# Estimation of the Noise Source Level of a Commercial Ship Using On-Board Pressure Sensors
## 2 Method and Validation
### Validation of the Method
For the hull pressure sensors, the result at the \"C0\" position were plotted. From the graphs, it can be seen that the result above 200 Hz can be used without correction. However, the TVR of ITC-1032 is uncalibrated at frequencies below 2 kHz. Therefore, the result at frequencies above 2 kHz is valid. It is also seen that the shaft rate components and the blade rate components have a strong effect on the spectrum below 2 kHz. These components may not be estimated well, even if the transducer covers this frequency range, as monopole-like sound propagation is assumed. The cavitation noise is dominant above 2 kHz, where a smooth curve is seen. The peaks around 20 kHz are the singing noise caused by structural resonances at the trailing edge of the propeller blades. \begin{table}
\begin{tabular}{c c c c} \hline
**Condition** & \(V\) **(m/s)** & \(\sigma_{n,0.7R}\) & \(n\) **(rps)** \\ \hline C1 & 7.0 & 3.06 & 44.04 \\ C2 & 7.0 & 2.82 & 44.16 \\ C3 & 7.0 & 2.61 & 44.30 \\ \hline \end{tabular}
\end{table} 
Table 2: Model test conditions. Figure 8: Measured transfer function between the source and sensor in the tunnel. To calculate the source level, a hexahedral source region was set around the propeller, which is shown in Figure 10. The ranges of \(y\) and \(z\) are defined symmetrically with respect to the propeller center, from \(-0.6D\) to \(0.6D\), where \(D\) is the propeller diameter. The range of \(x\) is defined from \(-0.1\) to \(0.1\) m. The spacing was set to be 0.01 m in all directions. The sound pressure measured from the sensors at the hull was converted using the Fourier transform. For the source-level estimation, the frequency range was set from 1 to 100 kHz with a spacing of 32 Hz. For the grid points in the source region, \(\alpha\) was calculated. Figure 11 shows the normalized values of \(\alpha\) in the source region at 8 and 40 kHz, for example. A strong noise source is seen between the 12 o'clock position and 1 o'clock position viewed from the stern side. In addition, it can be seen that the source was located downstream close to the propeller. At 8 kHz, the source was located in one region, whereas two main source regions are seen at 40 kHz. For this reason, \(\alpha\) is lower in the 40 kHz result. For comparison, the cavitation pattern for condition C3 is shown in Figure 12. Note that the angle is defined at the 12 o'clock position viewed from the stern side in the clockwise direction. It can be seen that the cavitation occurs strongly on the starboard side, which is also found in Figure 11. Thus, it can be said that the method reasonably describes the noise source. Figure 10: Assumed source region. Figure 9: Comparison between the measured signal and background noise: (**a**) hull sensor and (**b**) sensor at the tunnel bottom. From the \(\alpha\) map, the maximum value was found, and the corresponding source level was obtained. Having obtained the source level at all frequencies, the result was compared with the measured source level, which is shown in Figure 13 in terms of spectral density. Overall, the calculated source level agreed well with the measured one from 2 to 50 kHz. The differences below 2 kHz were due to the combination of the monopole assumption and the transducer characteristics. Above 50 kHz, the transfer function could not be applied, and thus, the predicted source level deviated from the measured one. From this, it can be said that the proposed method is valid. Comparisons were also made for the other conditions. This is illustrated in Figure 14. The same conclusion can be drawn, with few differences in the results. Figure 11: Normalized \(\alpha\) at (**a**) 8 kHz and (**b**) 40 kHz (condition C3). Figure 12: Cavitation pattern at (**a**) 20\({}^{\circ}\) and (**b**) 80\({}^{\circ}\) (condition C3). Figure 8:0 Comparison between measured and calculated source levels for condition C1.