# Leakage Effect on the Transmission Properties of a Duct Loaded with a Helmholtz Resonator
## 2 Acoustic Transmission of the DHR Device  
### Impedance Formulas  
As stated in Section 1, the investigation of the leakage effect of a single neck Helmholtz resonator can be attributed to the prediction of the transmission spectrum of a corresponding multi-neck resonator. A DHR device, that is a duct loaded by a Helmholtz resonator with \(n\) necks, is shown Figure 1. Of all the necks, we assumed that one was the main neck and the rest were the leak holes. In Figure 1, the height of the duct was \(w_{1}\). The total length and height of the resonator were \(h\) and \(w\), respectively. The height of each neck was \(w_{2}\) and the width of the necks were \(h_{21}\), \(h_{22}\),..., \(h_{2n}\), respectively. The height and the length of the cavity are \(w_{3}\) and \(h_{3}\), respectively. The distance between the inlet of DHR to the midpoint of the first neck was \(a_{1}\), the distance between the midpoints of the \(i\)th neck and the \((i+1)\)th neck (\(1\leq i\leq n-1\)) was \(a_{i+1}\), and the distance between the outlet of DHR to the midpoint of the \(n\)th neck was \(a_{n+1}\). \(p_{i}\), \(p_{r}\), and \(p_{t}\) were acoustic pressure of incident, reflected and transmitted waves, respectively. If we assume that the size of DHR is much smaller than the wavelength of the acoustic wave, then the air in the neck and the cavity can be treated as the lumped mechanical elements. In this way, the governing equations of the Helmholtz resonator with \(n\) necks can be written as  
\[M_{mi}\ddot{\xi}_{i}=p_{i}h_{2i}-K_{mi}\sum_{m=1}^{n}\xi_{m},\ 1\leq i\leq n \tag{1}\]  
where, \(M_{mi}\) is the mass of the air in the \(i\)th neck, \(K_{mi}\) is the equivalent spring stiffness of the cavity, and \(p_{i}\) is the average acoustic pressure acting on the \(i\)th neck. \(\xi_{i}\) and \(\ddot{\xi}_{i}\) are the displacement and the acceleration of the air column in the \(i\)th neck, respectively. Based on the analogy of electro-mechanics-acoustics, the acoustic mass of the \(i\)th neck \(M_{ai}\), and the acoustic capacity of the cavity \(C_{a}\) can be written as  
\[M_{ai}=\frac{M_{mi}}{h_{2i}^{2}},\ C_{a}=\frac{h_{2i}^{2}}{K_{mi}} \tag{2}\]  
Under the assumption of simple harmonic motion, the volume velocity \(U_{i}\) of the air column in the \(i\)th neck can be expressed as  
\[U_{i}=\frac{\bar{\xi}_{i}h_{2}}{j\omega}=j\omega\xi_{i}h_{2} \tag{3}\]  
where, \(j\) is the imaginary unit, \(\omega\) is the angular frequency of the acoustic wave. Substitution of Equations (2) and (3) in Equation (1) yields the following acoustic relation between the \(i\)th neck and the cavity  
\[\mathrm{j}\omega M_{ai}U_{i}+\frac{\sum\limits_{m=1}^{n}U_{m}}{\mathrm{j} \omega C_{a}}=p_{i} \tag{4}\]  
It will be seen from the end of this subsection that the relation described by Equation (4) is not accurate enough. Therefore, in what follows, a set of more accurate analytical formulas that reflect the acoustic characteristics of the system will be derived. The acoustic impedance of the \(i\)th neck is  
\[Z_{ni}=\frac{\rho_{0}c_{0}}{h_{2i}} \tag{5}\]  
The acoustic pressure in the cavity can be expanded in terms of normal modes [24], given by  
\[p(x,z)=\sum\limits_{s}\phi_{s}(z)[B_{s}^{+}e^{-\mathrm{j}k_{ss}(x-w_{2})}+B_{ s}^{-}e^{\mathrm{j}k_{\mathrm{\alpha}}(x-w_{2}-w_{3})}] \tag{6}\]  
where  
\[\phi_{s}(z)=\sqrt{2-\delta_{0s}}\cos[k_{zs}(z-\frac{h_{3}}{2})],\ s=0,1,2, \cdots,\infty \tag{7}\]  
is the \(s\)th normal mode, and satisfies the following orthogonal condition  
\[\int_{\sigma}\phi_{s}(z)\phi_{t}(z)\mathrm{d}z=\sigma\delta_{st} \tag{8}\]  
Figure 1: Schematic diagram of a DHR system. where \(\delta_{st}\) is Kronecker delta, and \(\sigma\) is the cross-sectional area of the cavity. \(B_{s}^{+}\) and \(B_{s}^{-}\) are the modal amplitudes corresponding to the components traveling in the positive and negative \(x\) directions, respectively. \(K_{xs}\) and \(K_{xs}\) are the wave number components in the \(x\) and \(z\) directions, respectively.