Delaney-Bazley Model

The Delaney-Bazley model is an empirical, one parameter model consisting of the static airflow resistivity \((\sigma)\) of the porous material.

The equations for the complex characteristic impedance \((Z_{c})\) and wavenumber \((k_{c})\) can be found below:

\[ Z_{c} = \rho_{0}c_{0}\Bigg[ 1+0.0571\left(\frac{\rho_{0}f}{\sigma}\right)^{-0.754}-j0.0870\left(\frac{\rho_{0}f}{\sigma}\right)^{-0.732}\Bigg] \]

\[ k_{c} = \frac{\omega}{c_{0}}\Bigg[1+0.0978\left(\frac{\rho_{0}f}{\sigma}\right)^{-0.700}-j0.1890\left(\frac{\rho_{0}f}{\sigma}\right)^{-0.595}\Bigg] \]

In the acoustipy implementation, the characteristic impedence and wavenumber are converted to the dynamic mass density \((\tilde{\rho})\) and dynamic bulk modulus \((\widetilde{K})\) via the equations below, as the Add_DB_Layer method calls the internal _calc_dynamics method for consistency with the other equivalent fluid models.

\[ \tilde{\rho} = \frac{Z_{c}k_{c}}{\omega} \]

\[ \widetilde{K} = \frac{{\omega}Z_{c}}{k_{c}} \]

The dynamic mass density and bulk modulus are then converted back to the characteristic impedence and wavenumber for use in the layer transfer matrix via:

\[ Z_{c} = \sqrt{\tilde{\rho}\widetilde{K}} \]

\[ k_{c} = {\omega}\sqrt{\frac{\tilde{\rho}}{\widetilde{K}}} \]

The model is valid in the frequency range defined below:

\[ 0.01 < {\frac{f}{\sigma}} < 1.00 \]

Model Parameters:

Using the following nomenclature --- Symbol = [Units] (name)

\[ \sigma = \Bigg[\frac{Pa*s}{m^2}\Bigg]\tag{static airflow resistivity} \]

Defining Other Symbols:

Using the following nomenclature --- Symbol = [Units] (name) or Symbol = equation = [Units] (name)

\[ \rho_{0} = \Bigg[\frac{kg}{m^3}\Bigg]\tag{air density} \]

\[ f = \Bigg[Hz\Bigg]\tag{linear frequency} \]

\[ \omega = 2{\pi}f = \Bigg[\frac{radians}{s}\Bigg]\tag{angular frequency} \]