Theory

The user-defined model has the form of the empirical models of Delany and Bazley (Delany and Bazley, 1970), Miki (Miki, 1990) and Komatsu (Komatsu, 2008): the characteristic impedance $Z_c$ and the propagation constant $k_c$ of the packing are power laws in one variable $X$ (Variable X of the user constants), equations (1)-(2). $C_1 \dots C_4$ belong to $k_c$ ( $C_1$ , $C_2$ its real part, $C_3$ , $C_4$ its imaginary part), $C_5 \dots C_8$ to $Z_c$ ( $C_5$ , $C_6$ real, $C_7$ , $C_8$ imaginary). From $Z_c$ and $k_c$ the program forms the equivalent density and bulk modulus that the acoustic solver gives the packing's elements, equation (3).

C1 is the coefficient of the real part of the propagation constant. It sets how much faster than in free gas the phase of a wave changes per metre in the packing (the real part of $k_c$ exceeds $\omega/c_0$ by $C_1 X^{-C_2}$ ): a larger $C_1$ means a slower sound speed in the packing.

Equations

Z_c = \rho_0 c_0\left[1 + C_5 X^{-C_6} - i\,C_7 X^{-C_8}\right]

(1)

k_c = \frac{\omega}{c_0}\left[1 + C_1 X^{-C_2} - i\,C_3 X^{-C_4}\right]

(2)

\rho_{eq} = \frac{Z_c k_c}{\omega}, \qquad K_{eq} = \frac{\omega Z_c}{k_c}

(3)

where	meaning	unit
$C_1$	This constant	-
$Z_c$	Characteristic impedance of the packing	Pa·s/m
$k_c$	Propagation constant (time dependence $e^{+i\omega t}$ )	1/m
$X$	$\rho_0 f/R$ or $2 - \log_{10}(f/R)$	-
$\rho_0, c_0$	Density and sound speed of the gas in the packing	kg/m³, m/s
$\omega$	Angular frequency	rad/s
$\rho_{eq}, K_{eq}$	Equivalent density and bulk modulus	kg/m³, Pa

Default and rule

Default: 0.122 (Miki's value)
Rule: 0.122: the user-defined set starts from Miki's constants. For comparison, C1 is 0.0978 in Delany-Bazley and 0.0004 in Komatsu (whose variable is $2 - \log_{10}(f/R)$ ).
Source: The model form and the built-in constant sets agree with Komatsu Komatsu (2008), Tables 1-2 and eqs. (15)-(18). User constants come from a fit to the measured $Z_c$ and $k_c$ of the material.

Valid range any real number; no validity range is checked

No validity range

The validity range of a user fit is not known, so the acoustic QA gate checks no frequency band. Keep the sweep inside the range of $X$ over which the constants were fitted.

Physical behaviour

Constants that make the real part of $Z_c$ or the decay of $k_c$ negative describe a material that creates energy; the passivity gate of the acoustic run (outgoing power must not exceed the incident power) can then fail.

Theory

Equations

Default and rule

Valid range any real number; no validity range is checked

Related 4

References 3