## 4 Simulation Results and Discussion
### HR for the Resonance Frequency of 1000 Hz  
Figure 7a shows an HR for the resonance frequency of 1000 Hz, and it has a large length-to-height ratio in the cavity cross-sectional area. The predicted transmission loss showed 22.8 dB and matched well with the target frequency of 1000 Hz, as shown in Figure 7.  
Figure 6: The effects on the transmission loss with the resonator arrangement: (**a**) serial arrangement of two resonators; (**b**) parallel arrangement of two resonators; (**c**) effect of the neck distance between two resonators in a serial arrangement; (**d**) effect of the offset distance between two resonators in parallel arrangement. Figure 7: The HR of 1000 Hz resonance frequency: (**a**) a three-dimensional model of HR; (**b**) predicted transmission loss. An identical study for the 500 Hz HR to identify the effects of shape changes while maintaining the same cross-sectional area using a scale factor ranging from 0.4 to 1.6 for the transverse length is performed. Note that details of the numerical study cases are summarized in Table 4.  
Figure 8 shows the simulation result. In Figure 8, it can be seen that, as happened to the 500 Hz HR, the shape variation causes the change in transmission loss. Figure 8a shows that the peak frequency can be changed outside a certain range of the cavity ratios. The magnitude can be increased with varying cavity ratios, but no certain trend can be observed. The frequency shift phenomenon outside a certain range can also be identified with varying neck and duct ratios, as shown in Figure 8b,c. No definite trend is identified in the magnitude for both varying neck and duct ratio cases, but in the case of duct ratio variation, the magnitude of the transmission loss can be increased with a small change in the duct ratio. In conclusion, the effect of the shape change of the neck and duct is less effective than the 500 Hz resonator. Using this result, the optimal model for the maximum transmission loss is designed, and 37.5 dB of the transmission loss, which was the same with the effect of the shape change of the cavity, is obtained, as shown in Figure 8d. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**(a) Cavity–Figure 8a**} \\ \hline
**Designation** & **Scale Factor** & **L\({}_{\text{c}}\) (mm)** & **H\({}_{\text{c}}\) (mm)** & **Cavity Ratio (L\({}_{\text{c}}\)/H\({}_{\text{c}}\))** & **A (mm\({}^{2}\))** \\ \hline Cavity 1 & 0.4 & 47 & 75 & 0.627 & 3525 \\ \hline Cavity 2 & 0.6 & 70.5 & 50 & 1.410 & 3525 \\ \hline Cavity 3 & 0.8 & 94 & 37.5 & 2.507 & 3525 \\ \hline Baseline & & 117.5 & 30 & 3.917 & 3525 \\ \hline Cavity 4 & 1.2 & 141 & 25 & 5.640 & 3525 \\ \hline Cavity 5 & 1.4 & 164.5 & 21.43 & 7.676 & 3525 \\ \hline Cavity 6 & 1.6 & 188 & 18.75 & 10.03 & 3525 \\ \hline \hline \multicolumn{5}{c}{**(b) Neck–Figure 8b**} \\ \hline
**Designation** & **Scale Factor** & **L\({}_{\text{n}}\) (mm)** & **H\({}_{\text{n}}\) (mm)** & **Neck Ratio (L\({}_{\text{n}}\)/H\({}_{\text{n}}\))** & **A (mm\({}^{2}\))** \\ \hline Neck 1 & 0.4 & 12 & 12.5 & 0.960 & 150 \\ \hline Neck 2 & 0.6 & 18 & 8  
The effect of HR arrangement, serial and parallel along the duct as shown in Figure 9a,b, is studied. In each arrangement, the effect of the distance between necks in a serial arrangement and the offset distance between necks in a parallel arrangement is compared. Note that the transmission loss of baseline serial and parallel arrangements along the duct is 48.17 dB and 15.52 dB, respectively. Figure 8: The shape variation effects on the transmission loss of the Helmholtz resonator with the resonance frequency of 1000 Hz: (**a**) variation in cavity shape; (**b**) variation in neck shape; (**c**) variation in duct shape; (**d**) comparison with the optimized and the baseline models.