Skip to content

Transfer Matrix Method

Three types of 2x2 transfer matrices are used to characterize porous materials \((T_{eq})\), air \((T_{air})\), and Maa microperforates \((T_{maa})\) in acoustipy.

The transfer matrices are also defined under 2 different sound field conditions -- normal and diffuse incidence.

Normal Incidence Matrices

\[ T_{eq} = \begin{bmatrix} \cos (k_{c}t) & jZ_{c}\sin(k_{c}t) \\ \frac{j}{Z_{c}}\sin(k_{c}t) & \cos (k_{c}t) \\ \end{bmatrix} \]
\[ T_{air} = \begin{bmatrix} \cos (k_{0}t) & jZ_{0}\sin(k_{0}t) \\ \frac{j}{Z_{0}}\sin(k_{0}t) & \cos (k_{0}t) \\ \end{bmatrix} \]
\[ T_{maa} = \begin{bmatrix} 1 & Z_{c}\\ 0 & 1 \\ \end{bmatrix} \]

where \(Z_{c}\) and \(k_{c}\) are the characteristic impedence and wavenumber of the material calculated by the ADD_XXX_Layer methods implemented in acoustipy. \(Z_{0}\) and \(k_{0}\) are the characteristic impedence and wavenumber of air and are determined via:

\[ Z_{0} = \rho_{0} c_{0} \]
\[ k_{0} = \frac{\omega}{c_{0}} \]

where \(\rho_{0}\) is the density of air, \(\omega\) is the angular frequency, and \(c_{0}\) is the speed of sound in air.

Diffuse Incidence Matrices

\[ T_{eq} = \begin{bmatrix} \cos (k_{x}t) & j\frac{Z_{c} k_{c}}{k_{x}}\sin(k_{x}t) \\ j \frac{j k_{x}}{Z_{c} k_{c}}\sin(k_{x}t) & \cos (k_{x}t) \\ \end{bmatrix} \]
\[ T_{air} = \begin{bmatrix} \cos (k_{x}t) & j\frac{Z_{c} k_{0}}{k_{x}}\sin(k_{x}t) \\ j \frac{k_{x}}{Z_{c} k_{0}}\sin(k_{x}t) & \cos (k_{x}t) \\ \end{bmatrix} \]
\[ T_{maa} = \begin{bmatrix} 1 & Z_{c}\cos(\theta)\\ 0 & 1 \\ \end{bmatrix} \]

where \(k_{x}\) is:

\[ k_{x} = \sqrt{k_{c}-k_{0}\sin(\theta)} \]

and \(\theta\) is the angle of incidence.

To obtain the total transfer matrix \((T_{t})\) of a multilayered structure, matrix multiplication is performed using the individual transfer matrices -- starting with the layer closest to the incident sound.

The illustration below is an example of a resistive screen on the face of a porous material, backed by a layer of air.

This is represented by:

\[ T_{t} = T_{screen} \times T_{porous} \times T_{air} \]