## 2 Materials and Methods
### Theory
#### 2.1.2 Determination of the Normal-Incidence Sound Absorption and Transmission Coefficient  
As mentioned above, the transfer function method (TFM) in accordance to ISO 10534-2 is the approach most often used for determination of the sound absorption coefficient in practice [29]. The method uses an impedance tube with a sound source connected to the one end of the tube and the test sample fixed in the tube at the other end. The complex sound pressure is measured by means of two flush-mounted fixed microphone positions in front of the specimen. The complex reflection coefficient \(r\) can then be determined by means of the acoustic wave field interference decomposition. The sound absorption coefficient (\(\alpha\)(-)) (Equation (5)) as well as other quantities like surface impedance (\(Z\) (Pa s/m)) and admittance (\(G\)(-)) of absorbing materials can be derived subsequently. The acoustic waves generated by the source below the cut-off frequency of the tube may be considered as plane waves. \[\alpha=\frac{W_{a}}{W_{in}}=\frac{W_{q}+W_{T}}{W_{in}}=1-|r\cdot r^{*}|=1-|r|^{ 2}=1-\left|\frac{\frac{p_{(x2)}}{p_{(x1)}}-e^{-jk_{0}s}}{e^{jk_{0}s}-\frac{p_{ (x2)}}{p_{(x1)}}}\right|^{2} \tag{5}\]  
where \(r^{*}\)(-) and \(s\) (m) are complex conjugate of the reflection coefficient and the distance between microphone positions 1 and 2, \(p_{(x1)}\), \(p_{(x2)}\) and \(k_{0}\) is sound pressure at coordinate \(x_{1}\) (m) and \(x_{2}\) (m) and wave number (m\({}^{-1}\)) (Figure 2). The measurement method proposed in this paper combines the laser doppler vibrometer and microphone measurement data (i.e., the surface displacement and known incident pressure \(p_{rms,1}\) at the plane at the surface of the specimen--specifically a membrane). The sound transmission and the absorption coefficient of the membrane are closely related. Whilst in Section 2.1.1 it was already detailed how the sound transmission \(nTL\) can be determined from sound pressure and vibration measurements, the absorption coefficient \(\alpha\) can also be determined in this way, as explained below. The sound absorption coefficient is defined as a ratio of energies (Equation (5)), where \(W_{a}\), \(W_{in}\), \(W_{q}\), \(W_{T}\) in (W) are, respectively, the absorbed sound power, incident sound power, dissipated sound power in the membrane and the transmitted sound power. Whereas the sound absorption in porous materials is caused by viscous and relaxation losses, in stretched membrane resonates it depends on the material properties, tension and the geometry of the membrane. If the excitation of the membrane is low (and non-linearities are neglected), the dissipation caused by dissipation of vibration energy (into heat) is also negligibly low. Therefore, in case of nonperforated membranes, the dissipation of energy out of the resonance region can be in most of cases neglected. The observed decrease in the sound reflection spectra is mainly related to the high sound transmission at the membrane resonance frequencies. By neglecting the dissipated energy in the membrane (which is often valid in the case of the single, thin membranes without internal dissipation), the absorption can be approximated by \(\tau\)(-), as defined by Equation (6):  
\[\tau=\frac{W_{T}}{W_{in}}\approx\frac{(\omega\cdotp_{0}\cdotc_{0}\cdotp\langle w \rangle)^{2}}{p_{i}^{2}} \tag{6}\]  
where \(W_{T}\) and \(W_{in}\) are the transmitted (obtained from the surface mobility at the receiving side of specimen) and incident sound power in (W) at the specimen surface and \(p_{i}\) is the sound pressure at the surface of incidence and can be expressed as (Equation (7)):  
\[p_{i}=\frac{p_{(x_{1})}e^{-j_{0}x_{1}}-p_{(x_{2})}e^{-j_{0}x_{2}}}{e^{-2j_{0}x _{1}}-e^{-2j_{0}x_{2}}} \tag{7}\]  
where \(p_{(x1)}\), \(p_{(x2)}\) and \(k_{0}\) is sound pressure at coordinate \(x_{1}\) (m) and \(x_{2}\) (m) and wave number (m\({}^{-1}\)), respectively. The pressure data \(p_{(x1)}\), \(p_{(x2)}\) can be obtained by the two microphones at the source side of the impedance tube.