Biot-Limp Model

The Biot-Limp model modifies the dynamic mass density \((\tilde{\rho}_{eq})\) determined via an equivalent fluid model such as: DB, DBM, JCA, JCAL, and JCAPL.

The acoustipy implementation follows from eq. 24 in Bécot and Jaouen.

\[ \frac{1}{\tilde{\rho}_{limp}} = \frac{1}{\phi\tilde{\rho}_{eq}}+\frac{\gamma^2}{\phi\tilde{\rho}} \]

where \(\gamma\) and \(\tilde{\rho}\) are defined as:

\[ \gamma = \frac{\rho_{0}}{\tilde{\rho}_{eq}}-1 \]

\[ \tilde{\rho} = \rho_{1}+\phi\rho_{0}-\frac{\rho_{0}^2}{\tilde{\rho}_{eq}} \]

The Add_Biot_Limp_Layer method then converts The modified dynamic mass density and bulk modulus to the characteristic impedence \((Z_{c})\) and wavenumber \((k_{c})\) for use in the layer transfer matrix.

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

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

Model Parameters:

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

\[ \rho_{1} = \Bigg[\frac{kg}{m^3}\Bigg]\tag{volumetric density} \]

The following parameters are used to find the dynamic mass density \((\tilde{\rho}_{eq})\) from the specified equivalent fluid model.

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

\[ \phi = \Bigg[unitless\Bigg]\tag{porosity} \]

\[ \tau = \Bigg[unitless\Bigg]\tag{tortuosity} \]

\[ \Lambda = \Bigg[{\mu}m\Bigg]\tag{viscous characteristic length} \]

\[ \Lambda^{\prime} = \Bigg[{\mu}m\Bigg]\tag{thermal characteristic length} \]

\[ k_{0}^{\prime} = \Bigg[m^2\Bigg]\tag{thermal permeability} \]

\[ \alpha_{0}^{\prime} = \Bigg[unitless\Bigg]\tag{thermal tortuosity} \]

\[ \alpha_{0} = \Bigg[unitless\Bigg]\tag{viscous tortuosity} \]

Defining Other Symbols:

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

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