# Leakage Effect on the Transmission Properties of a Duct Loaded with a Helmholtz Resonator
## 2 Acoustic Transmission of the DHR Device
### Reflection and Transmission Coefficients
Figure 2: The locations of the main neck and the leak holes used in simulations; (**a**) No leak hole; (**b**) One leak hole; (**c**) Two leak holes. where \(r_{2i}\) is the impedance of the \(i\)th neck given in Equation (31). Thus, the overall impedance \(Z_{h}\) (unit \(\rm{Pa}\cdot\rm{s}/\rm{m}^{2}\), same as below) of the DHR device can be written as  
\[Z_{h}=r_{\rm{neck}}+r_{1} \tag{62}\]  
where \(r_{1}\) is impedance of the cavity given in Equation (31). As shown in Figure 5, the imaginary part of the overall impedance \(Z_{h}\) changes with frequency and number of leak holes. With the increase of the number of leak holes, the slope of the impedance curve decreases, which will lead to an increase in frequency corresponding to the point with \(\rm{Im}(Z_{h})=0\). Therefore, we can conclude that the existence of the leak holes always results in an increase in resonance frequency of the DHR system. Next, the DHR device with one leak hole (see Figure 2b) is used to investigate the influences of different width of the leak hole on the transmission properties of the DHR device, the results are shown in Figure 6. In simulations, the width of the leak hole is taken as 1.5 mm, 2 mm, and 2.5 mm, respectively. The other parameters are the same as those in Figure 2b. As can be seen from Figure 6a,b, the amplitude and the phase shift of the transmitted wave predicted by the theoretical formulas are consistent with the COMSOL results. The increase of the width of the leak hole will lead to an increase of resonance frequency of the system. The resonance frequency of the system will shift to higher values with the increase of the width of the leak hole. Figure 4: Effects of the number of leak holes on the transmission properties of the DHR device. The solid lines represent the theoretical results, while the circle lines represent the COMSOL predictions. **(a)** Amplitude vs. frequency; **(b)** Phase shift vs. frequency. Figure 5: Overall impedance vs. frequency. Figure 6c shows the variations of the imaginary part of the impedance of the leak hole \(\text{Im}(Z_{m})\) with frequencies for different width of the leak hole. According to Equation (31), we can see that \(\text{Im}(Z_{m})\) increases with frequency and decreases with the width of leak hole. The reduction of \(\text{Im}(Z_{m})\) results in a decrease of the imaginary part of the overall impedance \(\text{Im}(Z_{h})\), which increases the resonance frequency, as shown in Figure 6d. The results shown in Figure 7 are obtained by adjusting the distance between the leak hole and the main neck \(a_{2}\) and keeping the position of the main neck unchanged. In simulations, the distance \(a_{2}\) is taken as 7 mm and 14 mm, respectively, and the other parameters used are the same as those in Figure 2b. As a contrast, one neck whose width is the sum of the main hole and the leak hole is also calculated in Figure 7, which is drawn in red lines. As shown in Figure 7, the distance between the necks has an effect on the amplitude and phase shift of the transmission wave. As the distance between the necks increases, the amplitude curves and the phase shift curves move right, and the resonance frequency of the system increases, as shown in Figure 7a,b. Although in traditional theory, the distance between the necks has no effect on the resonance frequency, both the theory in this paper and the simulation results do not accord with this conclusion. As the distances between the necks become coalescent neck, 7 mm and 14 mm, the resonance frequencies become 1878 Hz, 2224 Hz, and 2396 Hz, respectively, which is a substantial difference that cannot be ignored. From Appendix A, we can see that the distance between the necks affects the value of the transfer matrix \(\mathbf{M}_{2}\) and finally affects the amplitude and phase shift. Figure 6: Effects of the width of leak hole \(h_{21}\) on transmission properties of the DHR device with one leak hole (see Figure 2b). The solid lines correspond to theoretical results, while the circle lines correspond to COMSOL predictions.