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 (see equation above), 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 the expression for $Z_{sp}$ and the low frequency approximations, the fractional term at the denominator of equation right above, named hereafter $A$, can be rewritten as:

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

From the expressions of $Z_cs$ and $k_cs$ 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 the expression of $Z_s$ 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 the equation right above 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 the equation expressing $\rho_{cs}$ 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 the equation above at the first degree in $\omega$ reads:

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

On the other hand, the real part of $\widetilde{\rho}_{cs}$ 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 third and fourth equations of this document.