## 4 Results and Discussion
### Acoustic Attenuation along Exposed MDPE Pipe
Figure 8 further shows the frequency dependency of acoustic attenuation along the pipe based on the analytical modelling for the exposed MDPE pipe with the geometric and material properties in Table 1. It shows that acoustic attenuation along an exposed MDPE pipe increases with acoustic excitation frequency as predicted by Equations (1) and (2). Figure 8: Analytical predictions of the frequency dependency of acoustic signal attenuation along an exposed MDPE pipe. Figure 7: SNR (dB) of the measured acoustic signals at the four circumferential locations and at increasing distances along the exposed (empty) MDPE pipe. \begin{table}
\begin{tabular}{c c c} \hline
**Circumferential Location of** & **Measured Acoustic Attenuation** & **R2 Value** \\
**Acoustic Receivers** & **Using Linear Fit (dB/m)** & **R2 Value** \\ \hline
0 degrees & 0.8 & 82\% \\
90 degrees & 1.8 & 86\% \\
180 degrees & 0.7 & 72\% \\
270 degrees & 1.4 & 73\% \\ Average acoustic attenuation & \(1.2\pm 0.5\) dB/m & \\ \hline \end{tabular}
\end{table} 
Table 3: Average SNR (dB) at each circumferential position around and axial distance along the exposed (empty) MDPE pipe. For a water-filled pipe, a slight reduction (maximum of 0.02 dB/m) in acoustic attenuation occurs along the pipe at excitation frequencies above 60 Hz. This is understandable as the water within the pipe is stiffening the pipe wall (especially at higher frequencies), thus slightly reducing the acoustic attenuation (by reducing the acoustic wave number) along the pipe wall as noted by [24]. To evaluate the appropriateness of the existing analytical model, the results were compared with the experimental data. The acoustic excitation frequency and the elastic modulus of the MDPE pipe wall were required as input parameters for the analytical model. With a measured pipe wall temperature of 12.67 \(\pm\) 0.67 \({}^{\circ}\)C, the corresponding pipe wall elastic modulus was calculated as (8.36 \(\pm\) 0.10) \(\times\) 108 N/m\({}^{2}\)[30]. The analytical model predicted 0.06 dB/m attenuation at 38 Hz along the exposed MDPE pipe, which is a significantly lower value than the experimentally obtained average of 1.2 \(\pm\) 0.5 dB/m. This is not surprising as the analytical model only considers the pipe wall material loss factor as responsible for acoustic attenuation along the pipe. This demonstrates that the analytical model does not capture all the acoustic transmission mechanisms and, hence, a numerical model is needed to simulate this complex behaviour. For the numerical model, Figure 9a shows the predicted time domain response of the exposed MDPE pipe to an acoustic excitation with a frequency of 38 Hz (with acoustic measurement points at four circumferential positions and at 0, 2, 4 and 6 m along the pipe). From Figure 9a, the measured acoustic signal at 0 m from the acoustic transmitter is, as expected, dominated by high-frequency acoustic waves (due to the proximity to the acoustic transmitter) before being filtered out at increasing distances along the pipe. At 2, 4  
Figure 9: (**a**) Predicted time domain acoustic response and (**b**) predicted acoustic attenuation along the exposed MDPE pipe. and 6 m, the transmitted acoustic signal at 38 Hz is visible with a gradual attenuation at the increasing distances along the pipe. This acoustic signal attenuation can be calculated as  
\[Relative\ amplitude\ (\mathrm{dB})=20\times\mathrm{log}_{10}\frac{A_{x}}{A_{0}} \tag{5}\]  
where \(A_{0}\) (m/s\({}^{2}\)) and \(A_{x}\) (m/s\({}^{2}\)) are the acoustic signal amplitudes at 0 and x metres, respectively, from the acoustic transmitter. Using Equation (5), Figure 9b shows the relative acoustic amplitudes as well as the numerically predicted acoustic attenuation along the exposed MDPE pipe of 1.2 dB/m for all circumferential positions around the pipe. This result shows excellent agreement with the experimental results of 1.2 \(\pm\) 0.5 dB/m, indicating that the numerical model accurately predicts the acoustic signal attenuation along the exposed MDPE pipe.