho_{m}}}\int_{-\infty}^{t}\frac{\partial p/ \partial t^{\prime}}{\sqrt{t-t^{\prime}}}dt^{\prime}=-\
abla\mathbf{u_{m}} \tag{6}\]  
where \(\alpha_{\infty}\) is the static tortuosity, \(\Lambda\) is the viscous characteristic length, \(\Lambda^{\prime}\) is the thermal characteristic length and Pr is the Prandtl number. The behavior described in Equations (5) and (6) is clearly frequency dependent and, due to the convolution term, renders it difficult to be taken into account in time domain models. Furthermore, in the literature, the number of porous material models available for application in a time domain is relatively low. Indeed, some analytical models exist for rigid frame porous materials, but their high computational-cost make them un-practical for application and implementation. Regardless of the complexity in implementing Equations (5) and (6) in the time domain, efforts have been made to reduce the requirements of such parameters. [PERSON] et al. [12], developed an analytical method in which the acoustical behavior of porous materials can be represented with just three parameters, being the porosity, the median pore size, and the standard deviation in the pore size. They also show that it is not necessary to use the six non-acoustical parameters given in the JCAL model to represent the performance of porous materials. For this reason, reducing the initial, unknown JCAL parameters would allow for the easiest improvement in FDTD implementations. Most FDTD models dealing with porous materials found in the literature are based on the Zwikker and Kosten (ZK) equations [13], which, in fact, are the simplest way to have a porous absorber without excessive computational requirements. [PERSON] et al. [14], analyzing porous materials, make use of a variable called \"aerodynamic drag parameter\" (\(\lambda\)), which, in their model, is the only factor of air flow resistance through the material. As can be seen in the reference, this methodology involves one more equation per dimension, which generates the need for more computational resources. [PERSON] et al. [15], points out the use of a boundary condition for the reflections caused by a porous material, achieving frequency-dependent impedance via ZK equations using a two-parameter model.