## 4 Results and Discussions
### Location Results  
This section gives a preliminary introduction and discussion of the location results. The 20\({}^{\circ}\) leakage at sensor array position C for 1.5 mm is taken as an example. Figure 9 shows the results of the traditional beamforming algorithm and the FWMB algorithm proposed in this paper. For the traditional beamforming, a separate frequency band is usually used to approximate the wave speed information. In order to have more contrast with the FWMB algorithm, a narrow band similar to that in the FWMB algorithm was selected. Figure 9a shows the results in the case of the 120-130 kHz filtering. It can be seen that a maximum value could be obtained near 20\({}^{\circ}\), and a good \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency Band (kHz)** & **Phase Speed c\({}_{\text{cp}}\) (m/s)** & \
\begin{tabular}{c} **Weighting Factor of** \\ **1.5 mm** \\ \end{tabular} &
\begin{tabular}{c} **Weighting Factor of** \\ **2.0 mm** \\ \end{tabular} \\ \hline
100–110 & 1482 & 0.7679 & 0.9516 \\
110–120 & 1537 & 0.7693 & 0.9269 \\
120–130 & 1593 & 0.6198 & 0.8478 \\
130–140 & 1640 & 0.7351 & 0.8617 \\
140–150 & 1692 & 0.8770 & 1.0000 \\
150–160 & 1735 & 1.0000 & 0.9669 \\
160–170 & 1776 & 0.9451 & 0.6732 \\
170–180 & 1815 & 0.6996 & 0.5864 \\
180–190 & 1856 & 0.8209 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Phase speeds of different frequency bands of the A\({}_{0}\) mode signals. Figure 9: The results for the 20\({}^{\circ}\) with different algorithms. (**a**) Traditional beamforming with a 120–130 kHz filter, (**b**) traditional beamforming with a 150-160 kHz filter, and (**c**) result with the FWMB algorithm. result was obtained. However, the obtained result was 19\({}^{\circ}\), which, although very close to 20\({}^{\circ}\), still had some errors. In Figure 9b, another frequency band is used as an example. In this example, two peaks appeared, although there was also a significant peak at 20\({}^{\circ}\). However, a higher peak appeared at around 70\({}^{\circ}\) and misjudgment occurred. The reason for the above phenomenon is probably that the traditional beamforming only used one single narrow frequency band, and the energy carried by this frequency band was limited and not necessarily the largest. Therefore, there is a certain possibility that the superposition energy obtained from multiple angles was approximated, thereby obtaining multiple peaks. At the same time, the reflection was also a main interference factor for location. Though the energy was reduced for reflection, it was still a source of interference for positioning. The FWMB method can remove the effect of false peaks by superimposing more frequency band results. The above results show that the traditional beamforming algorithm is prone to errors because it uses less frequency band information and is less robust. To overcome the above problems, under the conditions of this paper, 64 sets of data overlay were used in the FWMB algorithm to improve the eight sets of data overlay in the traditional beamforming. Therefore, it can be seen in Figure 9c that the result graph of the FWMB algorithm has more fluctuations compared to traditional beamforming. Though the data are not smooth, more accurate results were obtained due to the weighted superposition of the multiple sets of frequency band data. In view of the above preliminary conclusions, we show the data of several other points as further displays. The results for the leakages in positions I and B are shown in Figure 10. The graphics use a circular coordinate system for easier display and understanding. The red curve and angle value represent the FWMB result, and the blue curve and angle value represent the traditional beamforming result. The frequency band in the label indicates that it is used in traditional beamforming. For the FWMB algorithm, the frequency band is adaptive and does not require manual selection.