# Green Shipping--Multifunctional Marine Scrubbers for Emission Control: Silencing Effect
## 2 Materials and Methods
### Transmission Loss Measurements and Calculations
#### 2.1.1 Experimental Techniques for Transmission Loss Measurements  
Three distinct techniques allow the TL to be measured for mufflers using an impedance tube: the decomposition method, the two-source method, and the two-load method. The decomposition method [19; 20] properly measures the acoustic properties in ducts [21] using a measurement set-up as depicted in Figure 1, where 1, 2, and 3 represent microphone locations. \(S_{ii}\), \(S_{rr}\), and \(S_{tt}\) represent the auto-spectra of the incident and the reflected and transmitted sound waves, respectively. The auto-spectra are defined as the product between a function and its complex conjugate; they provide the power distribution of the analyzed signal as a function of frequency. Considering a plane wave propagation and using two microphones at locations 1 and 2, the sound pressure can be decomposed into its incident and reflected waves. The microphone at location 3 directly measures the transmitted sound pressure. The decomposition theory provides an expression for the auto-spectrum \(S_{ii}\) of the incident wave, so the sound power of the incident and transmitted waves can be derived, and the TL can be expressed as follows [15]:  
\[\text{TL}=20\log_{10}\frac{p_{i}}{p_{t}}\ +\ 10\log_{10}\frac{S_{i}}{S_{o}}\ \ \, \tag{2}\]  
where \(p_{i}\) and \(p_{t}\) are the rms of the sound pressure of the incident and transmitted waves, respectively, and \(S_{i}\) and \(S_{o}\) are the cross-section area of the muffler inlet and outlet pipes, respectively. The major drawback of the decomposition method is that a fully anechoic termination is difficult to reproduce in experiments, affecting the reliability of the TL measurement. A muffler can also be modeled using the so-called four-pole parameters method [22]. The four parameters (\(A\), \(B\), \(C\), and \(D\)) relate the inlet pressure (\(p_{i}\)) and velocity (\(v_{i}\)) to the respective outlet values (\(p_{o}\), \(v_{o}\)), assuming a plane wave propagation. \[\left[ \begin{array}{c}p_{i}\\ v_{i}\end{array} \right]=\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]\left[ \begin{array}{c}p_{o}\\ v_{o}\end{array} \right]\ \, \tag{3}\]  
Two methods are available to calculate the four-pole parameters exploiting the transfer-matrix approach: the two-source method and the two-loads method [15]. When using the two-sources method, four microphones are needed. Consequently, the test has to be performed in two configurations to have the data required for TL parameter calculation according to Equation (4). In Figure 2 the two configurations (\(a\) and \(b\)) are reported. They are implemented by changing the source position while keeping the other pipe termination open. The numbers 1, 2, 3, and 4 represent the microphone locations. The muffler TL can be calculated as follows [16; 23]:  
\[\text{TL}=20\log_{10}\biggl{\{}\frac{1}{2}\biggl{|}A_{23}+\frac{B_{23}}{\rho c }+\rho c\ C_{23}+D_{23}\biggr{|}\biggr{\}}+10\log_{10}\frac{S_{i}}{S_{o}}\ \ \, \tag{4}\]  
where \(\rho\) is the fluid density, \(c\) is the speed of sound in the fluid medium, and \(A_{23}\), \(B_{23}\), \(C_{23}\), and \(D_{23}\) are the four-pole parameters between microphones 2 and 3. The specific expressions for the parameters are given in [15]. Figure 1: Measurement set-up according to the decomposition theory. Figure 3 shows the set-up of the two-load method; it requires two configurations and leads to the same results as the two-source process by changing the outlet impedance (expressed as load a and b in the picture) instead of the source location, resulting in the TL being calculated with Equation (4) [14]. Additionally, in Figure 3, numbers 1, 2, 3, and 4 denote the microphones locations. Different strategies allow the termination impedance to be changed: two terminations with different lengths, terminations with and without absorbing material, or even a closed and an open end. Of course, if the two loads are very similar, the results will be unstable.