## 3 Experiments and Simulations
### Test Methods
It is supposed that only plane waves propagate in the impedance tube shown in Figure 4, so that the acoustic pressure in the sound-source tube can be expressed as:  
\[p_{\rm s}(z)=P_{\rm t}\mathrm{e}^{i\mathrm{i}z}=P_{\rm r1}\mathrm{e}^{-i\mathrm{ k}z} \tag{5}\]  
Similarly, the acoustic pressure in the receiving tube is expressed as:  
\[p_{r}(z)=P_{\rm t}\mathrm{e}^{i\mathrm{k}(z-d)}=P_{\rm r2}\mathrm{e}^{-i\mathrm{ k}(z-d)} \tag{6}\]  
The amplitudes of the acoustic pressures \(P_{i}\), \(P_{r1}\), \(P_{t}\), and \(P_{r2}\) in the tube shown in Figure 4 can be obtained by simultaneously solving Equations (5) and (6) in terms of the acoustic pressures \(p_{1}\)-\(p_{4}\) measured in the impedance tube by the four microphones (1-4), giving:  
\[P_{i}=i\frac{p_{1}\mathrm{e}^{-i\mathrm{k}l_{1}}-p_{2}\mathrm{e}^{-i\mathrm{k }(l_{1}+s_{1})}}{2\sin ks_{1}} \tag{7}\]  
\[P_{r1}=i\frac{p_{2}\mathrm{e}^{i\mathrm{k}(l_{1}+s_{1})}-p_{1}\mathrm{e}^{i \mathrm{k}l_{1}}}{2\sin ks_{1}} \tag{8}\]  
\[P_{t}=i\frac{p_{3}\mathrm{e}^{i\mathrm{k}(l_{2}+s_{2})}-p_{4}\mathrm{e}^{i \mathrm{k}l_{2}}}{2\sin ks_{2}} \tag{9}\]  
\[P_{r2}=i\frac{p_{4}\mathrm{e}^{-i\mathrm{k}l_{2}}-p_{3}\mathrm{e}^{-i\mathrm{ k}(l_{2}+s_{2})}}{2\sin ks_{2}} \tag{10}\]  
We further suppose there is a linear relationship among the amplitudes \(P_{i}\), \(P_{r1}\), \(P_{t}\), and \(P_{r2}\), which can be expressed in the form:  
\[\left[ \begin{array}{c}P_{i}\\ P_{r1}\end{array} \right]=\mathbf{T}\left[ \begin{array}{c}P_{t}\\ P_{r2}\end{array} \right] \tag{11}\]  
where \(\mathbf{T}\) denotes the scattering matrix of the test sample:  
\[\mathbf{T}=\left[ \begin{array}{cc}t_{11}&t_{12}\\ t_{21}&t_{22}\end{array} \right] \tag{12}\]  
where \(t_{11}\), \(t_{12}\), \(t_{21}\), and \(t_{22}\) are the unknowns and represent the linear relationship between \(P_{i}\), \(P_{r1}\), and \(P_{t}\), \(P_{r2}\). They can be obtained by solving Equation (11) in terms of Equations (7)-(10). As can be seen in Equations (3) and (4), the STL value can be calculated by finding the ratio of the amplitudes of the transmitted acoustic pressure in the receiving tube and the incident acoustic pressure in the sound-source tube. If one attempts to solve Equation (11), however, it is generally found that no solution can be obtained based on results from a single experiment. This is because the reflected acoustic waves in the receiving tube will penetrate the specimen and return to the sound-source  
Figure 5: Schematic diagram showing the components of the impedance tube. tube. In order to overcome this problem, it is necessary to carry out two experiments using the same specimen, but two different acoustic ends. Theoretically, any two ends could be used but, generally speaking, the greater the difference in the acoustic properties of the two different ends that are used, the better the effect. At present, this kind of 'double-load' measurement technique is the one mainly used when measuring the STL values of materials using an impedance tube. The double-load method requires two kinds of boundary condition to be set at the tail end of the tube (e.g., condition '\(a\)' might correspond to having a back plate in position, and condition '\(b\)' to having no back plate). Under such circumstances, the expressions generated using Equation (11) for the two experiments can be combined into a single equation of the form:  
\[\left[ \begin{array}{c}p_{ia}\\ p_{r1a}\\ p_{ib}\\ p_{r1b}\end{array} \right]=\left[ \begin{array}{cccc}\text{t}_{11}&\text{t}_{ 12}&0&0\\ \text{t}_{21}&\text{t}_{22}&0&0\\ 0&0&\text{t}_{11}&\text{t}_{12}\\ 0&0&\text{t}_{21}&\text{t}_{22}\end{array} \right]\left[ \begin{array}{c}p_{ta }\\ p_{r2a}\\ p_{tb}\\ p_{r2b}\end{array} \right] \tag{13}\]  
Using Equation (13), the scattering matrix (**T**) of the test material can now be obtained.