Wilson's model[Wil93], which involves only 4 parameters, is developed to match the middle frequency behavior (or transition behavior, when the viscous and thermal boundary layers are of the order of the pore size) while models by Johnson, Koplik & Dashen for visco-inertial effects and models by Champoux-Allard or Champoux-Allard-Lafarge for thermal effects are developed to match he low and high frequency behaviors of the material.

[Wil93] K. Wilson, Relaxation-matched modeling of propagation through porous media, including fractal pore structure, J. Acoust. Soc. Am. 94(2), 1993, pp. 1136-1145

With a $\exp(+j\omega t)$ time convention, Wilson's model writes:

\[ \widetilde{\rho}(\omega) = \rho_{\infty} \displaystyle{\frac{(1+j\omega\tau_{\textrm{vor}})^{1/2}} {(1+j\omega\tau_{\textrm{vor}})^{1/2}-1}} \] \[ \widetilde{K}(\omega) = K_{\infty} \displaystyle{\frac{(1+j\omega\tau_{\textrm{ent}})^{1/2}} {(1+j\omega\tau_{\textrm{ent}})^{1/2}+\gamma-1}} \]

where $\rho_{\infty}$, $\tau_{vor}$, $K_{\infty}$ and $\tau_{ent}$ are the 4 parameters of Wilson's model, $\gamma$ is the ratio of specific heats and $j$ is $+\sqrt{-1}$.

Some links between Wilson's model and the Johnson-Champoux-Allard-[Lafarge] ones can be found when identifying low or high frequency behaviors of Wilson's model with Johnson-Champoux-Allard-[Lafarge] ones (See [Wil93,P006,OP08]):

\[ \begin{align} \rho_\infty &=\displaystyle\frac{\rho_0 \alpha_\infty}{\phi} \\ \tau_{vor} &=\displaystyle\frac{2\rho_0\alpha_\infty}{\phi\sigma} \\ K_\infty &=\displaystyle\frac{\gamma P_0}{\phi} \end{align} \]

Panneton & Olny [P006,OP08] have also proposed experimental procedures to characterize Wilson's parameters.

[P006] R. Panneton, X. Olny,Acoustical determination of the parameters governing viscous dissipation in porous media, J. Acoust. Soc. Am. 119(4), 2006, pp. 2027-2040

[OP08] X. Olny, R. Panneton, Acoustical determination of the parameters governing thermal dissipation in porous media, J. Acoust. Soc. Am. 123(2), 2008, pp. 814-824