# 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.