Kirchhoff first developed a theory of sound propagation in tubes with circular cross-section accounting for both viscosity and thermal conductivity of the air. Zwikker & Kosten then restricted this theory to a narrow frequency range and a narrow radius range. It thus appears that visco-inertial and thermal effects can be treated separately using a complex mass density and a complex bulk modulus functions denoted as $\widetilde{\rho}$ and $\widetilde{K}$ respectively hereafter.
From Zwikker & Kosten work the wave equation for the acoustic pressure wave inside a tube is:
\[
\Delta p + \omega^{2}\displaystyle{\frac{\widetilde{\rho}}{\widetilde{K}}}p = 0
\]
This equation is analogous to the Helmholtz equation used to described the sound propagation in free air (without any dissipation). However, for porous media, the mass density $\widetilde{\rho}$ and the bulk modulus $\widetilde{K}$ are complex functions of the frequency and of the pore shape.
Motionless skeleton models aim at providing expressions of $\widetilde{\rho}$ and $\widetilde{K}$ for the acoustics frequency spectrum and for given pore shapes.
Obviously, the wave equation for motionless skeleton materials can be recovered from Biot's theory. For such a material, the displacement vector $\underline{u}$ of the skeleton and its strain tensor $\underline{\underline{\varepsilon}}$ are equal to zero. Under these displacements and strains conditions, the system of equations describing the motion of a porous material reduces to one equation (i.e. only one compressional wave: $P_{2}$, can propagate in the fluid phase):
\[
\Delta p + \omega^{2}\displaystyle{\frac{\widetilde{\rho}_{22}}{\widetilde{R}}}p = 0
\]
This last equation is equivalent to the first reported one.
Overview of the different motionless skeleton models
Three classes of models (i.e. expressions of $\widetilde{\rho}$ and $\widetilde{K}$ as functions of the frequency and of the pore shape) can be listed. Empirical models which usually require to know a small number of parameters (or information). They are very popular and still very used in spite of their restrictive limits. Analytical models are valid for porous materials with simple pore morphologies: parallel cylindrical pores with a singular cross-section (circular, square, triangular). Finally, semi-phenomenological models have been developed for more complicated pore morphologies. For these latter models, only the asymptotic behaviors are known. A behavior between these asymptotes is assumed without it has been mathematically proven.
The figure below shows the growing complexity of propagation models assuming a motionless skeleton since Zwikker & Kosten. A PDF version of this image is available for download.
Some of the models reported above are further described on APMR.
The references used above are reported here after.
[DB70] Delany M. E. and Bazley E. N.,
Acoustical properties of fibrous absorbent materials, Applied
Acoustics 3, 1970, pp. 105-116 [Mik90b] Miki Y.,
Acoustical properties of porous materials - Modifications of
Delany-Bazley models, J. Acoust. Soc. Jpn (E). 11(1), 1990, pp. 19-24 [Mik90a] Miki Y.,
Acoustical properties of porous materials - Generalizations of empirical models,
J. Acoust. Soc. Jpn (E). 11(1), 1990, pp. 25-28 [Wil93] K. Wilson, Relaxation-matched modeling of
propagation through porous media, including fractal pore
structure, J. Acoust. Soc. Am. 94(2), 1993, pp. 1136-1145 [JKD87] Johnson D. L., Koplik J. and Dashen R.,
Theory of dynamic permeability and tortuosity in fluid-saturated porous media,
J. Fluid Mech., 176, 1987, pp. 379-402 [CA91] Champoux Y. and Allard J.-F.,
Dynamic tortuosity and bulk modulus in air-saturated porous media,
J. Appl. Phys., 70, 1991, pp. 1975-1979 [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 [LLAT97] Lafarge D., Lemarinier P., Allard, J-F. and Tarnow V.,
Dynamic compressibility of air in porous structures at audible frequencies,
J. Acoust. Soc. Am. 102 (4), 1997, pp. 1995-2006
