## 2 Analytical Model
### Displacement Fields  
The NL equation is the fundamental equation governing wave motion in elastic and homogeneous media. If the media is subjected to a non-equilibrium local force \(\mathbf{f}\), the NL equation can be written in vector form [24] as  
\[\left(\lambda+2\mu\right)\mathbf{\
abla}(\mathbf{\
abla}\mathbf{\cdot}\mathbf{u})-\mu\mathbf{ \
abla}\times\mathbf{\
abla}\times\mathbf{u}+\mathbf{f}=\rho\frac{\partial^{2}\mathbf{u}}{ \partial t^{2}}, \tag{10}\]  
where \(\mathbf{u}\) is the displacement vector, \(\lambda\) and \(\mu\) are Lame constants, and \(\rho\) is the density of the media. The displacement field in cylindrical coordinates is specified by three potentials: the scalar potential \(\Phi\) for the P wave, and two vector potentials, \(X\hat{e}_{2}\) for the SH wave and \(\
abla\hat{e}_{2}\) for the SV wave. Previously, we adapted the model proposed by [PERSON] and [PERSON] [19], expressed by  
\[\mathbf{u}=\mathbf{\
abla}\Phi+\mathbf{\
abla}\times(X\hat{e}_{2})+a\mathbf{\
abla}\times\mathbf{ \
abla}\times(\
abla\hat{e}_{2}), \tag{11}\]  
because the three components were easily separated. The displacement vector can be described by the displacement components in the \((r,\theta,z)\) coordinates  
\[\mathbf{u}=u_{r}\hat{\mathbf{r}}+u_{\theta}\hat{\mathbf{\theta}}+u_{z}\hat{\mathbf{z}}, \tag{12}\]  
where  
\[u_{r}=\frac{\partial\Phi}{\partial r}+\frac{1}{r}\frac{\partial X}{\partial \theta}+a\frac{\partial^{2}\Psi}{\partial r\partial z}, \tag{13}\]  
\[u_{\theta}=\frac{1}{r}\frac{\partial\Phi}{\partial\theta}-\frac{\partial X}{ \partial r}+\frac{a}{r}\frac{\partial^{2}\Psi}{\partial\theta\partial z}, \tag{14}\]  
\[u_{z}=\frac{\partial\Phi}{\partial z}-a\left(\frac{\partial^{2}\Psi}{\partial r ^{2}}+\frac{1}{r}\frac{\partial\Psi}{\partial r}+\frac{1}{r^{2}}\frac{ \partial^{2}\Psi}{\partial\theta^{2}}\right). \tag{15}\]  
Turbulence is generated through the pinhole, but it is maximized at the leaking orifice due to edge discontinuities. Since the hole cross-sectional area is very small compared to the cylinder surface, the force due to turbulent outflow at the orifice surface can be treated as PS. As the PS, the force \(\mathbf{f}\) due to FRS is expressed as follows  
\[\mathbf{f}=\mathbf{P}_{w}\delta(\mathbf{x}-\mathbf{x}_{0})e^{-i\omega t}. \tag{16}\]  
In Equation (16), \(\omega\) is the angular frequency that transmits FRS energy from the PS to the cylinder and resonates with the energy of cylinder materials. As a solution to the delta function, [PERSON]'s function \(g(x;\mathbf{x}_{0})\) is defined as  
\[\
abla^{2}g(x;\mathbf{x}_{0})=\delta(\mathbf{x}-\mathbf{x}_{0}). \tag{17}\]  
In cylindrical coordinates, Equation (17) is expressed as  
\[\
abla^{2}g(r,\theta,z;r_{0},\theta_{0},z_{0})=\frac{\delta(r-r_{0})\delta( \theta-\theta_{0})\delta(z-z_{0})}{r}, \tag{18}\]  
and the solution of Equation (18) is given as [18]  
\[g(r,\theta,z;r_{0},\theta_{0},z_{0})=A_{v1}g_{r}(r;r_{0})g_{\theta}(\theta; \theta_{0})g_{z}(z;z_{0}), \tag{19}\]  
where \(A_{v1}\) is the coupling constant, expressed as the first root of the Bessel function in \(g_{r}\) and the integer \(v\) of the aperiodicity in \(g_{\theta}\). To make the position of the PS the new origin, as shown in Figure 1b, the coordinates \((r,\theta,z;r_{0},\theta_{0},z_{0})\) are replaced by \((\xi,\theta,\eta;0)\), defined as  
\[\mathbf{u}=\mathbf{\
abla}\Phi+\mathbf{\
abla}\times(X\hat{e}_{2})+a\mathbf{\
abla}\times\mathbf{ \
abla}\times(\
abla\hat{e}_{2}), \tag{11}\]0  
where the PS is located on the \(x_{i}\) axis, \(r_{0}=x_{0i}\). Previously [18], we derived the Green's function responsible for the PS located inside the cylinder. Assuming that the Green's function is periodic (\(v=0\))  
\[\mathbf{u}=\mathbf{\
abla}\Phi+\mathbf{\
abla}\times(X\hat{e}_{2})+a\mathbf{\
abla}\times\mathbf{ \
abla}\times(\
abla\hat{e}_{2}), \tag{11}\]1  
\[G_{01}=-\frac{1}{2\kappa_{z}}A_{01}J_{0}(\kappa_{z}\xi_{0}) \begin{cases}e^{- \kappa_{z}\eta}(0\leq\eta\leq l-z_{0})\\ e^{\kappa_{z}\eta}\quad\quad(-z_{0}\leq\eta<0)\end{cases}