# Leakage Effect on the Transmission Properties of a Duct Loaded with a Helmholtz Resonator
## 2 Acoustic Transmission of the DHR Device
### Impedance Formulas
They can be expressed as  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]0  
where \(k\) is the wave number, \(k=\omega/c_{0}\), \(c_{0}\) is the sound speed. Only the plane wave component is considered. In this case, substitute \(s=0\) into Equation (9), the wave number components in the \(z\) and \(x\) directions can be written as \(k_{z0}=0\) and \(k_{x0}=k\), respectively. From Equation (8), we have \(\delta_{00}=1\) and \(\phi_{0}(z)=1\). Now, Equation (6) can be recast as  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]1  
For the time-harmonic wave, the velocity \(u\) and the acoustic pressure \(p\) are related by  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]2  
where \(\rho_{0}\) is the air density, and \(\rho_{0}c_{0}\) is the characteristic impedance of the air. Substituting Equation (10) into Equation (11), the particle velocity can be obtained as  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]3  
Considering rigid wall boundary condition \(u=0\) at \(x=w_{2}+w_{3}\), we have  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]4  
The average velocity of the air column in the \(i\)th neck at \(x=w_{2}\) can be written as  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]5  
where \(U_{i}(x)\) is the volume velocity of the air column in the \(i\)th neck. At \(x=w_{2}\), the velocity distribution given by Equation (12) should be equal to that generated by the motions of the air columns in the necks. Therefore, substituting Equation (14) into Equation (12), and integrating along the \(z\) direction, we obtain  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]6  
where \(z_{m}\) is the \(z\) coordinate of the midpoint of the \(m\)th neck, and \(z_{C}\) is the \(z\) coordinate of the midpoint of the cavity. According to Equation (15), the second relation between \(B_{0}^{+}\) and \(B_{0}^{-}\) is of the form  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]7  
Based on Equations (13) and (16), we can solve \(B_{0}^{+}\) and \(The average acoustic pressure field \(\overline{p}_{i}(x)\) at the junction of the \(i\)th neck and cavity can be defined as  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]8  
After substituting Equation (17) into Equation (10), the average acoustic pressure field at \(x=w_{2}\) can be obtained from Equation (18), given by  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]9  
After obtaining the relation between \(\overline{p}_{i}(x)\) and \(U_{i}(x)\) at \(x=w_{2}\), the acoustic impedance of the cavity corresponding to the \(i\)th neck can be written as  
\[M_{ai}=\frac{M_{mi}}{h_{2i}^{2}},\ C_{a}=\frac{h_{2i}^{2}}{K_{mi}} \tag{2}\]0  
According to the impedance transfer formula, the effective acoustic impedance at the midpoint of the duct can be expressed as  
\[M_{ai}=\frac{M_{mi}}{h_{2i}^{2}},\ C_{a}=\frac{h_{2i}^{2}}{K_{mi}} \tag{2}\]1  
where \(\widetilde{w}_{2i}=w_{2}+>\alpha_{h}h_{2i}\) is the corrected height of the \(i\)th neck [25] and \(\alpha_{h}\) is the correction coefficient, which is dependent on the geometric parameters of the neck. Substituting Equation (20) into Equation (21), we obtain  
\[M_{ai}=\frac{M_{mi}}{h_{2i}^{2}},\ C_{a}=\frac{h_{2i}^{2}}{K_{mi}} \tag{2}\]2  
In order to predict the transmission coefficient more accurately, the radiation impedance due to vibration of the air column in the neck should be considered. In fact, the neck plays the role of acoustic capacity as well as acoustic mass.