When a porous material has a stiffness, or a weight, of order of
magnitude much greater than the one of the air and if it is excited by
an acoustic loading at a frequency higher than the
phase decoupling frequency, the material
skeleton can be considered as rigid and motionless.
The displacement vector
$\underline{u}$
and the strain tensor
$\underline{\underline{\varepsilon}}$
are equal to zero. Only one compressional wave
($P_{2}$)
can propagate in the fluid phase of the material.
Under these displacements and strains conditions, the system of
equations describing the motion of a porous material reduced to one equation:
| $\Delta p + \omega^{2}\displaystyle{\frac{\widetilde{\rho}_{22}}{\widetilde{R}}}p = 0$ | (1) |
which is analogous to the Helmholtz equation and can be rewritten:
| $\Delta p + \omega^{2}\displaystyle{\frac{\widetilde{\rho}}{\widetilde{K}}}p = 0$ | (2) |
Three classes of models for the prediction of the sound propagation in acoustical porous materials can be listed.
The empirical models usually require to know a small number of parameters. They are very popular and still very used
in spite of the fact they have restrictive limits.
The analytical models are valid for porous materials with simple pore
morphologies: parallel cylindrical pores with a singular cross-section
(circular, square, triangular).
Note that the expression "equivalent fluid" is widely used to qualify porous materials with rigid and motionless skeletons. This confusing expression, which usually refers to the homogeneized medium and not its fluid phase, should not be employed for the sake of clarity.
The models described on APMR are the following.
- Empirical models
-
- Delany-Bazley [DB70] 1 parameter
- Delany-Bazley-Miki [DB70 Mik90a] 1 parameter
- Semi-phenomenological models
-
- Wilson model [Wil93] 4 parameters
- Johnson-Champoux-Allard model [JKD87,CA91] 5 parameters
- Johnson-Champoux-Allard-Lafarge model [JKD87, CA91, LLAT97] 6 parameters
- Johnson-Champoux-Allard-Pride-Lafarge model [JKD87, CA91, LLAT97] 8 parameters
References
[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
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