At low frequencies: $|k_{cs}h| \ll 1$ and $|k_{ca}d| \ll 1$ so that $\tan(k_{cs}h)$ and $\tan(k_{ca}d)$ reduce at the first order to $k_{cs}h$ and $k_{ca}d$ respectively. An approximate value of the frequency above which these conditions are no more verified can be obtained replacing $k_{cs}$ and $k_{ca}$ with the wave number in free air. At such low frequencies and away from frequency bands around acoustic cavity resonances given by equation (\ref{eq.f_cavity_resonances}), the normal surface impedance of the system can be approximated with:

\begin{equation}
Z_{s} \simeq Z_{cs} \frac{-jZ_{sp}+Z_{cs}k_{cs}h}{-jZ_{cs} \left( 1 + \displaystyle\frac{jZ_{sp}k_{cs}h}{Z_{cs}} \right)}
\label{eq.Zs_intermediate}
\end{equation}

Making use of equation (\ref{eq.Zsp}) and the low frequency approximations, the fractional term at the denominator of equation (\ref{eq.Zs_intermediate}), named hereafter $A$, can be rewritten as:

\begin{equation}
A = \frac{Z_{ca}k_{cs}h}{k_{ca}Z_{cs}d}
\end{equation}

From equations (\ref{eq.Zc}) and (\ref{eq.kc}) which are general expressions and can be applied to $Z_{cs}$ as well as $Z_{ca}$, $A$ reads:

\begin{equation}
A = \frac{K_{ca} h}{K_{cs}d}
\end{equation}

The asymptotic limits for low and high frequencies of the bulk modulus of the screen $K_{cs}$ are $P_0 / \phi$ and $\gamma P_0 / \phi$ respectively ($\gamma$ is the ratio of the specific heats in air and $P_0$ refers to the static atmospheric pressure). These limits are respectively $P_0$ and $\gamma P_0$ for the bulk modulus of the air in the cavity $K_{ca}$. Since the porosity of the screen $\phi$ is smaller than 1, $K_{cs}$ is of the same order of magnitude or large compared to $K_{ca}$. Assuming the thickness of the screen is much smaller than the thickness of the air cavity ($h \ll d$), $A$ is finally found to be negligible compared to 1 and equation (\ref{eq.Zs_exact}) can be rewritten as:

\begin{equation}
Z_{s} \simeq Z_{sp} + j\omega\widetilde{\rho}_{cs}h
\label{eq.Zs_approximation}
\end{equation}

where $\omega$ denotes the angular frequency of the incident acoustic wave.

From equation (\ref{eq.Zs_approximation}) we deduced the expression of the screen dynamic mass density $\widetilde{\rho}_{cs}$:

\begin{equation}
\widetilde{\rho}_{cs} \simeq \frac{Z_{s}-Z_{sp}}{j\omega h}
\label{eq.rho_cs_from_Z}
\end{equation}

The meaning of $\widetilde{\rho}_{cs}$ is the same as for porous media: it represents the apparent mass density of the air in the screen. This value differs from its static real value of 1.2 kg.m$^{-3}$ due to visco-inertial effects inside the screen and due to the flow distortion effects on both sides of the screen.

When $Z_s$ is measured, all quantities on the right hand side of equation (\ref{eq.rho_cs_from_Z}) are known. Thus $\widetilde{\rho}_{cs}$ can be assessed.

The general expression of $\widetilde{\rho}_{cs}$ given by Johnson et al. [JKD87] is:

\begin{equation}
\widetilde{\rho}_{cs}(\omega) =
\frac{\alpha_{\infty}\rho_{0}}{\phi}
\left[ 1 - j\frac{\sigma\phi}
{\omega\rho_{0}\alpha_{\infty}}
\sqrt{ 1 + j\frac{4\alpha_{\infty}^{2}\eta\rho_{0}\omega}
{\sigma^{2}\Lambda^{2}\phi^{2}}
} \ \right] \ .
\label{eq.rho_cs_from_JCA}
\end{equation}

