## 5 Approximate Model for Optimal Location of Additional Source  
In order to further investigate the effect of the arrangement of the additional source on the noise reduction effect in large spaces behind the barrier and acquire the optimal location of the additional source, the SPLs around the top of the barrier are calculated and compared when the additional source is located at different locations (the location of the additional source changed from 0.1 m to 1.7 m at an interval of 0.1 m). For the convenience of discussion, the SPLs at receiver \(R\) are mainly concerned, where the \(R\) is located at 1 m away from the top of the barrier, the angle between the line connecting \(R\) and the apex of the barrier and the horizontal plane is 30\({}^{\circ}\), and the sound waves must pass through this region when propagating into a large space behind the barrier. In the calculation process, the primary source is located at 2 m away from the center of the barrier. Figure 7a shows the results when the primary source is located at 0.5 m above the ground. Figure 7b shows the results when the primary source is located at 1 m above the ground. It can be seen from Figure 7 that when the primary source is located at 0.5 m above the ground, the additional source should preferably be located at 0.2 m above the ground, and when the primary source is located at 1 m above the ground, the additional source should preferably be located at 0.4 m above the ground. This indicates that the optimal location of the additional source is highly related to the height of the primary source.  
The height of the barrier is a key factor in the sound diffraction. To investigate the effect of the barrier height on the location of the additional source, the SPLs are also calculated when the height of the barrier is 1.5 m. The calculation results are shown in Figure 8. It can be seen that when the primary source is located at 0.5 m above the ground, the additional source should be located at 0.2 m above the ground, the same as when the height of the  
Figure 7: The overall SPLs at \(R\). (**a**) The primary source is located at 0.5 m above the ground. (**b**) The primary source is located at 1 m above the ground.  
barrier is 2 m. However, when the primary source is located at 1 m above the ground, the additional source should be located at 0.2 m above the ground, which is different from when the height of the barrier is 2 m. It indicates that the optimal location of the additional source is also related to the height of the barrier.  
According to the basic theory of acoustics and the analysis mentioned above, the optimal location of the additional source is related to the height of the primary source and the barrier, and the higher the location of the primary source or the higher the barrier is, the higher the optimal location of the additional source is. The optimal location of the additional source can be regarded as approximately proportional to the product of the height of the primary source and the height of the barrier. Thus, a model is established to acquire the optimal location of the additional source so as to reduce the noise in a large space. The model can be expressed as follows.  
\[S_{h}=a\times P_{h}\times B_{h} \tag{1}\]  
where \(S_{h}\) is the height of the additional source, \(a\) is a coefficient, \(P_{h}\) is the height of the primary source, and \(B_{h}\) is the height of the barrier. \(a\approx 0.2\) according to the calculation results mentioned above.