## 2 Mathematical Solution  
### Concentrated Forces  
In general, the CF as the PS excitation of the AE is described as follows:  
\[\mathbf{f}=P\delta(\mathbf{x}-\mathbf{x}_{0})e^{-i\omega t}, \tag{1}\]  
where \(\mathbf{P}\) is the force vector acting at \(\mathbf{x}_{0}\), and \(\omega\ (=2\pi\
u)\) is the angular frequency that transfers the AE energy from the PS to a solid space. In Equation (1), the delta function is associated with [PERSON]'s function \(g(\mathbf{x};\mathbf{x}_{0})\), providing the spatial distribution of the CF at a given time. Since \(\
abla^{2}g(\mathbf{x};\mathbf{x}_{0})=\delta(\mathbf{x}-\mathbf{x}_{0})\), the CF generated by the PS can be rewritten in terms of vectors as the following:  
\[\mathbf{f}=P\
abla^{2}g(\mathbf{x};\mathbf{x}_{0})e^{-i\omega t},\] (2a) or \[\mathbf{f}=\{\
abla[\
abla\cdot\mathbf{P}g(\mathbf{x};\mathbf{x}_{0})]-\
abla\times[\
abla \times\mathbf{P}g(\mathbf{x};\mathbf{x}_{0})]\}e^{-i\omega t}. \tag{2b}\]  
The derivation of the [PERSON]'s function for the cylindrical cell structure [22; 23; 24] is summarized in Appendix A. Applying the periodicity of \(2\pi\ (v=0)\) to Equations (A18)-(A20) gives the following:  
\[g(\xi,\vartheta\eta;0)=G_{0}(\eta)J_{0}(\kappa_{\eta}\xi), \tag{3}\]  
\[G_{0}=-\frac{1}{2\kappa_{z}}A_{0}J_{0}\big{(}\kappa_{\eta}\xi_{0}\big{)}\times \begin{cases}e^{\kappa_{\eta}\eta}(0\leq\eta\leq l-z_{0})\\ e^{-\kappa_{\eta}\eta}(-z_{0}\leq\eta\leq 0)\end{cases} , \tag{4}\]  
\[A_{0}=\frac{1}{\pi\Big{[}e^{\kappa_{\eta}z_{0}}+e^{-k_{\eta}(l-z_{0})}-2\Big{]} }\times\frac{2}{\Big{\{}a^{2}\big{[}J_{1}\big{(}\kappa_{\eta}a\big{)}\big{]}^ {2}-b^{2}\big{[}J_{1}\big{(}\kappa_{\eta}b\big{)}\big{]}^{2}\Big{\}}}, \tag{5}\]  
where the PS is located at \(\xi=0,\vartheta=0\) and \(\eta=0\). In Equation (3), the variable \(\xi\) corresponds to the distance between the PS and a given point. Since the thickness is much shorter than the diameter of the inner circle, the value of \(\xi\) can be given as the length of the arc (\(\xi_{P}\)) around the outer circle (Figure A1):  
\[\xi\cong\xi_{P}=\frac{\pi\theta}{180}a=\frac{\pi(\vartheta-90)}{90}a, \tag{6}\]  
where the PS and the detector are located on the outer surface of the cylinder. (Notes. Corrects a typographical error in \(\xi_{P}\) inserted in Figure 2b of Ref. [22].) In Refs. [20; 22], the value of \(\kappa_{z}\), defined as Equation (A4), was determined by applying boundary conditions to the Bessel function \(J_{0}(\kappa_{z}r)\) in Equation (A7). For the outer surface, \(J_{0}(\kappa_{z}r)|_{r=a-\tau_{0}}=0\) gives \(\kappa_{z}=r_{0\eta}/(a-\tau_{0})\). On the other hand, in Ref. [22], the value of \(\kappa_{z}\) was determined experimentally from the angular dependence of the AE amplitude. The subscripts \(v\) and of \(A_{vn}\) and \(G_{vn}\) represent the azimuthal constant and the nth-order root in the boundary conditions, respectively. In this study, the value of \(\kappa_{\eta}\) was determined using the external excitation method; therefore, the subscript \(n\) was omitted in Equations (3)-(5). The FRS is the most effective excitation source for AE due to fluid leakage through a pinhole. For turbulent flow with a radial turbulent velocity of \(v_{r}^{\prime}\) through a circular hole in the cylindrical structure, the CF is exerted on the wall of the pinhole by the FRS. Denoting the fluctuating velocities along the axial (z) and tangential (\(\theta\)) directions as \(v_{z}^{\prime}\) and \(v_{\theta}^{\prime}\), respectively, the FRS at low Mach numbers is defined as follows:  
\[T_{rz}^{\prime}=-\rho v_{r}^{\prime}v_{z}^{\prime}\text{and }T_{r\theta}^{ \prime}=-\rho v_{r}^{\prime}v_{t}^{\prime}, \tag{7}\]  
where \(\rho\) is the density of the charged gas.