# Leakage Effect on the Transmission Properties of a Duct Loaded with a Helmholtz Resonator
## 2 Acoustic Transmission of the DHR Device
### Acoustic Transfer Relations in the Duct  
Acoustic pressure and volume velocity at the junction between the \(i\)th neck and the duct satisfy the following equation  
\[\frac{p_{i}}{U_{i}(0)}=Z_{hi} \tag{28}\]  
Substituting Equation (25) into Equation (28), we have  
\[p_{i}=-j\frac{\rho_{0}c_{0}}{h_{3}\tan(kw_{3})}\sum_{m=1}^{n}U_{m}(0)+j\frac{ \rho_{0}c_{0}}{h_{2i}}\tan(k\widetilde{w}_{2i})U_{i}(0)+j\frac{\rho_{0}c_{0}}{ w_{1}h_{2i}^{2}}\frac{\sin(kh_{2i})-kh_{2i}}{k^{2}}U_{i}(0) \tag{29}\]  
Equation (29) can be simplified as  
\[p_{i}=r_{2i}U_{i}(0)+r_{1}\sum_{m=1}^{n}U_{m}(0) \tag{30}\]  
where  
\[r_{1}=-j\frac{\rho_{0}c_{0}}{h_{3}\tan(kw_{3})},\ r_{2i}=j\frac{\rho_{0}c_{0}} {h_{2i}}\tan(k\widetilde{w}_{2i})+j\frac{\rho_{0}c_{0}}{w_{1}h_{2i}^{2}}\frac {\sin(kh_{2i})-kl_{2i}}{k^{2}} \tag{31}\]  
Writing Equation (30) into the matrix form, we obtain  
\[\left[ \begin{array}{c}p_{1}\\ p_{2}\\ \vdots\\ p_{n}\end{array} \right]=\mathbf{R}\left[ \begin{array}{c}U_{1}(0)\\ U_{2}(0)\\ \vdots\\ U_{n}(0)\end{array} \right] \tag{32}\]  
where \(\mathbf{R}\) is a matrix of order \(n\times n\), given by  
\[\mathbf{R}=\left[ \begin{array}{cccc}r_{1}+r_{21}&r_{1}&\ldots&r_{1}\\ r_{1}&r_{1}+r_{22}&\cdots&r_{1}\\ \vdots&\vdots&\ddots&\vdots\\ r_{1}&r_{1}&\cdots&r_{1}+r_{2n}\end{array} \right] \tag{33}\]  
Let \(\mathbf{R}_{1}=\mathbf{R}^{-1}\), Equation (32) can be reduced to  
\[\left[ \begin{array}{c}U_{1}(0)\\ U_{2}(0)\\ \vdots\\ U_{n}(0)\end{array} \right]=\mathbf{R}_{1}\left[ \begin{array}{c}p_{1}\\ p_{2}\\ \vdots\\ p_{n}\end{array} \right] \tag{34}\]where  
\[U_{i}(0)=\sum\limits_{m=1}^{n}\mathbf{R}_{1}(i,m)p_{m} \tag{35}\]  
Now, consider the case that the incident acoustic wave enters the duct at the entrance \(z=0\). Let \(p_{\text{in}}\) and \(U_{\text{in}}\) denote the incident acoustic pressure and volume velocity, respectively. Similarly, let \(p_{\text{out}}\) and \(U_{\text{out}}\) denote the acoustic pressure and volume velocity at the exit \(z=a_{1}+a_{2}+\cdots+a_{n+1}\), respectively. For \(0\leq z\leq a_{1}\), acoustic pressure and volume velocity can be expressed as  
\[p_{A_{1}}(z)=A_{1}^{+}\text{e}^{-\text{j}kz}+A_{1}^{-}\text{e}^{\text{j}kz}\,\ U_{A_{1}}(z)=\frac{A_{1}^{+}\text{e}^{-\text{j}kz}-A_{1}^{-}\text{e}^{ \text{j}kz}}{R_{w_{1}}} \tag{36}\]  
where \(A_{1}^{+}\) and \(A_{1}^{-}\) are, respectively, the transfer coefficients of the acoustic wave along the \(+z\) and \(-z\) direction. \(R_{w_{1}}\) is the acoustic impedance of the duct, given by  
\[R_{w_{1}}=\frac{\rho_{0}c_{0}}{w_{1}} \tag{37}\]  
For \(\sum\limits_{m=1}^{i-1}a_{m}\leq z\leq\sum\limits_{m=1}^{i}a_{m}(2\leq i\leq n)\), acoustic pressure and volume velocity can be written as  
\[p_{A_{i}}(z)=A_{i}^{+}\text{e}^{-\text{j}k(z-\sum\limits_{m=1}^{i-1}a_{m})}+A _{i}^{-}\text{e}^{\text{j}k(z-\sum\limits_{m=1}^{i-1}a_{m})},U_{A_{i}}(z)= \frac{A_{i}^{+}\text{e}^{-\text{j}k(z-\sum\limits_{m=1}^{i-1}A_{m})}-a_{i}^{- }\text{e}^{\text{j}k(z-\sum\limits_{m=1}^{i-1}a_{m})}}{R_{w_{1}}} \tag{38}\]  
Similarly, for \(\sum\limits_{m=1}^{n}a_{m}\leq z\leq\sum\limits_{m=1}^{n+1}a_{m}\), we have  
\[p_{A_{n+1}}(z)=A_{n+1}^{+}\text{e}^{-\text{j}k[z-\sum\limits_{m=1}^{n}a_{m}] }+A_{n+1}^{-}\text{e}^{\text{j}k[z-\sum\limits_{m=1}^{n}a_{m}]},U_{A_{n+1}}(z )=\frac{A_{n+1}^{+}\text{e}^{-\text{j}k[z-\sum\limits_{m=1}^{n}a_{m}]}-A_{n+1 }^{-}\text{e}^{\text{j}k[z-\sum\limits_{m=1}^{n}a_{m}]}}{R_{w_{1}}} \tag{39}\]  
The continuity conditions of acoustic pressure and volume velocity at \(z=0\) are  
\[p_{\text{in}}=p_{A_{1}}(0),\ U_{\text{in}}=U_{A_{1}}(0) \tag{40}\]  
Substituting Equation (36) into Equation (40), we obtain  
\[\left[ \begin{array}{c}A_{1}^{+}\\ A_{1}^{-}\end{array} \right]=\mathbf{M}_{1}\!\left[ \begin{array}{c}p_{\text{in }}\\ U_{\text{in}}\end{array} \right] \tag{41}\]  
where  
\[\mathbf{M}_{1}=\left[ \begin{array}{cc}\frac{1}{2}&\frac{R_{w_{1}}}{2}\\ \frac{1}{2}&\frac{R_{w_{1}}}{2}\end{array}