The low frequency expression of $\widetilde{\rho}_{cs}$ obtained as the Taylor series of equation (\ref{eq.rho_cs_from_JCA}) at the first degree in $\omega$ reads:

\begin{equation}
\widetilde{\rho}_{cs}(\omega) = \frac{\alpha_{\infty}\rho_{0}}{\phi}
\left( 1 + \frac{2\alpha_{\infty}\eta}{\sigma\Lambda^{2}\phi}\right)-j\frac{\sigma}{\omega}
\end{equation}

which could also be re-written:

\begin{equation}
\widetilde{\rho}_{cs}(\omega) = \frac{\alpha_{\infty}\rho_{0}}{\phi}
\left( 1 + \frac{\alpha_{\infty}}{4}\right)-j\frac{\sigma}{\omega}
\label{eq.rho_cs_from_JCA_at_low_freq}
\end{equation}

using the fact that $\sigma=8\eta/(\phi r^{2})$ and $\Lambda=r$ for the perforation geometry assumed.

On the one hand, it appears in equation (\ref{eq.rho_cs_from_JCA_at_low_freq}) that the imaginary part of $\widetilde{\rho}_{cs}$, $\textrm{Im}(\widetilde{\rho}_{cs})$, can be used to estimate the static air flow resistivity $\sigma$:

\begin{equation}
\sigma = -\omega \ \textrm{Im}(\widetilde{\rho}_{cs})
\label{eq.sigma_from_rho}
\end{equation}

On the other hand, the real part of (\ref{eq.rho_cs_from_JCA_at_low_freq}) is given by:

\begin{equation}
\textrm{Re}(\widetilde{\rho}_{cs}) =
\frac{\alpha_{\infty}\rho_{0}}{\phi}\left( 1 + \frac{\alpha_{\infty}}{4}\right)
\label{eq.real_part_of_rho_cs}
\end{equation}

with $\alpha_{\infty}$ given by equations (\ref{eq.alpha_infty_from_AS07}) and (\ref{eq.varepsilon_from_Ingard}).

Equation (\ref{eq.real_part_of_rho_cs}) can be re-written as a polynomial expression of order 6 in $E=\sqrt{\displaystyle\phi}$ replacing $\alpha_{\infty}$ by its expression as a function of $\phi$ and $r$. $r$ is then replaced by its expression as a function of $\phi$ and $\sigma$: $r=\sqrt{8\eta/(\sigma\phi)}$. Indeed $\sigma$ is a parameter which can be directly measured (see ISO 9053 \cite{ISO9053}) or estimated from equation (\ref{eq.sigma_from_rho}). The possible values for $E$ are then retrieved as roots of the polynomial. Only the real positive value contained in the open interval $]0 - 1[$ is physical. Finally, $\phi$ is calculated from the value of $E$ as $\phi = E^{2}$. It should be highlighted here than among the numerous characterizations done by the authors only one admissible solution was found per screen. In the case of multiple admissible solutions, only one porosity value would correctly predict the acoustical behavior of the screen and in particular the normal sound absorption of the screen backed by an air gap. In this latter case, optical measurements can also be done to discriminate between possible solutions. For optical measurements of the porosity, a fine photography of the facing micro-structure is first obtained using e.g a magnifying glass. A color threshold is then applied to convert the color picture to a binary image (black and white image, for instance black for air and white for the facing skeleton ). The porosity is then deduced from the ratio of the black-pixel count over the total-pixel count.

To check the consistency of characterization results one might compare the simulations of the surface properties for the screen and air gap system to the measurements.

The surface impedance of the screen + air gap cavity system, $Z_s$, can be recovered from estimated parameters $\phi$ and $r$ using its low frequency approximation (cf. equation (\ref{eq.Zs_approximation})) and making use of equations (\ref{eq.rho_cs_from_JCA_at_low_freq}), (\ref{eq.alpha_infty_from_AS07}) and (\ref{eq.varepsilon_from_Ingard}):

\begin{equation}
Z_s \simeq Z_{sp} + j\omega\frac{\alpha_{\infty}\rho_{0}}{\phi}\left(1+\frac{\alpha_{\infty}}{4}\right)h + \sigma h
\end{equation}