Macroscopic properties from microstructure
Once the micro-structure is defined it remains to compute the macroscopic properties of a material samples e.g. its sound absorption coefficient or sound transmission loss.
These macroscopic properties are derived from the information on the micro-structure by solving two linearized and frequency-dependent problems in the harmonic regime: (i) the Navier-Stokes equation with a local incompressibility condition to solve the dynamic visco-inertial problem and (ii) the heat equation to solve the dynamic thermal problem.
From a practical point of view, two quantities are computed from which all macro-properties can be deduced. These two characteristic quantities of the material are the dynamic mass density $\widetilde{\rho}_{eq}$ and the dynamic bulk modulus $\widetilde{K}_{eq}$.
Analytical calculation
For a small number of micro-structure morphologies (slit-like pores, cylindrical pores with circular, triangle or square cross-sections, cylindrical or spherical solid inclusions...) the visco-thermal problem can be solve analytically.
See our dedicated page on motionless skeleton models for expressions of $\widetilde{\rho}_{eq}$ and $\widetilde{K}_{eq}$ in such particular cases.
Direct numerical computation
For most micro-structures (in particular for the first two ones illustrated in this document), numerical computations are required.
The visco-thermal dissipative problem being frequency dependent, one computation is required for each studied frequency. Thus, this direct numerical approach is memory and time consumming.
Hybrid numerical computation
This approach allows to determine the acoustic behavior of a material in a wide frequency range while requiring only three numerical computations.
The original idea by is to estimate the parameters of semi-phenomenological models, for example the Johnson-Champoux-Allard-Pride-Lafarge model, from the micro-structure morphology and 3 asymptotic behaviors.