# Low-Frequency Sound Absorption in Air Pipelines and Water-Filled Pipelines Based on Metamaterials
## 2 Design Paradigm and Theoretical Method
### Theory of the Asymmetric Absorption Metamaterials  
In the scope of this research, losses caused by viscosity and thermal effects can be represented by complex wave number and impedance as follows [41]:  
\[k_{e}=\frac{\omega}{c_{0}}\bigg{(}1+\frac{\beta}{s_{e}}(1+(\gamma-1)/\chi\bigg{)}, \tag{1}\]  
\[Z_{e}=\frac{\rho_{0}c_{0}}{S_{e}}(1+\frac{\beta}{s_{e}}(1-(\gamma-1)/\chi)), \tag{2}\]  
where \(s_{e}(=r_{e}/\delta)\) is a defined parameter and \(S_{e}(=\pi r_{e}^{2})\) is cross-sectional area size (\(e=\mathrm{n}\) for the neck and \(e=\mathrm{c}\) is for the cavity). \(\delta=\sqrt{2\mu/\rho_{0}\omega}\) is the viscous boundary layer thickness, with \(\mu=1.568\times 10^{-5}\mathrm{m}^{2}/\mathrm{s}\) being the viscosity of air. \(\chi=\sqrt{\mathrm{Pr}}\) is the Prandtl number at atmospheric pressure with \(\mathrm{Pr}=0.702\), \(\beta=(1-j)/\sqrt{2}\) (\(j\) is the imaginary unit), and \(\gamma=1.4\) is the heat capacity ratio of air. The impedance of HR can be articulated as follows [41]  
\[Z_{\mathrm{HR}}=-\mathrm{j}\frac{\cos(k_{\mathrm{n}}l_{\mathrm{n}})\cos(k_{ \mathrm{c}}l_{\mathrm{c}})-Z_{\mathrm{n}}k_{\mathrm{n}}\Delta l\cos(k_{ \mathrm{n}}l_{\mathrm{c}})\sin(k_{\mathrm{c}}l_{\mathrm{c}})/Z_{\mathrm{c}}- Z_{\mathrm{n}}\sin(k_{\mathrm{n}}l_{\mathrm{n}})\sin(k_{\mathrm{c}}l_{\mathrm{c}})/Z_{ \mathrm{c}}}{\sin(k_{\mathrm{n}}l_{\mathrm{n}})\cos(k_{\mathrm{c}}l_{\mathrm{ c}})/Z_{\mathrm{n}}-k_{\mathrm{n}}\Delta l\sin(k_{\mathrm{n}}l_{\mathrm{n}})\sin(k_{ \mathrm{c}}l_{\mathrm{c}})/Z_{\mathrm{c}}+\cos(k_{\mathrm{n}}l_{\mathrm{n}}) \sin(k_{\mathrm{c}}l_{\mathrm{c}})/Z_{\mathrm{c}}}, \tag{3}\]  
where \(l_{c}\) and \(l_{n}\) represent the values of the length of the neck and the cavity. \(\Delta l\) refers to the adjusted length calculated from the summation of two individual adjusted lengths \(\Delta l=\Delta l_{1}+\Delta l_{2}\). Here, \(\Delta l_{1}\) and \(\Delta l_{2}\) originate from the pressure radiations at the inconsistency transitioning from the neck area to the cavity portion of the HR and to the pipe, each individually [41]  
\[\Delta l_{1}=0.82\bigg{[}1-1.35\frac{r_{n}}{r_{c}}+0.31\bigg{(}\frac{r_{n}}{r _{t}}\bigg{)}^{3}\bigg{]}r_{n}, \tag{4}\]  
\[\Delta l_{2}=0.82\bigg{[}1-0.235\frac{r_{n}}{r_{t}}-1.32\bigg{(}\frac{r_{n}}{r _{t}}\bigg{)}^{2}+1.54\bigg{(}\frac{r_{n}}{r_{t}}\bigg{)}^{3}-0.86\bigg{(} \frac{r_{n}}{r_{t}}\bigg{)}^{4}\bigg{]}r_{n}, \tag{5}\]  
where \(r_{t}=\sqrt{S_{t}/\pi}\) is the effective radius of the pipe, and \(S_{t}\) is the area of the pipes. The entire performance of the acoustical system can be characterized using the transfer matrix method. The comprehensive transfer matrix in Figure 2b is \(T=T_{\mathrm{HR-1}}T_{\mathrm{tube}}T_{\mathrm{HR-2}}\), where \(T_{\mathrm{HR-i}}\) and \(T_{\mathrm{tube}}\) are the representations of the transfer matrices for HR-\(i\) and the tube segment that exists between a pair of HRs [41]:  
\[T_{\mathrm{HR-i}}= \begin{bmatrix}1&0\\ 1/Z_{\mathrm{HR-i}}&1\end{bmatrix} , \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(I_{\mathrm{ni}}\) **(mm)** & \(r_{\mathrm{i}}\) **(mm)** & \(I_{\mathrm{ci}}\) **(mm)** & \(D_{\mathrm{i}}\) **(mm)** \\ \hline HR-1 & 11.5 & 6 & 53 & 46 \\ HR-2 & 11 & 6 & 60 & 46 \\ HR-3 & 12 & 6 & 58 & 46 \\ HR-4 & 11 & 6 & 68 & 46 \\ HR-5 & 11 & 6 & 48 & 46 \\ HR-6 & 10 & 6 & 55 & 46 \\ HR-7 & 5.5 & 1 & 20 & 16 \\ HR-8 & 6.5 & 1 & 25 & 16 \\ HR-9 & 5 & 1 & 20 & 16.5 \\ HR-10 & 4.5 & 1 & 16 & 16 \\ HR-11 & 4.5 & 1 & 14 & 14.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical dimensions of Helmholtz resonators. \[T_{tube}= \begin{bmatrix}\cos(kl)&jZ_{0}\sin(kl)\\ j\sin(kl)/Z_{0}&\cos(kl)\end{bmatrix} , \tag{7}\]  
The transmittance and reflectance are obtained as  
\[T=\left|\frac{2e^{ikd}}{T_{11}+T_{12}/Z_{0}+T_{21}Z_{0}+T_{22}}\right|^{2}, \tag{8}\]  
\[R=\left|\frac{T_{11}+T_{12}/Z_{0}-T_{21}Z_{0}-T_{22}}{T_{11}+T_{12}/Z_{0}+T_{21 }Z_{0}+T_{22}}\right|^{2}, \tag{9}\]  
Subsequently, the absorption can be determined as follows: \(A=1-R-T\).