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:
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:
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]:
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.
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