## 3 Numerical Analysis
### Calculation Results  
Firstly, the sound field in the observation area is calculated when the primary source is located at (4.5 m, 0.5 m) and the additional source is located at 0.3 m, 0.5 m, 1 m, 1.5 m, and 2 m above the ground. Figure 2 shows the calculation results at the frequency of 800 Hz. The results at the frequency of 500 Hz, 630 Hz, and 1000 Hz are also calculated, and they showed the same characteristics as at 800 Hz. It can be seen from Figure 2 that the barrier shows good performance, and the sound pressure behind the barrier is significantly reduced. The sound waves behind the barrier propagate forward with the apex of the barrier as the center, and the directivity of the sound field is similar. However, when the additional source is located at the top of the barrier, the noise reduction in the large space behind the barrier is significantly worse than otherwise, as shown in Figure 2d. In addition, the sound field behind the barrier is also calculated when the primary source is located at 1 m above the ground; the calculation results agree with the results shown in Figure 2. It can be concluded that in order to obtain good performance in the large space behind the barrier, it is not appropriate to locate the additional source at the top of the barrier or a position higher than the barrier. Secondly, in order to analyze the effect of the additional source location on the sound field behind the barrier more conveniently, observation points were set at the top of the barrier, and the directivity of the sound field over the top of the barrier is analyzed when the additional source is located at different positions. The SPLs at the central frequency of one-third octave were compared in the frequency range between 500 Hz and 5000 Hz. The results showed that the effect of the location of the additional source on the sound field is almost the same whether located at the low or middle part of the barrier in the frequency range over 2500 Hz. Moreover, the active noise control technology is mainly applied to the control of low-frequency noise. Therefore, the following analysis focuses on the effect of the additional source location on the sound field behind the barrier in the frequency range of 500 Hz-1000 Hz. When the primary source is located at 0.5 m above the ground, the directivity of the sound field over the barrier is calculated as shown in Figure 3, where the additional source is located at 0.3 m, 0.5 m, and 1 m above the ground. The right half of the graph (0-90\({}^{\circ}\)-180\({}^{\circ}\)) represents the space behind the barrier, which is the main concern. It can be seen from Figure 3 that the effect of the additional source on the sound field behind the barrier is closely related to the location of the additional source and the frequency of the sound waves. At the frequency of 500 Hz, the SPLs in the space behind the barrier are the smallest when the additional source is located at 0.5 m above the ground. The SPLs are almost the same when the additional source is located at 0.3 m and 1 m above the ground. At the frequency of 630 Hz, the SPLs are the smallest in the region of 0 to 150\({}^{\circ}\) when the additional source is located at 0.3 m above the ground. When the additional source is located at 0.5 m or 1 m above the ground, the SPLs are almost the same in the region of 150\({}^{\circ}\) to 180\({}^{\circ}\) and smaller than in other cases. At the frequency of 800 Hz, the SPLs are the smallest in the region of 0 to 120\({}^{\circ}\) when the additional source is located at 0.3 m above the ground. In the region of 120\({}^{\circ}\) to 160\({}^{\circ}\), the SPLs are the smallest when the additional source is located at 0.5 m above the ground. At the frequency of 1000 Hz, the SPLs are the smallest in the region of 0 to 150\({}^{\circ}\) when the additional source is located at 0.3 m above the ground. In the region of 150\({}^{\circ}\) to 180\({}^{\circ}\), the SPLs are the smallest when the additional source is located at 0.5 m above the ground.