## 2 Materials and Methods
### Theory
#### 2.1.1 Determination of Normal Incidence Sound Transmission Loss
\[nTL=-20 log_{10}\left|H_{32}\left(1-r_{2}r_{b}e^{2 jk_{0}D_{2}}\right)\frac{e^{jk_ {0}L_{1}}+r_{1}e^{-jk_{0}L_{1}}}{e^{-jk_{0}L_{2}}+r_{b}e^{jk_{0}L_{2}}}\right| \tag{2}\]  
where \(H_{32}\), \(D_{2}\), \(L_{1}\), \(L_{2}\) and \(j\) is the transfer function between microphones 2 and 3, the distance between the termination and the sample, the distance between microphone 2 and the sample on the source side and microphone 3 and the sample on the termination side, respectively, and imaginary unit (Figure 1). An interesting approach was recently developed by [PERSON] [57], who proposed the WD-based \(nTL\) method using only two upstream microphones (the so-called upstream tube wave field decomposition UWD). One of the methods that is used in this paper as a reference for the determination of \(nTL\) was derived from the work by [PERSON] [58]. Unlike other methods, it takes into account the multiple reflection form the back of the impedance tube. The method was developed originally for the sound transmission loss determination in ducts systems (Equation (3)). \[nTL=10 log_{10}\frac{S_{11}+S_{22}-2C_{12}cosk_{0}s+2Q_{12}sink_{0}s}{S_{33}+S_ {44}-2C_{34}cosk_{0}s+2Q_{34}sink_{0}s} \tag{3}\]  
where \(S_{11}\), \(S_{22}\), \(S_{33}\), and \(S_{44}\) are auto spectra (Pa\({}^{2}\)) of the total acoustic pressure at microphone positions 1, 2, 3 and 4 (Figure 1), \(Q_{12}\) and \(Q_{34}\) are imaginary parts and \(C_{12}\) and \(C_{34}\) are the real parts of pressure cross spectra between microphones 1 and 2 or 3 and 4 and \(s\) (m) is distance between microphone pairs 1 and 2 or 3 and 4, respectively. The method is precisely described in Reference [58]. Thus far, an overview of several of the most well-known methods for \(nTL\) determining has been mentioned. As mentioned earlier, motivated by the idea of [PERSON] et al. [5], the presented work wants to prove the theory that if the incident sound pressure and the vibration vector of membrane measured by means of the chosen vibrometry technique is known, the sound transmission loss as well as absorption can be determined. In the case of an impedance tube, the radiated sound power into the downstream part of the tube, or the sound pressure at a specific distance from the vibrating surface (depending on the termination), can be determined from the spatially averaged measured displacement of  
Figure 1: The scheme of the impedance tube with indication of all necessary variables to determine \(nTL\). Upstream tube (Left), downstream tube (Right). Where \(p_{(x1-4)}\) and Mic 1–4 denote the microphone positions, where the sound pressure is measured related to the \(x\)-coordinate, \(p_{irms}\) and \(p_{trms}\) are effective values of the incident and transmitted sound pressure at the surface of the specimen (derived from the measurement data \(p_{(x1-4)}\)), \(s\) is distance between microphone positions and \(L_{1}\), \(L_{2}\) are distances from of the microphone to the specimen surface. the vibrating surface. The normal component of the wall velocity is equal to the particle velocity at the wall surface in the case of the perpendicular plane wave excitation [59]. The measured specimen can be considered as a non-baffled piston, assuming a simple \(\rho\)-\(c\) impedance. Motivated by the work of [PERSON] [20] (hereinafter referred to as the Mobility-based Method--MM), the sound transmission loss can be subsequently determined as (Equation (4)):  
\[nTL=20 log_{10}\frac{p_{irms}}{|j\cdot\omega\cdot p_{0}\cdot c_{0}\cdot\langle w \rangle|} \tag{4}\]  
where \(<\)\(w\)\(>\) is the spatially averaged displacement measured at the specimen surface at the receiving side, \(p_{irms}\) (Pa) is the rms value of the incident sound pressure at the \(z\)-coordinate of the upstream side of the specimen, \(c_{0}\) is speed of the sound (m/s) and \(\rho_{0}\) density of the air (kg/m\({}^{3}\)).