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.

$$\frac{1}{{\tilde{\rho }}_{limp}}=\frac{1}{\varphi {\tilde{\rho }}_{eq}}+\frac{{\gamma }^2}{\varphi \tilde{\rho }}$$

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

$$\gamma =\frac{{\rho }_0}{{\tilde{\rho }}_{eq}}-1$$

$$\tilde{\rho }={\rho }_1+\varphi {\rho }_0-\frac{{\rho }_0^2}{{\tilde{\rho }}_{eq}}$$

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.

$$Z_c=\sqrt{{\tilde{\rho }}_{limp}\tilde{K}}$$

$$k_c=\omega \sqrt{\frac{{\tilde{\rho }}_{limp}}{\tilde{K}}}$$

Model Parameters:

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

$$\begin{array}{}\text{(volumetric density)}& {\rho }_1=[\frac{kg}{m^3}]\end{array}$$

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

$$\begin{array}{}\text{(static airflow resistivity)}& \sigma =[\frac{Pa\ast s}{m^2}]\end{array}$$

$$\begin{array}{}\text{(porosity)}& \varphi =[unitless]\end{array}$$

$$\begin{array}{}\text{(tortuosity)}& \tau =[unitless]\end{array}$$

$$\begin{array}{}\text{(viscous characteristic length)}& \mathrm{\Lambda }=[\mu m]\end{array}$$

$$\begin{array}{}\text{(thermal characteristic length)}& {\mathrm{\Lambda }}^{\prime }=[\mu m]\end{array}$$

$$\begin{array}{}\text{(thermal permeability)}& k_0^{\prime }=[m^2]\end{array}$$

$$\begin{array}{}\text{(thermal tortuosity)}& {\alpha }_0^{\prime }=[unitless]\end{array}$$

$$\begin{array}{}\text{(viscous tortuosity)}& {\alpha }_0=[unitless]\end{array}$$

Defining Other Symbols:

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

$$\begin{array}{}\text{(air density)}& {\rho }_0=[\frac{kg}{m^3}]\end{array}$$