## 4 Results and Discussion
### Variation of the Sound Field along the Radial Direction  
To explore the variation of the sound field in the radial direction, measurements were performed before and after the hydrophone bracket was raised by 1.5 cm, as shown schematically in Figure 10. The results of these measurements are shown in Figure 11.  
Figure 10: Raising of hydrophone bracket. Figure 9: Variation of the main frequency average power spectrum along the axial direction: **(a)** 4.2 kHz source; **(b)** 5.2 kHz source. The frequency band between the first- and second-order normal frequencies was analyzed. Before lifting, the 2# and 3# hydrophones were at the same distance from the axis, and similarly for the 1# and 4# hydrophones. Therefore, in Figure 11a, the curves of the power spectra from the 2# and 3# hydrophones are the same, as are those from the 1# and 4# hydrophones. After lifting, the 3# hydrophone is nearest to the axis, followed in order by the 2#, 4#, and 1# hydrophones, which is consistent with the increasing strengths of the respective power spectra. This is in accordance with the theoretical radial dependence on the Bessel function \(J_{0}(k_{r}r)\) in Equation (8). The following is a quantitative analysis of the radial distribution. The distance between each pair of hydrophones is 50 mm, and the hydrophone bracket is initially at the radial center position. After lifting, the distances of the 3#, 2#, 4#, and 1# hydrophones from the axis are 10, 40, 60, and 90 mm, respectively. At the cutoff frequency, \(k_{r}=k_{0}\), depending on the distance \(r\) from the axis, the theoretical differences between the sound pressure self-spectrum measured by the 3# hydrophone and those measured by the other three hydrophones can be calculated, and the results are compared with the experimental measurements in Table 5. For convenience of exposition, the distance between the 3# hydrophone and the axis is set as \(r_{0}\), and the differences between the sound pressure power spectrum of the 1#, 2#, and 4# hydrophones and the 3# hydrophone as \(X_{1}\), \(X_{2}\), and \(X_{3}\), respectively. There is good agreement between theory and experiment, and the radial distribution of the normal wave in the tube is quantitatively confirmed to follow the Bessel function behavior. The reasons for the error are as follows. In the experiment, the magnitude of the lifting was controlled manually, and not very accurately, which is the main source of error: If the lifting range were slightly larger, and the 3# and 4# hydrophones closer to the axis, the amplitude would be higher. If the 1# and 2# hydrophones were further away from the axis, the amplitude would be smaller, which would lead to an increase in \(X_{2}\); according to the properties of the Bessel function, the closer the value of the function is to the axis, the slower is its rate of change. Therefore, the power spectrum at the 4# hydrophone would increase more if it were lifted, so the value of \(X_{3}\) would be reduced. If the 1# hydrophone were closer to the upper slot and the outside medium were air (which can be regarded as an absolutely soft boundary), the sound would be totally reflected, so the amplitude would become  
Figure 11: Results of hydrophone measurements at different depths 5.05 m from the source: (**a**) Before and (**b**) after lifting of hydrophone bracket. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Power Spectral** & **Difference between \(r_{0}\)** & **Theoretical Power Spectral** & **Experimental Power** \\
**Difference** & **and \(r\)** & **Difference (dB)** & **Spectral Difference (dB)** \\ \hline \(X_{1}\) & 80 & 12.1 & 11.1 \\ \hline \(X_{2}\) & 30 & 1.6 & 2 \\ \hline \(X_{3}\) & 40 & 3.8 & 3.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of hydrophone measurements and theoretical values at different radial locations. higher, leading to a decrease in \(X_{1}\).