## 2 Materials and Methods
### Theoretical Foundation  
In the cylindrical coordinate system \((r,\theta,z)\), the elastodynamic behavior of an infinite multilayered elastic cylindrical shell under normally incident plane wave excitation is investigated using the displacement potential function method. Let the wave vector of the incident wave be \(k\). The \(z\)-direction displacement \(U_{z}=0\) in the elastic cylindrical structure can be fully characterized using two scalar potential functions \(\phi\) (dilatational) and \(\psi_{z}\) (shear), which satisfy the Navier-Cauchy equation:  
\[\Big{(}\
abla^{2}+k_{d}^{2}\Big{)}\phi(r,\theta)=0,\Big{(}\
abla^{2}+k_{s}^{2 }\Big{)}\psi_{z}(r,\theta), \tag{1}\]  
where  
\(\phi\) = Dilatational potential function. \(\psi_{z}\) = Shear potential function. \(k_{d}=\omega\sqrt{\rho/(\lambda+2\mu)}\) = Dilatational wavenumber. \(k_{s}=\omega\sqrt{\rho/\mu}\) = Shear wavenumber. \(\lambda\), \(\mu\) = Lane constants of the material. \(\omega\) = Angular frequency. As depicted in Figure 1, a plane wave is incident on an infinitely long multilayered elastic cylindrical shell, inducing scattered waves. The internal solutions within the elastic cylinder derived through the separation of variables method are expressed as follows:  
\[\phi_{m}(r,\theta,t)=p_{0}e^{-i\omega t}\sum_{n=0}^{\infty}i^{n}\varepsilon_{n} \Big{[}A_{n}^{(I)}\mathrm{J}_{n}(k_{dn}r)+E_{n}^{(I)}\mathrm{Y}_{n}(k_{dn}r) \Big{]}\mathrm{cos}\,n\theta, \tag{2}\]  
\[\psi_{z_{m}}(r,\theta,t)=p_{0}e^{-i\omega t}\sum_{n=0}^{\infty}i^{n}\varepsilon _{n}\Big{[}B_{n}^{(I)}\mathrm{J}_{n}(k_{dn}r)+F_{n}^{(I)}\mathrm{Y}_{n}(k_{dn} r)\Big{]}\mathrm{sin}\,n\theta, \tag{3}\]  
where  
\(A_{n}^{(I)},E_{n}^{(I)},B_{n}^{(I)},F_{n}^{(I)}=\) Mode coefficients for layer \(l\). \(\mathrm{J}_{n},\mathrm{Y}_{n}=\) Bessel and Neumann functions of order \(n\)[21]. \(r,\theta=\) Cylindrical coordinates. The displacements and stresses must satisfy the following function:  
\[u_{r}=\frac{\partial\phi}{\partial r}+\frac{1}{r}\frac{\partial\psi_{z}}{ \partial\phi},u_{\theta}=\frac{\partial\phi}{\partial\theta}-\frac{\partial \psi_{z}}{\partial r}, \tag{4}\]  
\[\tau_{rr}=-\lambda_{e}k_{d}^{2}\phi+2\mu_{e}\bigg{[}\frac{\partial^{2}\phi}{ \partial r^{2}}+\frac{\partial}{\partial r}\bigg{(}\frac{1}{r}\frac{\partial \psi_{z}}{\partial\theta}\bigg{)}\bigg{]}, \tag{5}\]  
\[\tau_{r\theta}=-\mu_{e}\bigg{[}2\frac{\partial^{2}}{\partial r\partial\theta} \frac{\phi}{r}-\bigg{(}\frac{2}{\partial r^{2}}-\frac{1}{r}\frac{\partial \psi_{z}}{\partial r}-\frac{1}{r^{2}}\frac{\partial^{2}\psi_{z}}{\partial \theta^{2}}\bigg{)}\bigg{]}. \tag{6}\]  
The exterior and interior acoustic fields of the cylinder can be expressed as follows:  
\[p_{out}(r,\theta,t)=p_{0}e^{-i\omega t}\sum_{n=0}^{\infty}i^{n}\varepsilon_{ n}\Big{[}\mathrm{J}_{n}(kr)+b_{n}\mathrm{H}_{n}^{(1)}(kr)\Big{]}\mathrm{ cos}\,n\theta, \tag{7}\]  
\[p_{in}(r,\theta,t)=p_{0}e^{-i\omega t}\sum_{n=0}^{\infty}i^{n}\varepsilon_{n} \mathrm{J}_{n}(k_{0}r)\mathrm{cos}\,n\theta. \tag{8}\]  
By applying the continuity conditions for stress and displacement, we derive the system of equations for an infinitely long cylindrical shell with an exterior fluid medium and an interior air-filled cavity. The matrix dimensionality is dependent on the number of layers. For instance, a two-layer shell corresponds to a \(6\times 6\) matrix, three-layer to \(10\times 10\)  
Figure 1: Plane wave incident on an infinitely long multilayered elastic cylindrical shell. four-layer to 14 \(\times\) 14, and so forth. The solution to the system is given by \(b_{n}=-B_{n}/D_{n}\), with matrix elements detailed in Appendix A. When a plane wave is incident normally on an infinitely long cylindrical shell, the scattered sound pressure \(p_{s}\) can be expressed as follows:  
\[p_{s}(r,\theta)=p_{0}e^{-i\omega t}\sum_{n=0}^{\infty}i^{n}\varepsilon_{n}b_{n }\mathrm{H}_{\mathrm{n}}^{(1)}(k_{0}r)\cos(n\theta), \tag{9}\]  
where  
\(b_{n}=\) Scattering coefficients. \(\mathrm{H}_{\mathrm{n}}^{(1)}=\) Hankel function of first kind.