## 4 Experimental Verification  
### Test Method  
The two-load method is easy to operate because it requires only a change in resistance at the outlet of the muffler without the need to adjust the sound source position at the inlet. Hence, this method is used in the experiment for this section. The experiment incorporated two distinct impedance boundary conditions at the muffler's exhaust outlet: an open-end condition and a condition with a fully anechoic termination. Corresponding equation  
Figure 9: Schematic diagram of fixed constraints.  
sets were established through the two conditions. Subsequently, the outcomes from these systems were introduced, enabling the determination of the four-pole parameters and the subsequent derivation of the transmission loss [23]. The test principle of the two-load method is illustrated in Figure 10.  
In the two-load test method shown in Figure 10, the relationship between the acoustic waves at both ends of the muffler is expressed as follows:  
\[\begin{bmatrix}p_{i}\\ p_{r}\end{bmatrix}=\left[\begin{array}{cc}T_{11}&T_{12}\\ T_{21}&T_{22}\end{array}\right]\begin{bmatrix}p_{i}\\ p_{r^{\prime}}\end{bmatrix} \tag{41}\]  
where \(p_{i}\) and \(p_{r}\) on the left represent the upstream incident wave and reflected wave, respectively. \(p_{t}\) and \(p_{r^{\prime}}\) represent the downstream transmitted wave and reflected wave, respectively. \(T_{11}\), \(T_{12}\), \(T_{21}\), and \(T_{22}\) are the quadrupole parameters of the transmission matrix.  
According to the definition of transmission loss and the test principle of the two-load method, the outlet of the muffler must be a non-reflective end, meaning that \(p_{r^{\prime}}\) = 0. Therefore, the transmission loss using the two-load method can be expressed as follows:  
\[TL=20\log_{10}\left(\left.\frac{|p_{i}|}{|p_{t}|}\right|_{p_{r^{\prime}}=0} \right)=20\log_{10}|T_{11}| \tag{42}\]  
As seen from Equation (42), under general test conditions, it is difficult to achieve a non-reflective end downstream of the muffler. Therefore, establishing two sets of equations by varying the loads at the outlet of the muffler is necessary to solve for \(T_{11}\) in Equation (42).  
For Figure 10a, where the outlet is open, the equation set is  
\[\begin{bmatrix}p_{ia}\\ p_{rb}\end{bmatrix}=\begin{bmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{bmatrix}\begin{bmatrix}p_{ta}\\ p_{r^{\prime}b}\end{bmatrix} \tag{43}\]  
For Figure 10b, where the outlet is a total silenced end, the equation set is as follows:  
\[\begin{bmatrix}p_{tb}\\ p_{rb}\end{bmatrix}=\begin{bmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{bmatrix}\begin{bmatrix}p_{tb}\\ p_{r^{\prime}b}\end{bmatrix} \tag{44}\]  
Figure 10: Diagram of the two-load method. (**a**) Without a silenced end, (**b**) With a silenced end.  
A combination of Equations (43) and (44) results in the following equation:  
\[T_{11}=\frac{p_{ia}p_{r^{b}}-p_{ib}p_{r^{\prime}a}}{p_{ia}p_{r^{\prime}b}-p_{tb}p_{ r^{\prime}a}} \tag{45}\]  
where subscripts \(a\) and \(b\) of \(p\) represent the two different terminal loads of the muffler. During the two-load test, two microphones are placed in front of and at the back of the muffler to separate the incident and reflected waves. The sound pressure signals measured by the four microphones can be used to distinguish between the sound pressures of the advancing wave and the reflected wave by changing the loads at the muffler outlet. These measurements are then substituted into Equation (45) to calculate \(T_{11}\). Finally, the transmission loss of the muffler can be determined using \(T_{11}\)[24; 25].