Theory

At high frequency the viscous boundary layers in the pores are thin and the gas moves as an inviscid fluid around the fibres; its effective density tends to $\alpha_\infty\rho_0/\phi$ . A straight-pore material has $\alpha_\infty = 1$ ; fibrous packings are close to 1 (about 1.0-1.1), foams higher. In the JCA/JCAL density (Johnson et al., 1987) the tortuosity multiplies the inertial term and also appears in the viscous correction, equation (1), and in the estimate of Viscous characteristic length (JCA / JCAL) when that is blank, equation (2). It does not enter the bulk modulus.

Equations

\rho_{eq} = \frac{\alpha_\infty\rho_0}{\phi}\left[1 + \frac{R\,\phi}{i\omega\rho_0\alpha_\infty}\sqrt{1 + i\,\frac{4\alpha_\infty^2\eta\rho_0\omega}{R^2\Lambda^2\phi^2}}\right]

(1)

\Lambda = \sqrt{\frac{8\,\eta\,\alpha_\infty}{R\,\phi}}\ \ \text{(when the viscous length is blank)}

(2)

where	meaning	unit
$\alpha_\infty$	Tortuosity (this setting)	-
$\rho_{eq}$	Equivalent density of the packing	kg/m³
$\rho_0, \eta$	Density and viscosity of the local gas	kg/m³, Pa·s
$\phi$	Porosity	-
$R$	Flow resistivity at the local temperature	Pa·s/m²
$\Lambda$	Viscous characteristic length	m
$\omega$	Angular frequency	rad/s

Default and rule

Default: 1.0
Rule: 1.0 (straight pores), a close approximation for fibrous wools.
Source: The equivalent density follows Johnson et al. Johnson et al. (1987) in the form given by Atalla and Panneton Atalla and Panneton (2000). The typical values quoted above are indicative; they are not taken from a cited source.

Valid range ≥ 1; warning above 3

Values

$\alpha_\infty \ge 1$ (refused below 1).

High tortuosity

Above 3 the acoustic QA gate warns: high for a fibrous packing (typically 1.0-1.5).

Verified

Checked in a full acoustic run of a packed resonator (2026-11-11): every quality check passes; with estimated parameters the TL agrees with Miki within about 1 dB below 1 kHz and falls 7-9 dB below it at 2 kHz.

Tortuosity (JCA / JCAL)

Theory

Equations

Default and rule

Valid range ≥ 1; warning above 3

References 2

Theory

Equations

Default and rule

Valid range ≥ 1; warning above 3

Related 5

References 2