## Appendix A Derivation of the \(A,b,c\) and \(D\) in the TM
 \tag{16}\]  
According to the orthogonality property of eigenfunctions, Equation (16) is transformed with \(m=m^{\prime}\) and \(n=n^{\prime}\) to the following equation:  
\[\iint_{S_{i}}\overline{v}\cos\left(\frac{m\pi x}{b_{i}}\right)\cos\left(\frac {n\pi y}{h}\right)\mathrm{d}x\mathrm{d}y=-\frac{1}{\rho_{0}\omega}k_{z,mn,j} \iint_{S_{i}}\cos^{2}(\frac{m\pi x}{b_{i}})\cos^{2}(\frac{n\pi y}{h})\mathrm{ d}x\mathrm{d}y\{C_{+,mn}K_{1}(-\beta_{i,mn})-C_{-,mn}K_{1}(\beta_{i,mn})\}. \tag{17}\]  
The coefficients \(C_{+,mn}\) and \(C_{-,mn}\) can be calculated from Equations (15) and (17) as follows:  
\[C_{+,mn}=-\rho_{0}c\overline{v}\frac{kK_{1}(\beta_{o,mn})}{k_{z,mn,i}W(k_{z, mn})}\frac{\iint_{S_{i}}\cos(\frac{m\pi x}{b_{i}})\cos(\frac{n\pi y}{h})\mathrm{d}x \mathrm{d}y}{\iint_{S_{i}}\cos^{2}(\frac{m\pi x}{b_{i}})\cos^{2}(\frac{n\pi y} {h})\mathrm{d}x\mathrm{d}y} \tag{18}\]  
\[C_{-,mn}=\frac{C_{+,mn}K_{1}(-\beta_{o,mn})}{K_{1}(\beta_{o,mn})} \tag{19}\]  
where  
\[W(k_{z,mn})=K_{1}(-\beta_{i,mn})K_{1}(\beta_{o,mn})-K_{1}(\beta_{i,mn})K_{1}(- \beta_{o,mn}). \tag{20}\]  
Substituting Equations (18) and (19) into Equation (11), the 3D solution of the pressure is obtained as follows:  
\[p(x,y,z)=-\rho_{0}c\overline{v}\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{V(k _{z,mn},z)}{W(k_{z,mn},z)}\frac{k}{k_{z,mn,j}}\frac{\cos(\frac{m\pi x}{h}) \cos(\frac{n\pi y}{h})\iint_{S_{i}}\cos(\frac{m\pi x}{h})\cos(\frac{n\pi y}{h} )\mathrm{d}x\mathrm{d}y}{\iint_{S_{i}}\cos^{2}(\frac{m\pi y}{h})\cos^{2}(\frac {n\pi y}{h})\mathrm{d}x\mathrm{d}y} \tag{21}\]where  
\[V(k_{z,mn},z)=K_{1}(\beta_{o,mn})K_{0}(-\beta_{mn})+K_{0}(\beta_{mn})K_{1}(-\beta_ {o,mn}). \tag{107}\]  
Substituting Equation (106) with \(z=0\) into Equation (16) to obtain the average pressure at the inlet as follows:  
\[p_{i} =\frac{1}{S_{i}}\iint_{S_{i}}\bigg{\{}-\rho_{0}c\overline{c} \overline{v}\sum\limits_{m=0}^{\infty}\sum\limits_{n=0}^{\infty}\frac{V(k_{z,mn },0)}{W(k_{z,mn})}\frac{k}{k_{z,mn},i}\frac{\cos(\frac{n\pi\pi\imath}{n\imath}) \iint_{S_{i}}\cos(\frac{n\pi\imath\imath}{n\imath})\cos(\frac{n\pi\imath\imath} {n\imath})\mathrm{d}x\mathrm{d}y}{\iint_{S_{i}}\cos^{2}(\frac{n\pi\imath\imath }{n\imath})\cos^{2}(\frac{n\pi\imath\imath}{n\imath})\mathrm{d}x\mathrm{d}y} \bigg{\}}\mathrm{d}x\mathrm{d}y \tag{108}\] \[=-\rho_{0}c\overline{c}_{1}\bigg{\{}\underbrace{\frac{V(k_{z,mn },0)}{W(k_{z,mn})}\frac{2k}{k_{z,mn},i}\bigg{(}\frac{2}{n\imath\imath}\Psi_{ \hbar 1}\bigg{)}^{2}}_{E_{13}}}\] \[=-\rho_{0}c\overline{c}_{1}\bigg{\{}\underbrace{\frac{V(k_{z,mn },i,0)}{W(k_{z,mn})}\frac{4k}{k_{z,mn},i}\bigg{(}\frac{4}{mn\pi^{2}}\Psi_{ \hbar 1}\Psi_{\hbar 1}\bigg{)}^{2}}_{E_{14}}}\] \[=-\rho_{0}c\overline{c}_{1}E_{1}\]  
where  
\[C_{+,mn}=-\rho_{0}c\overline{v}\frac{kK_{1}(\beta_{o,mn})}{k_{z,mn,i}W(k_{z, mn})}\frac{\iint_{S_{i}}\cos(\frac{m\pi x}{b_{i}})\cos(\frac{n\pi y}{h})\mathrm{d}x \mathrm{d}y}{\iint_{S_{i}}\cos^{2}(\frac{m\pi x}{b_{i}})\cos^{2}(\frac{n\pi y} {h})\mathrm{d}x\mathrm{d}y} \tag{18}\]0  
\[C_{+,mn}=-\rho_{0}c\overline{v}\frac{kK_{1}(\beta_{o,mn})}{k_{z,mn,i}W(k_{z, mn})}\frac{\iint_{S_{i}}\cos(\frac{m\pi x}{b_{i}})\cos(\frac{n\pi y}{h})\mathrm{d}x \mathrm{d}y}{\iint_{S_{i}}\cos^{2}(\frac{m\pi x}{b_{i}})\cos^{2}(\frac{n\pi y} {h})\mathrm{d}x\mathrm{d}y} \tag{18}\]1  
\[C_{+,mn}=-\rho_{0}c\overline{v}\frac{kK_{1}(\beta_{o,mn})}{k_{z,mn,i}W(k_{z, mn})}\frac{\iint_{S_{i}}\cos(\frac{m\pi x}{b_{i}})\cos(\frac{n\pi y}{h})\mathrm{d}x \mathrm{d}y}{\iint_{S_{i}}\cos^{2}(\frac{m\pi x}{b_{i}})\cos^{2}(\frac{n\pi y} {h})\mathrm{d}x\mathrm{d}y} \tag{18}\]2  
The quantity \(E_{11}\) is obtained with \(m=0\) and \(n=0\). \(E_{12}\) is obtained with \(m\geq 1\) and \(n=0\). \(E_{13}\) is obtained with \(m=0\) and \(n\geq 1\). Additionally, \(E_{14}\) is obtained with \(m\geq 1\) and \(n\geq 1\).