Transmission Loss

The frequency-dependent transmission coefficients \((\tau)\) are calculated directly from the total transfer matrix \((T_{t})\) of a multilayered structure. These coefficients are a measure of how much sound passes through 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 transmission coefficients can then be used to calculate transmission loss \((TL)\), which is also a frequency dependent metric.

The acoustipy implementation for both cases can be found here.

Normal Incidence

Starting from the total transfer matrix:

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

Then, the transmission coefficients \((\tau)\) are:

\[ \tau = \frac{2e^{jk_{0}t}}{T_{11}+\frac{T_{12}}{Z_{0}}+Z_{0}T_{21}+T_{22}} \]

where \(t\) is the total thickness of the structure and \(k_{0}\) is the wavenumber:

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

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

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

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

and \(\rho_{0}\) is the density of air.

Then the transmission loss is:

\[ TL = 10\log_{10}\frac{1}{|\tau|^2} \]

Diffuse Incidence

Under the diffuse sound field condition, the calculation of the transmission coefficients is:

\[ \tau = \frac{2e^{jk_{0}t}}{T_{11}+\frac{T_{12}\cos\theta}{Z_{0}}+\frac{Z_{0}T_{21}}{\cos\theta}+T_{22}} \]

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

\[ T = \frac{\sum \tau\cos(\theta)\sin(\theta)}{\sum \cos(\theta)\sin(\theta)} \]

the transmission loss is then calculated in the same manner as the normal incidence condition:

\[ TL = 10\log_{10}\frac{1}{|T|^2} \]