Variable X of the user constants

The dimensionless variable in which the user-defined packing model's power laws are written: either $\rho_0 f/R$ (the Delany-Bazley and Miki form) or $2 - \log_{10}(f/R)$ (the Komatsu form). Used only when Acoustic (poroacoustic) model of the packing is "user defined".

Last updated 2026-11-04

Theory

The empirical equivalent-fluid models write the characteristic impedance and propagation constant of a fibrous packing as power laws in one variable, equations (1)-(2). Delany and Bazley (Delany and Bazley, 1970) and Miki (Miki, 1990) used $f/\sigma$ , which the program writes with the gas density as $X = \rho_0 f/R$ (dimensionless); Komatsu (Komatsu, 2008) used $2 - \log_{10}(f/R)$ with $f/R$ in m³/kg, a variable without the gas density. Select the form in which the user constants User constant C1 ... C8 were fitted. With the Komatsu form the constants of a power law in a variable that falls with frequency have negative exponents (Komatsu: $C_2 = C_6 = -6.2$ , $C_4 = C_8 = -4.1$ ).

Equations

Z_c = \rho_0 c_0\left[1 + C_5 X^{-C_6} - i\,C_7 X^{-C_8}\right], \qquad k_c = \frac{\omega}{c_0}\left[1 + C_1 X^{-C_2} - i\,C_3 X^{-C_4}\right]

(1)

X = \frac{\rho_0 f}{R} \qquad \text{or} \qquad X = 2 - \log_{10}\frac{f}{R}

(2)

where	meaning	unit
$X$	Model variable (this setting)	-
$\rho_0, c_0$	Density and sound speed of the gas in the packing	kg/m³, m/s
$f$	Frequency	Hz
$R$	Flow resistivity at the local temperature	Pa·s/m²
$C_1 \dots C_8$	User constants	-

Default and rule

Default: ρ₀ f / R (Delany-Bazley, Miki form)
Rule: $\rho_0 f/R$ .
Source: The two forms are those of the program's built-in models, as given by Komatsu Komatsu (2008).

Valid range ρ₀ f / R | 2 − log₁₀(f / R)

No validity check

A user-defined model has no known validity range; the acoustic QA gate does not flag any frequency.

Komatsu form

$X = 2 - \log_{10}(f/R)$ becomes 0 or negative for $f/R \ge 100$ m³/kg, where a non-integer power is undefined; keep the sweep and resistivity inside the range of your fit.

Theory

Equations

Default and rule

Valid range ρ₀ f / R | 2 − log₁₀(f / R)

Related 4

References 3