## 2 Theoretical Models
### Solving for the Conductance Function of a Thin Underwater Cylindrical Shell  
Establish the (\(r\), \(\varphi\), \(z\)) cylindrical coordinate system on the middle surface of the thin cylindrical shell, and let the origin of the coordinates lie at the centre of the circle on the left face of the cylindrical shell, \(r\) and \(z\) are the distances from the spatial point \(\sigma(r\), \(\varphi\), \(z\)) to the central axis and the left face of the thin cylindrical shell respectively, and \(\varphi\) is the angle at which \(\sigma(r\), \(\varphi\), \(z)\) deviates from the vertical direction; the transformation relationship between cylindrical coordinates and Cartesian coordinates is: \(x=r\text{cos}\varphi\), \(y=r\text{sin}\varphi\), \(z=z\). Set the length of the thin cylindrical shell as \(L\), the thickness as \(d\) and the radius of the middle surface as a (\(d/a\leq 0.05\)); the density, Poisson's ratio and modulus of elasticity of the cylindrical shell material are set as \(\rho\),\(\mu\),\(E\), respectively. Apply a simple harmonic excitation force \(F_{z}\)-e\({}^{\text{i}\omega}\) along the axial direction (z coordinate axis direction) at the inner side of the thin cylindrical shell \(c_{\text{e}}(q_{\varphi},z_{\text{e}})\).Let \(u(z\), \(\varphi\), \(t)=U(z\), \(\varphi)\)-e\({}^{\text{i}\omega t}\), \(v(z\), \(\varphi\), \(t)=V(z\), \(\varphi)\)-e\({}^{\text{i}\omega t}\), \(w(z\), \(\varphi\), \(t)=W(z\), \(\varphi)\)-e\({}^{\text{i}\omega t}\) denote the vibrational displacement of the point \(\sigma(\varphi,z)\) on the cylindrical shell in the \(z\),\(\varphi\),and \(r\) coordinate directions, respectively, then \(u\),\(\varphi\) and \(w\) satisfy the following differential equations of motion [25]. \[\left\{ \begin{array}{c}S_{1}(u,v,w)+\rho d\cdot\partial^{2}u/\partial t^{2}= F_{z}\cdot\delta(\varphi-q_{\text{e}})\cdot\delta(z-z_{\text{e}})\cdot\text{e}^{ \text{i}\omega t}\\ S_{2}(u,v,w)+\rho d\cdot\partial^{2}v/\partial t^{2}=0\\ S_{3}(u,v,w)+\rho d\cdot\partial^{2}w/\partial t^{2}=-p(r,\varphi,z)|_{r=a} \end{array} \right. \tag{7}\]  
where \(p\) is the radiated hydroacoustic pressure generated on the outside of the thin cylindrical shell due to its vibration; \(S_{1}\), \(S_{2}\), \(S_{3}\) are differential operators, and  
\[\left\{ \begin{array}{c}S_{1}(U,V,W)=-\frac{Ed}{1-\mu^{2}}\Big{[}\Big{(} \frac{\partial^{2}}{\partial z^{2}}+\frac{1-\mu}{2\mu^{2}}\frac{\partial^{2} }{\partial z^{2}}\Big{)}U+\frac{1+\mu}{2a}\frac{\partial^{2}V}{\partial z \partial\varphi}+\frac{u}{a}\frac{\partial V}{\partial z}\Big{]}\\ S_{2}(U,V,W)=-\frac{Ed}{1-\mu^{2}}\Big{[}\frac{1+\mu^{2}}{2a}\frac{\partial^{2 }U}{\partial z\partial\varphi}+\Big{(}\frac{1-\mu^{2}}{2}\frac{\partial^{2} }{\partial z^{2}}+\frac{1}{a^{2}}\frac{\partial^{2}}{\partial z^{2}}\Big{)}V +\frac{1}{a^{2}}\frac{\partial}{\partial z}\Big{(}1-\frac{\mu^{2}}{2}\
abla^ {2}\Big{)}W\Big{]}\\ S_{3}(U,V,W)=\frac{Ed}{1-\mu^{2}}\Big{[}\frac{u}{a}\frac{\partial U}{\partial z }+\frac{1}{a^{2}}\frac{\partial}{\partial\varphi}\Big{(}1-\frac{d^{2}}{12} \
abla^{2}\Big{)}V+\Big{(}\frac{1}{a^{2}}+\frac{d^{2}}{12}\
abla^{4}\Big{)}W \Big{]}\end{array} \right. \tag{8}\]  
where \(\
abla^{2}=\frac{\partial^{2}}{\partial z^{2}}+\frac{1}{a^{2}}\frac{\partial ^{2}}{\partial z^{2}}\), \(\
abla^{4}=\frac{\partial^{4}}{\partial z^{2}}+\frac{2}{a}\frac{\partial^{4}} {\partial z^{2}}+\frac{1}{a^{2}}\frac{\partial^{4}}{\partial z^{2}}+\frac{1}{ a^{2}}\frac{\partial^{4}}{\partial z^{4}}\). The sound pressure (\(p=P(r\), \(\varphi\), \(z\))-e\({}^{\text{i}\omega t}\)) of the water sound field outside the thin cylindrical shell satisfies the Helmholtz equation [26]. \[\
abla^{2}P+k^{2}P=0 \tag{9}\]  
where \(k\) is the number of waves, \(k=\omega/c_{0}\), \(c_{0}\) is the wave speed (\(c_{0}\approx 1500\) m/s in water). On the dividing surface between the thin cylindrical shell and the external hydroacoustic field, there are the following boundary conditions:  
\[\left.\frac{\partial^{P(1)}}{\partial r}\right|_{r=a}=\left\{