## 2 Fundamental Equation of the Spectral Element Method
### Potential Energy Equation of Displacement Field  
According to [PERSON]'s theorem, the displacement field can be represented by the sum of scalar potential \(\varphi\) and vector potential \(\delta\)[33,34], as shown below:  
\[\overset{\rightarrow}{u}=\
abla\varphi+\
abla\overset{\rightarrow}{\delta} \tag{4}\]  
The axially symmetric vector potential \(\delta\) only has a single component \(\delta_{\theta}\). In this study, \(u\) represents the displacement in the \(r\) direction, \(w\) represents the displacement in the \(z\) direction, and the relationship between displacement and potential is as follows:  
\[u=\frac{\partial\varphi}{\partial r}-\frac{\partial\delta}{\partial z} \tag{5}\]  
\[w=\frac{\partial\varphi}{\partial z}+\frac{1}{r}\frac{\partial\sigma}{\partial r} \tag{6}\]  
By substituting Equations (5) and (6) into Equation (2), and combining Equations (1) and (3), we obtain:  
\[\frac{\partial^{2}\varphi}{\partial r^{2}}+\frac{1}{r}\frac{\partial\varphi}{ \partial r}+\frac{\partial^{2}\varphi}{\partial z^{2}}=\frac{1}{c_{p}^{2}} \frac{\partial^{2}\varphi}{\partial t^{2}} \tag{7}\]\[\frac{\partial^{2}\delta}{\partial r^{2}}+\frac{1}{r}\frac{\partial\delta}{ \partial r}+\frac{\partial^{2}\delta}{\partial z^{2}}-\frac{\delta}{r^{2}}=\frac {1}{c_{s}^{2}}\frac{\partial^{2}\delta}{\partial t^{2}} \tag{8}\]  
where \(c_{p}=\left(\left(\lambda+2G\right)/\rho\right)^{1/2}\) and \(c_{s}=\left(G/\rho\right)^{1/2}\) represent the compression and shear wave velocities, respectively. Applying a Fourier transform to Equations (7) and (8) can convert the expressions from the time domain to the frequency domain for further solution:  
\[\frac{\partial^{2}\widetilde{\varphi}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial\widetilde{\varphi}}{\partial r}+\frac{\partial^{2}\widetilde{ \varphi}}{\partial z^{2}}=-\frac{1}{c_{p}^{2}}\omega^{2}\widetilde{\varphi} \tag{9}\]  
\[\frac{\partial^{2}\widetilde{\delta}}{\partial r^{2}}+\frac{1}{r}\frac{ \partial\widetilde{\delta}}{\partial r}+\frac{\partial^{2}\widetilde{\delta} }{\partial z^{2}}-\frac{\widetilde{\delta}}{r^{2}}=-\frac{1}{c_{s}^{2}}\omega ^{2}\widetilde{\delta} \tag{10}\]  
Since the function \(\widetilde{\varphi}\) is related to \(r\) and \(z\), the function can be constructed by multiplying two separate functions \(r\) and \(z\). \[\widetilde{\varphi}=\widetilde{R}\bullet\widetilde{Z} \tag{11}\]  
In this equation, the function \(\widetilde{R}\) and the function \(\widetilde{Z}\) represent those related only to \(r\) and \(z\), respectively. By substituting Equation (11) into Equation (9) and simplifying both sides, two independent ordinary differential equations in terms of \(\widetilde{R}\) and \(\widetilde{Z}\) are derived, as follows:  
\[\frac{\mathrm{d}^{2}\widetilde{R}}{\mathrm{d}r^{2}}+\frac{1}{r}\frac{\mathrm{ d}\widetilde{R}}{\mathrm{d}r}+k^{2}\widetilde{R}=0 \tag{12}\]  
\[\frac{\mathrm{d}^{2}\widetilde{Z}}{\mathrm{d}z^{2}}+\Bigg{(}\frac{\omega^{2}}{ c_{p}^{2}}-k^{2}\Bigg{)}\widetilde{Z}=0 \tag{13}\]  
Assuming, and using the chain rule, Equation (12) can be transformed into a Bessel equation, as shown below:  
\[u=\frac{\partial\varphi}{\partial r}-\frac{\partial\delta}{\partial z} \tag{5}\]0  
The solution form of Equation (14) consists of a sum of the first kind Bessel function \(J_{0}\) and the second kind Bessel function \(Y_{0}\). When \(r\) equals zero, \(Y_{0}\) is singular, which means it will go to infinity. However, because the oscillation at the origin is not infinite, the solution of \(Y_{0}\) should be discarded. The function for \(\widetilde{R}_{r}\) can be simplified as:  
\[u=\frac{\partial\varphi}{\partial r}-\frac{\partial\delta}{\partial z} \tag{5}\]1  
in which \(A_{1}\) is a constant that can be determined by boundary conditions; \(k\) indicates the radial wavenumber. For the purpose of obtaining a discretized solution, it is necessary to introduce a boundary condition. This paper assumes that at the boundary where \(r=R\), the oscillation amplitude completely vanishes.