# Mathematical expressions

Using the $+j\omega t$ time convention, expressions of the characteristic impedance and the wave number obtained by Delany and Bazley are:

APMR

Acoustical Porous Material Recipes

Using the $+j\omega t$ time convention, expressions of the characteristic impedance and the wave number obtained by Delany and Bazley are:

\[
Z_{c} = \rho_{0} c_{0}
\Bigg[ 1+9.08 \left(10^3\frac{f}{\sigma}\right)^{-0.75}
- j 11.9 \left(10^3\frac{f}{\sigma}\right)^{-0.73} \Bigg]
\]

\[
k = \displaystyle{\frac{\omega}{c_{0}}}
\Bigg[ 1 + 10.8 \left(10^3\frac{f}{\sigma}\right)^{-0.70}
- j 10.3 \left(10^3\frac{f}{\sigma}\right)^{-0.59} \Bigg]
\]

where $\rho_{0}$ and $c_{0}$ are the density of air and the sound speed in air,$\omega = 2\pi f$ is the angular frequency and $\sigma$ is the static air flow resistivity in the wave direction of propagation (expressed in N.m^{-4}.s).

Boundaries, proposed by the authors, for the validity of these power law expressions are:

\[
0.01 < \displaystyle{\frac{f}{\sigma}} < 1.00
\]

This empirical model, which can provide reasonable estimations of $k$ and $Z_{c}$ in the approximative frequency range defined above is still widely used for its simplicity: only one parameter, $\sigma$, is needed to describe the acoustic behavior of a material.

A previous fit of the experimental data [DB69] is later used by Allard and Champoux [AC92]:

\[
Z_{c} = \rho_{0} c_{0}
\Bigg[ 1+0.0571 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.754}
- j 0.0870 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.732} \Bigg]
\]

\[
k = \displaystyle\frac{\omega}{c_{0}}
\Bigg[ 1 + 0.0978 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.700}
- j 0.1890 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.595} \Bigg]
\]

with the same boundaries as above.

Both expressions of $k$ and $Z_{c}$ give very close results.

In the case of multiple layers, Miki [Mik90] noticed that the real part of the surface impedance when computed with the Delany-Bazley model sometimes becomes negative at low frequencies denoting a non-physical result. From Delany and Bazley measurements data, Miki proposes modifications of the expressions above to correct the surface impedance behavior.

It is highly recommended to use the revised expressions reported by Miki to compute $\textcolor{white}{k}$ and $\textcolor{white}{Z_{c}}$

Follow the link comparison_DB70_DBM90.m to download a Matlab/GNU Octave script which compares the model by Delany-Bazley and the model by Delany-Bazley-Miki:

% comparison of models by Delany-Bazley % and Delany-Bazley-Miki % % M. E. Delany and E. N. Bazley, % Acoustical properties of fibrous absorbent materials, % Applied Acoustics (3), 1970, pp. 105-116 % % Y. Miki % Acoustical properties of porous materials % - modifications of Delany-Bazley models - % J. Acoust. Soc. Jpn. (E), 11 (1), 1990, pp. 19-24 % % Copyleft 2006 luc.jaouen@matelys.com % cf. APMR on the web, % PropagationModels/MotionlessSkeleton/DelanyBazleyMikiModel.html % for more information