## 2 Theoretical Models
### Solving for the Conductance Function of a Thin Underwater Cylindrical Shell
\begin{array}{c} \rho_{0}\omega^{2}\sum_{i=0,j=1}^{\infty}\sum_{l=1}^{3}q_{ijl}^{(1)}C_{ijl}^{ \text{av}}\cos[i(\varphi-\varphi_{\epsilon})]\sin\Big{(}\frac{i\tau z}{L} \Big{)}\ (0\leq z\leq L)\\ 0\ (z<0\ \text{or}\ z>L)\end{array} \right. \tag{10}\]  
where \(\rho_{0}\) is the sound field medium density. The vibration displacements U, V and W of a thin cylindrical shell can be expressed as a superposition of vibration patterns  
\[\left\{ \begin{array}{c}U=\sum_{i,j=0}^{\infty}q_{ij}\cdot U_{ij}=\sum_{i,j= 0}^{\infty}q_{ij}\cdot A_{ij}(z)a_{j}(z)\cdot\cos[i(\varphi-\varphi_{\epsilon} )]\\ V=\sum_{i,j=1}^{\infty}q_{ij}\cdot V_{ij}=-\sum_{i,j=1}^{\infty}q_{ij}\cdot B_{ ij}(z)\beta_{j}(z)\cdot\sin[i(\varphi-\varphi_{\epsilon})]\\ W=\sum_{i,j=0}^{\infty}q_{ij}\cdot W_{ij}=\sum_{i,j=0}^{\infty}q_{ij}\cdot C_{ ij}(z)\gamma_{j}(z)\cdot\cos[i(\varphi-\varphi_{\epsilon})]\end{array} \right\} \tag{11}\]  
where \(q_{ij}\) is the modal influence factor; \(U_{ij}\), \(V_{ij}\), \(W_{ij}\) are the vibration functions, \(A_{ijl}a_{j}(z)\), \(B_{ij}\beta_{j}(z)\) and \(C_{ij}\gamma_{j}(z)\) are determined by the boundary conditions of the cylindrical thin shell. Apply the variational separation method and the Fourier integral method to the Helmholtz equation (Equation (9)) and note that \(P(r,\,\varphi,z)\) is periodic with respect to \(\varphi\); it can be shown that \(P\) has the following form of a series solution:  
\[P(r,\varphi,z)=\frac{1}{2\pi}\sum_{i=0,j=1}^{\infty}D_{ij}\cos[i(\varphi- \varphi_{\rm e})]\int_{-\infty}^{+\infty}H_{i}^{(2)}(k_{r}r)\cdot\widetilde{ \delta}_{j}(\tau)\cdot{\rm e}^{-\mathrm{j}\tau z}\mathrm{d}\tau \tag{12}\]  
where \(H_{i}^{(2)}(k_{r}r)\) is the second class i-order Hankel function, and \(k_{r}{}^{2}=k^{2}-\tau^{2}\). Substituting the above equation and Equation (3) of Equation (11) into the boundary conditions (Equation (10)) and apply a Fourier transformation to the variable \(z\) on both sides of the equation. By comparing the coefficients it is possible to obtain a solution for the sound pressure level expressed in terms of the modal influence factor \(q_{ij}\) and the vibration shape function \(C_{i\overline{j}}\gamma\left(z\right)\)  
\[P(r,\varphi,z)=\frac{\rho_{0}\omega^{2}}{2\pi}\sum_{i=0,j=1}^{\infty}q_{ij} \cos[i(\varphi-\varphi_{\rm e})]\int_{-\infty}^{+\infty}\frac{H_{i}^{(2)}(k_{ r}r)}{k_{r}\Big{[}\mathrm{d}H_{i}^{(2)}(k_{r}r)/\mathrm{d}(k_{r}r)\Big{]}_{r=a}} \widetilde{f}_{ij}(\tau)\cdot{\rm e}^{-\mathrm{j}\lambda\tau}\mathrm{d}\tau \tag{13}\]  
where \(\widetilde{f}_{ij}(\tau)=\int_{0}^{L}C_{i\overline{j}}\cdot\gamma_{j}(z)\cdot \mathrm{e}^{\mathrm{j}\tau z}\mathrm{d}z\). To obtain the final modal impact factor \(q_{ij}\), let \(r=a\) in Equation (13) and substitute it into Equation (8), and then use the orthogonality of the vibration function to obtain a linear system of equations for the modal impact factor \(q_{ij}\)  
\[M_{ij}\left(\frac{2}{ij}-\omega^{2}\right)q_{ij}+\sum_{l=1}^{\infty}\chi_{il} q_{il}=F_{z}\cdot A_{ij}a_{j}(z_{\rm e})\ (i,j=0,1,2,\ldots) \tag{14}\]  
where \(M_{i\overline{j}}\) and \(\omega_{i\overline{j}}\) are the modal masses and modal frequencies of the cylindrical thin shell corresponding to the modes \(U_{i\overline{j}}\), \(V_{i\overline{j}}\) and \(W_{i\overline{j}}\), respectively. Organizing Equation (14) into matrix form, that is, \(\mathbf{G}\mathbf{Q}=F_{z}\), \(\mathbf{Q}\) is a column vector consisting of the modal influence factor \(q_{ij}\), \(\mathbf{G}\) is a list square of the corresponding coefficients of \(q_{ij}\), and \(F_{z}\) is a modal force vector consisting of \(F_{z}A_{ij}a_{j}(z_{\rm e})\) on the right-hand side of each equation of Equation (14), then \(\mathbf{Q}=\mathbf{G}^{-1}F_{z}\).