Delany-Bazley model
From a large number of measurements on fibrous materials with porosities close to 1.00, Delany and Bazley [DB70] have proposed empirical expressions for the values of the complex wave number $k$ ($\gamma/j$ in [DB70]) and characteristic impedance $Z_{c}$ for such materials:
|
$Z_{c} = \rho_{0} c_{0}
\Bigg[ 1+9.08 \left(10^3\frac{f}{\sigma}\right)^{-0.75}
- j 11.9 \left(10^3\frac{f}{\sigma}\right)^{-0.73} \Bigg]$ $k = \displaystyle{\frac{\omega}{c_{0}}} \Bigg[ 1 + 10.8 \left(10^3\frac{f}{\sigma}\right)^{-0.70} - j 10.3 \left(10^3\frac{f}{\sigma}\right)^{-0.59} \Bigg]$ |
(1) |
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:
| $0.01 < \displaystyle{\frac{f}{\sigma}} < 1.00$ | (2) |
This empirical model, which can provide reasonable estimations of $k$ and $Z_{c}$ in the approximative frequency range defined by eq. (2), is still widely used for its simplicity: only one parameter, $\sigma$, is needed to describe the acoustic behavior of a material.
Another fit of experimental data is given by Allard and Champoux [AC92]:
|
$Z_{c} = \rho_{0} c_{0}
\Bigg[ 1+0.0571 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.754}
- j 0.0870 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.732} \Bigg]$ $k = \displaystyle\frac{\omega}{c_{0}} \Bigg[ 1 + 0.0978 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.700} - j 0.1890 \left(\frac{\rho_{0}f}{\sigma}\right)^{-0.595} \Bigg]$ |
(3) |
with boundaries:
| $0.01 < \displaystyle{\frac{\rho_{0} f}{\sigma}} < 1.00$ | (4) |
The expressions of $k$ and $Z_{c}$ given by all these authors 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
$k$
and
$Z_{c}$
References
[DB70] Delany M. E. and Bazley E. N.,
Acoustical properties of fibrous absorbent materials, Applied
Acoustics 3, 1970, pp. 105-116
[AC92] Allard J.-F. and Champoux Y.,
New empirical equations for sound propagation in rigid frame
fibrous materials, J. Acoust. Soc. Am. 91(6), 1992, pp. 3346-3353
[Mik90] Miki Y.,
Acoustical properties of porous materials - Modifications of
Delany-Bazley models, J. Acoust. Soc. Jpn (E). 11(1), 1990, pp. 19-24
.
The