# 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  
The reflected and transmitted waves can be expressed as [26]  
\[p_{r}=\sum\limits_{s}r_{s}\mathrm{e}^{-\mathrm{j}k\beta_{s}x+ka_{s}z}(z<0),\ p_{t}=\sum\limits_{s}t_{s}\mathrm{e}^{-\mathrm{j}k \beta_{s}x-ka_{s}(z-\sum\limits_{m=1}^{n+1}a_{m})}\ (z>\sum\limits_{m=1}^{n+1}a_{m}) \tag{50}\]  
where \(p_{r}\) and \(p_{t}\) are the reflected and transmitted acoustic pressure, respectively. \(r_{s}\) and \(t_{s}\) are the reflection and transmission coefficients of the plane wave in \(s\) modes. \(ka_{s}\) and \(k\beta_{s}\) are the wave number components in the vertical and horizontal directions, which satisfy \(k\beta_{s}=k\sin\theta_{i}+2\pi\mathrm{s}/w\) and \(-\mathrm{j}ka_{s}=k\sqrt{1-\beta_{s}^{2}}\), where, \(\theta_{i}\) is the incidence angle of the acoustic wave. In this paper, we take \(s=0\) and \(\theta_{i}=0\). In this case, we have \(\beta_{0}=0\) and \(\alpha_{0}=\mathrm{j}\). Now, Equation (50) can be reduced to  
\[p_{r}=r_{0}\mathrm{e}^{\mathrm{j}kz}(z<0),\ p_{t}=t_{0}e^{-\mathrm{j}k(z- \sum\limits_{m=1}^{n+1}a_{m})}(z>\sum\limits_{m=1}^{n+1}a_{m}) \tag{51}\]  
At \(z=0\), by the continuity of velocity, we can get  
\[u_{i}+u_{r}=\left\{ \begin{array}{cc}u_{a}&-w_{1}\leq x\leq 0\\ 0&\mathrm{others}\end{array} \right. \tag{52}\]  
where \(u_{i}\), \(u_{r}\), and \(u_{a}\) are acoustic velocities of incident wave, reflected wave and wave in the duct, respectively. Considering Equations (51) and (11), and the momentum conservation law, Equation (52) can be reduced to  
\[\mathrm{j}kr_{0}=-\mathrm{j}\rho_{0}c_{0}\left\{ \begin{array}{cc}u_{a}&-w_{ 1}\leq x\leq 0\\ 0&\mathrm{others}\end{array} \right. \tag{53}\]  
The transmittance predictions are performed to capture the influences of leakage by using the developed theoretical formulas. The results are compared with those obtained by COMSOL software. In COMSOL simulations, the pressure acoustic module is used, and the acoustic FEM mesh is shown in Figure 3. All the walls are taken as hard boundaries, and the periodic boundary condition is applied in the direction perpendicular to the incident wave. As shown in Figure 4, the theoretical results (solid lines) from Equation (60) are in very good agreement with the COMSOL predictions (circle lines), which confirms the good accuracy of the theoretical formulas. In Figure 4, the frequency corresponding to the point with zero transmittance represents the resonance frequency of the DHR system, which is the key parameter in the design of the acoustic absorbers. Since all the holes share one cavity, the DHR device exhibits only one non-zero resonance frequency, at which the transmission phase undergoes an abrupt jumping. From the perspective of sound absorption, when the frequency of transmitted wave approaches the resonance point, nearly perfect reflection appears with a significant decrease of transmittance. It can be seen from Figure 4 that the resonance frequency of the system increases with the number of leak holes. This means that additional leak holes in the Helmholtz resonator always leads to an increase of resonance frequency of the system. Obviously, the change of the resonance frequency can be used as an important criterion for acoustic leak detection. In Figure 4, the region below the resonance frequency is called the high transmission band, which is usually utilized to control the transmitted wavefront of the acoustic metamaterials, so that anomalous refraction, acoustic focusing, and other functions can be realized. In fact, the influence of leakage on resonance frequency can be explained by the theory of acoustic impedance. Note that all the neck impedances can be considered as parallel, so the reciprocal of the overall impedance \(r_{\text{neck}}\) of the necks can be written as the sum of the reciprocal of impedance of each neck, given by  
\[r_{\text{neck}}=1/\sum_{i=1}^{n}\left(1/r_{2i}\right) \tag{61}\]  
Figure 3: COMSOL acoustic finite element model (FEM) of the DHR device with two leak holes.