## 4 Simulation
### Procedure for Forming a Transmission-Traveling Wave Field  
The second phase is to verify whether the adaptive algorithm helps the impedance tube form a traveling wave field in the transmission section successfully or not by controlling the secondary source. The frequency range tested in the simulation is 100-2500 Hz. The wavelength calculation equation is as follow:  
\[\lambda=\frac{c}{f}. \tag{29}\]  
where \(\lambda\) is wavelength, \(c\) is the sound speed, and \(f\) is frequency. The range of wavelength in water is from 0.6 m to 15 m, and the range of wavelength in steel is from 2.44 m to 61 m. Owing to the page limitation, only the sound field of the transmission section (before and after control of the adaptive algorithm) in the frequency of 2500 Hz and the transformation of the sound signals of hydrophones 3 and 4 in the above two working conditions (without sample and with steel sample) are displayed. Initially, with the primary sound source working alone, the acoustic information (sound pressure amplitude and phase in the transmission section) can be collected in the transmission section, whose detailed information is shown in Figures 6 and 7. Figure 5: (**a**) 3D model of impedance tube without sample in software. (**b**) 3D model of impedance tube with steel sample in software. Figure 6: (**a**)The information of sound pressure amplitude in impedance tube downstream without sample at 2500 Hz. (**b**) The information of sound pressure phase in impedance tube downstream without sample at 2500 Hz. The sound field style is a standard standing wave field. The acoustic information (amplitude and phase) is verified as the tube length in the figure when the primary source works alone. Next, the secondary source begins to work with the adaptive algorithm. The output varies with time until the error signal in the adaptive algorithm is minimized to zero. The transformation of sound signals from hydrophones 3 and 4 can be achieved, as shown in Figures 8 and 9. Lastly, the acoustic information of the sound tube is collected again. The secondary source output is kept stable, as shown in Figures 10 and 11. The adaptive algorithm applied to the water-filled sound tube to generate a traveling wave is demonstrated to be effective through comparing the above acoustic information in the transmission section before and after controlling by using the adaptive algorithm. By comparing Figures 7 and 11, it can be observed that the variation of sound field distribution in water before and after the adaptive control in transmission section. Before the adaptive active control, the sound field changes dramatically with distance, which is close to standing wave field. After control, the sound field fluctuates in a very small range, which is almost a traveling wave field in the downstream of tube. Figure 8: **(a)**The signal of hydrophone 3 changes with time in the impedance tube without sample at 2500 Hz. **(b)** The signal of hydrophone 4 changes with time in the impedance tube without sample at 2500 Hz. Figure 7: **(a)** The information of sound pressure amplitude in sound tube inner with steel sample at 2500 Hz. **(b)** The information of sound pressure phase in sound tube inner downstream with steel sample at 2500 Hz. Figure 11: (**a**) The information of sound pressure amplitude in sound tube inner with steel sample at 2500 Hz. (**b**) The information of sound pressure phase in sound tube inner with steel sample at 2500 Hz. Figure 10: (**a**) The information of sound pressure amplitude in sound tube inner with steel sample at 2500 Hz. (**b**) The information of sound pressure phase in sound tube inner with steel sample at 2500 Hz. Figure 9: (**a**) The signal of hydrophone 3 changes with time in the impedance tube with steel sample at 2500 Hz. (**b**) The signal of hydrophone 4 changes with time in the sound tube with steel sample at 2500 Hz. With the test frequency changes, the combination of different hydrophones is selected.