JCAPL Model

The Johnson-Champoux-Allard-Pride-Lafarge model is an eight parameter model consisting of the static airflow resistivity \((\sigma)\), porosity \((\phi)\), tortuosity \((\tau)\), viscous characteristic length \((\Lambda)\), thermal characteristic length \((\Lambda^\prime)\), thermal permeability \((k_{0}^\prime)\), thermal tortuosity \((\alpha_{0}^\prime)\), and viscous tortuosity \((\alpha_{0})\).

The acoustipy implementation for the JCA, JCAL, and JCAPL models are all based on the implementation from APMR. The equations described below can be found in the _calc_dynamics method.

Dynamic Mass Density

\[ \tilde{\rho} = \frac{\rho_{0}\tilde{\alpha}(\omega)}{\phi} \]

\[ \tilde{\alpha}(\omega) = \tau\Bigg[1+\frac{\tilde{F}(\omega)}{j\bar{\omega}}\Bigg] \]

\[ \tilde{F}(\omega) = 1-P+P\sqrt{1+\frac{M}{2P^2}j\bar{\omega}} \]

\[ \bar{\omega} = \frac{{\omega}{\rho_{0}}{\tau}}{{\sigma}{\phi}} \]

\[ M = \frac{8{\eta}{\tau}}{{\sigma}{\phi}{\Lambda}^2} \]

\[ P = \frac{M}{4\Bigg(\frac{\alpha_{0}}{\tau}-1\Bigg)} \]

Dynamic Bulk Modulus

\[ \widetilde{K} = \frac{{\gamma}P_{0}}{{\phi}\tilde{\beta}(\omega)} \]

\[ \tilde{\beta}(\omega) = \gamma-(\gamma - 1)\Bigg[1+\frac{\tilde{F^{\prime}}(\omega)}{j\bar{\omega^\prime}}\Bigg]^{-1} \]

\[ \tilde{F^{\prime}}(\omega) = 1-P^{\prime}+P^{\prime}\sqrt{1+\frac{M^{\prime}}{2P^{\prime{2}}}j\bar{\omega^{\prime}}} \]

\[ \bar{\omega^{\prime}} = \frac{{\omega}{\rho_{0}}{Pr}{k_{0}^\prime}}{{\eta}{\phi}} \]

\[ M^{\prime} = \frac{8{k_{0}^\prime}}{{\phi}{\Lambda^{\prime{2}}}} \]

\[ P^{\prime} = \frac{M^\prime}{4\Bigg(\alpha_{0}^\prime-1\Bigg)} \]

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

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

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

Model Parameters:

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

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

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

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

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

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

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

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

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

Defining Other Symbols:

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

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

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

\[ \eta = \Bigg[Pa*s\Bigg]\tag{viscosity of air} \]

\[ \Pr = \Bigg[unitless\Bigg]\tag{Prandtl Number} \]