Reflection

The frequency-dependent reflection coefficients \((R)\) are calculated directly from the total transfer matrix \((T_{t})\) of a multilayered structure. These coefficients are a measure of how much sound is reflected off the surface of a structure.

The coefficients can be calculated under both normal and diffuse sound field conditions. Under a normal incidence sound field, the sound impinges on the surface from a single, perpendicular angle. In the diffuse field case, the incident sound theoretically strikes the surface of the material from all possible angles -- though the acoustipy implementation defaults to angles between 0 and 79, as seen in literature on the topic.

The acoustipy implementation for both cases can be found here.

Normal Incidence

Starting from the total transfer matrix:

\[ T_{t} = \begin{bmatrix} T_{11} & T_{12}\\ T_{21} & T_{22} \\ \end{bmatrix} \]

First, the surface impedence \((Z_{s})\) is calculated:

\[ Z_{s} = \frac{T_{11}}{T_{21}} \]

Then the reflection coefficients are:

\[ R = \frac{Z_{s}-Z_{0}}{Z_{s}-Z_{0}} \]

where \(Z_{0}\) is the characteristic impedence of air:

\[ Z_{0} = \rho_{0} c_{0} \]

and \(\rho_{0}\) is the density of air and \(c_{0}\) is the speed of sound in air.

Diffuse Incidence

Under the diffuse sound field condition, the calculation of surface impedence \((Z_{s})\) is the same as the normal incidence condition.

The reflection coefficients at each angle are then:

\[ r = \frac{Z_{s}\cos(\theta)-Z_{0}}{Z_{s}\cos(\theta)+Z_{0}} \]

which yields a vector of shape \([f, \theta]\). To collapse this vector to shape \([f,1]\), Paris' formula is used as shown below.

\[ R = \frac{\sum r\cos(\theta)\sin(\theta)}{\sum \cos(\theta)\sin(\theta)} \]