## 4 Simulation Results and Discussion
### Simulation Parameter Setting and Criteria of the Control Performance  
In this 2D simulation, the primary sources are single-frequency plane waves with frequencies ranging from 400 Hz to 1100 Hz at a frequency interval of 100 Hz. The source strength of the primary field was \(q_{p}=1\) m\({}^{3}\)s\({}^{-1}\) and the strength vector (dimension \(1\times 1\)) of one incident plane wave was \(\mathbf{q}_{p}=[q_{p}]\). The maximum power of the secondary loudspeakers was \(P_{\text{max}}=2\) m\({}^{6}\)s\({}^{-2}\) and the speed of sound was considered as \(c=343\) ms\({}^{-1}\). Target area 1 is set in the region of the backward-scattered sound, which is a 1 m \(\times\) 1 m square area with 169 uniformly distributed error points. Target area 2 is set in the forward-scattered sound region, which is a rectangular 1 m \(\times\) 2 m area with 325 evenly distributed error points. The spacing between the error points was 0.083 m and the radius of the cylinder was a = 0.38 m. The infinite line of monopoles is treated as a secondary loudspeaker. Eight secondary loudspeakers were placed uniformly around the cylinder at a distance of 0.05a from the cylinder surface in Figure 1a. Eight optimized placed secondary loudspeakers were selected from 64 uniformly placed candidate secondary loudspeakers at a distance of 0.05a from the cylinder surface in Figure 1b. Thirty-two constraint points were placed uniformly between target area 1 and target area 2 at a distance of 2.53a from the cylinder surface in Figure 1b. To observe the noise reduction in the far field, 360 uniformly placed far-field observing points were located at a distance of 1000a from the center of the cylinder. To observe the noise reduction in the target area, 63,001 near-field observing points were evenly distributed in a 5 m \(\times\) 5 m square area in the x-y plane centered at the center of the cylinder; the spacing between the near-field observing points was 0.02 m. The noise reduction for one observing point is defined as:  
\[NR_{ob}=10\log\left(\frac{\left|P_{ob\_ANCoff}\right|^{2}}{\left|P_{ob\_ANC \_m}\right|^{2}}\right) \tag{10}\]  
where \(P_{ob\_ANCoff}\) and \(P_{ob\_ANCon}\) are the sound pressures before and after the active noise control for one observing point, respectively. The noise reduction for the entire target area can be described as:  
\[NR_{\text{e}}=10\log\left(\frac{\left\|\mathbf{P}_{e\_ANCoff}\right\|^{2}_{ 2}}{\left\|\mathbf{P}_{e\_ANCon}\right\|^{2}_{2}}\right)=10\log\left(\frac{ \left\|\mathbf{Z}_{pe\_sca}\mathbf{q}_{p}\right\|^{2}_{2}}{\left\|\mathbf{Z}_ {pe\_sca}\mathbf{q}_{p}+\mathbf{Z}_{ee}\mathbf{q}_{s}\right\|^{2}_{2}}\right) \tag{11}\]  
where \(NR_{\text{e}}\) is the ratio of the squared sum of the sound pressure before and after active noise control for 494 error points, \(\mathbf{P}_{e\_ANCoff}=\left[ \begin{array}{cc}P_{e\_ANCoff},&\dots,&P_{e494 \_ANCoff}\end{array} \right]\) is the sound pressure matrix (dimension \(494\times 1\)) for the error points before noise control and \(\mathbf{P}_{e\_ANCon}=\left[ \begin{array}{cc}P_{e\_ANCon},&\dots,&P_{e494 \_ANCon}\end{array} \right]\) is the sound pressure matrix (dimension\(494\times 1\)) for the error points after noise control.