APMR

Acoustical Porous Material Recipes

Check the model with the Matlab scripts and data provided by the authors of this model.

From three independent parameters (including the open porosity $\phi$), the model developed by Horoshenkov et al. provides relations to compute the remaining parameters of a Johnson-Champoux-Allard-Lafarge (JCAL) model (which are no more considered as independent of each other).

The computations of the equivalent JCAL parameters are indicated below.

$\bullet$ the high frequency limit of the dynamic tortuosity :
\[
\alpha_\infty = \exp\Bigg( 4 \Big[\sigma_s ln(2)\Big]^2 \Bigg)
\]
$\bullet$ the static air-flow resistivity :
\[
\sigma = \eta/k_0
\]
$\eta$ being the dynamic viscosity of air and $k_0$ the static permeability :
\[
k_0 = \displaystyle\frac{\phi\bar{s}^2}{8\alpha_\infty} \exp \bigg( 6 \Big[\sigma_s ln(2)\Big]^2 \bigg)
\]
$\bullet$ the viscous characteristic length :
\[
\Lambda = \bar{s} \exp\bigg( -\displaystyle\frac{5}{2} \Big[\sigma_s ln(2)\Big]^2\bigg)
\]
$\bullet$ the thermal characteristic length :
\[
\Lambda' = \bar{s} \exp\bigg( \displaystyle\frac{3}{2} \Big[\sigma_s ln(2)\Big]^2\bigg)
\]
$\bullet$ the static thermal permeability :
\[
k_0' = \displaystyle\frac{\phi\bar{s}^2}{8\alpha_\infty} \exp \bigg( -6 \Big[\sigma_s ln(2)\Big]^2 \bigg)
\]
This model has been successfully compared to measurements on granular, foams and even fibrous materials.

[HGD16] K. V. Horoshenkov, J.-P. Groby, O. Dazel,*Asymptotic limits of some models for sound propagation in porous media and the assignment of the pore characteristic lengths*, J. Acoust. Soc. Am. 139 (5), 2016, pp. 2463–2474.

[HHG19] K. V. Horoshenkov, A. Hurrell, J.-P. Groby,*A three-parameter analytical model for the acoustical properties of porous media* J. Acoust. Soc. Am. 145 (4), 2019, pp. 2512–2517.

[HHG19] K. V. Horoshenkov, A. Hurrell, J.-P. Groby,

Note that, in [HGD16], the authors also propose Padé approximations for the computation of the dynamic mass density and dynamic bulk compressibility (the inverse of the dynamic bulk modulus).

The choice of the equations to compute the dynamic mass density or the dynamic bulk modulus does not make much of a difference to the prediction, except the Padé approximations correct mistakes for the low-frequency behaviors of the dynamic mass density and bulk modulus as discussed here.

[HGD16] K. V. Horoshenkov, J.-P. Groby, O. Dazel,*Asymptotic limits of some models for sound propagation in porous media and the assignment of the pore characteristic lengths*, J. Acoust. Soc. Am. 139 (5), 2016, pp. 2463–2474.