# Acoustic Pressure Amplification through In-Duct Sonic Black Holes
## 2 Materials and Methods
### Transfer Matrix Formulation  
The SBH acoustic sensing properties under plane wave propagation conditions will now be examined using the Transfer Matrix Method (TMM) valid up to the first duct cut-on frequency \(f_{c}\). The total transfer matrix \(\mathbf{T}\) between the SBH outlet and inlet is obtained from the product of the individual transfer matrices \(\mathbf{T}_{n}\) related to each SBH unit cell, made up of a sidebranch cavity and the adjacent ring wall. It reads  
\[\mathbf{T}=\prod_{n=1}^{N}\mathbf{T}_{n}\,=\, \begin{bmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{bmatrix} . \tag{5}\]  
The continuity of the acoustic pressure \((p_{n}=p_{n+1})\) and acoustic flow rate \((u_{n}=Y_{\text{cav},n}p_{n+1}+u_{n+1})\) across \(n^{\text{th}}\) cavity-ring yields a relationship between the input and output variables, \([p_{n}\ u_{n}]^{\text{T}}=\mathbf{T}_{n}[p_{n+1}\ u_{n+1}]^{\text{T}}\), such that  
\[\mathbf{T}_{n}= \begin{bmatrix}\cos(k_{0}w)&\text{j}\frac{Z_{0}}{S}\text{sin}( k_{0}w)\\ \frac{S}{Z_{0}}\text{sin}(k_{0}w)&\cos(k_{0}w)\end{bmatrix}\begin{bmatrix}\cos (k_{0}w_{p})&\text{j}\frac{Z_{0}}{S}\text{sin}(k_{0}w_{p})\\ \text{j}\frac{S}{Z_{0}}\text{sin}(k_{0}w_{p})&\cos(k_{0}w_{p})\end{bmatrix} \begin{bmatrix}1&0\\ Y_{\text{cav},n}&1\end{bmatrix}  \tag{6}\]  
where the \(n\)-th cavity sidebranch admittance is given by \(Y_{\text{cav},n}=S_{\text{cav}}y(x_{n})/Z_{0}\) and \(y(x_{n})\) is obtained from Equations (3a)-(3c) for C-SBH, O-SBH and MPP-SBH. \(S_{\text{cav}}\) reads \(2\pi Rw\) (resp. \(2\pi(R-D_{n})w\)) for outer (resp. inner) SBHs. Also, \(S=\pi R^{2}\) (resp. \(\pi\,(R-D_{n})^{2}\)) for outer (resp. inner) SBHs. Visco-thermal effects are accounted for through complex frequency-dependent sound speed and air density quantities obtained from the Johnson-Champoux-Allard-Lafarge model [25; 26]. The fraction of the incident power dissipated by the rigidly-backed one-port C-SBH is deduced from its reflection coefficient \(r\) as \(\eta=1-|r|^{2}\) with \(r=(z_{0}T_{12}-T_{22})/(z_{0}T_{12}-T_{22})\) and \(z_{0}=S/Z_{0}\). As for the opened two-port SBHs, one obtains \(\eta=1-|r|^{2}-|t|^{2}\) with \(r=[(T_{11}+z_{0}T_{12})-(T_{21}+z_{0}T_{22})]/[(T_{11}+z_{0}T_{12})+(T_{21}+z_ {0}T_{22})]\) and \(t=(1+r)/(T_{11}+z_{0}T_{12})\) the transmission coefficient. In both cases, the SBH specific input impedance is obtained from \(z_{\text{in}}=\widetilde{r}+\widetilde{y}\widetilde{n}=(1+r)/(1-r)\) with \(\widetilde{r}\) (resp. \(\widetilde{m}\)) the specific input resistance (resp. reactance). The ability of the SBH sensors to amplify the magnitude of an incident plane wave is described by the pressure gain factor defined as \(\text{PG}(x_{n};\omega)=p_{\text{cav-back},n}/\,p_{\text{rigid},n}\) with \(p_{\text{cav-back},n}\) (resp. \(p_{\text{rigid},n}\)) the acoustic pressure at the back of the \(n^{\text{th}}\) cavity (resp. in the rigid duct at the same axial position \(x_{n}\) as the \(n^{\text{th}}\) cavity). The pressure at the cavity back,\(p_{\text{cav-back},n}\), is related to the pressure at the cavity mouth, \(p_{\text{cav},n}\), as follows  
\[p_{\text{cav-back},n}=p_{\text{cav},n}\frac{J_{0}(k_{0}d_{\text{max}})H_{1}(k_ {0}d_{\text{max}})-J_{1}(k_{0}d_{\text{max}})H_{0}(k_{0}d_{\text{max}})}{J_{0 }(k_{0}d_{\text{min}})H_{1}(k_{0}d_{\text{max}})-J_{1}(k_{0}d_{\text{max}})H_ {0}(k_{0}d_{\text{min}})}, \tag{7}\]  
with \((d_{\text{max}},d_{\text{min}})=(R+D_{n},R)\) for outer O- and MPP-SBHs and \((d_{\text{max}},d_{\text{min}})=(R,R-D_{n})\) for inner C-SBHs, \(J_{0}\), \(J_{1}\), \(H_{0}\) and \(H_{1}\) being Bessel and Hankel functions of zeroth and first orders. In the case of squared cross-sectional silencers, Equation (7) turns into \(p_{\text{cav-back},n}=p_{\text{cav},n}\tan(k_{0}D_{n})\). Note that Equation (7) in the cylindrical (resp. square) configurations assumes locally reacting silencers with normal plane wave propagation in the cavities, neglecting azimuthal (resp. tangential) wave propagation. The latter effects have been observed in the experiment, as discussed in Section 4.