## 2 [PERSON]'s Function  
The body force arising from the point source located at \(x_{0}\) is mathematically formulated in terms of an oscillating impulse with natural frequencies of the material in a given geometry, as follows:  
\[\mathbf{f}=\mathbf{P}(t)\delta(\mathbf{x}-\mathbf{x}_{0})e^{-i\omega t}, \tag{1}\]  
where \(\mathbf{P}(t)\) is the CF vector, \(\delta(\mathbf{x}-\mathbf{x}_{0})\) is the delta function, and \(\omega\) is the predominant angular frequency \((\omega=2\pi\
u)\) of AE. [PERSON]'s function \(g(\mathbf{x};\mathbf{x}_{0})\), as the solution of the delta function, is defined as  
\[\bar{\
u}^{2}g(\mathbf{x};\mathbf{x}_{0})=\delta(\mathbf{x}-\mathbf{x}_{0}). \tag{2}\]  
The force can be rewritten as  
\[\mathbf{f}=\bar{\
u}^{2}\mathbf{P}g(\mathbf{x};\mathbf{x}_{0})=\mathbf{V}[\mathbf{V}\cdot\mathbf{P}g(\mathbf{x };\mathbf{x}_{0})]-\mathbf{V}\times[\mathbf{V}\times\mathbf{P}g(\mathbf{x};\mathbf{x}_{0})]. \tag{3}\]  
In cylindrical coordinates, Equation (2) is expressed as  
\[\bar{\
u}^{2}g(\mathbf{r},\theta,\mathbf{z};\mathbf{r}_{0},\theta_{0},\mathbf{z}_{0})=\frac{ \delta(\mathbf{r}-\mathbf{r}_{0})\delta(\mathbf{\theta}-\theta_{0})\delta(\mathbf{z}-\mathbf{z}_{0 })}{\mathbf{r}}. \tag{4}\]  
If \(\mathbf{r}-\mathbf{r}_{0}\
eq 0,\theta-\theta_{0}\
eq 0\),  
\[\bar{\
u}^{2}g(\mathbf{x};\mathbf{x}_{0})=0. \tag{5}\]  
[PERSON]'s function is separable in a cylindrical coordinate system as  
\[g(\mathbf{x};\mathbf{x}_{0})=g_{r}(\mathbf{r},\mathbf{r}_{0})g_{\theta}(\mathbf{\theta};\theta_{0} )g_{z}(\mathbf{z};\mathbf{z}_{0}). \tag{6}\]Substituting the Laplacian in cylindrical coordinates into Equation (5) gives the following:  
\[\frac{r^{2}}{g_{r}}\frac{\partial^{2}g_{r}}{\partial r^{2}}+\frac{r}{g_{r}}\frac{ \partial g_{r}}{\partial r}+\frac{1}{g_{\theta}}\frac{\partial^{2}g_{\theta}}{ \partial\theta^{2}}+\frac{r^{2}}{g_{z}}\frac{\partial^{2}g_{z}}{\partial z^{2} }=0. \tag{7}\]  
Letting  
\[\frac{1}{g_{z}}\frac{\partial^{2}g_{z}}{\partial z^{2}}=\kappa_{z}^{2}, \tag{8}\]  
\[\frac{1}{g_{\theta}}\frac{\partial^{2}g_{\theta}}{\partial\theta^{2}}=-v^{2}, \tag{9}\]  
then Equation (7) has the form  
\[r^{2}\frac{\partial^{2}g_{r}}{\partial r^{2}}+r\frac{\partial g_{r}}{\partial r }+\Big{(}\kappa_{z}^{2}r^{2}-v^{2}\Big{)}g_{r}=0. \tag{10}\]  
The solution of Equation (8) can be formulated in two regions:  
\[g_{z}(z;z_{0})=\left\{ \begin{array}{ll}A_{z}(z_{0})e^{\kappa z}&\quad 0<z<z_{0} <l\\ B_{z}(z_{0})e^{-\kappa z}&\quad 0<z_{0}<z<l\end{array} \right. \tag{11}\]  
The continuity and discontinuity principles are applied to find coefficients \(A_{z}\) and \(B_{z}\). The continuity principle states that [PERSON]'s function is continuous around the point source, i.e.,  
\[g_{z}(z;z_{0})\big{|}_{z=z_{0}^{+}}=\left.g_{z}(z;z_{0})\right|_{z=z_{0}^{-}}. \tag{12}\]  
Equation (12) leads to  
\[A_{z}(z_{0})e^{\kappa_{z}z_{0}}=B_{z}(z_{0})e^{-\kappa_{z}z_{0}}. \tag{13}\]  
Although [PERSON]'s function is discontinuous at the point source, the difference in [PERSON]'s function between the adjacent two points, \(z_{0}\pm\varepsilon\), around the point source is unity. When \(\varepsilon\) becomes 0, the discontinuity principle leads to  
\[\left.\frac{\partial g_{z}(z;z_{0})}{\partial z}\right|_{z=z_{0}^{+}}-\left. \frac{\partial g_{z}(z;z_{0})}{\partial z}\right|_{z=z_{0}^{-}}=1. \tag{14}\]  
As a result, we obtain  
\[-\kappa_{z}B_{z}(z_{0})e^{-\kappa_{z}z_{0}}-\kappa_{z}A_{z}(z_{0})e^{\kappa z _{0}}=1. \tag{15}\]  
From Equations (13) and (15),  
\[A_{z}(z_{0})=-\frac{1}{2\kappa_{z}}e^{-\kappa_{z}z_{0}}\ \ \text{and}\ \ \ B_{z}(z_{0})=-\frac{1}{2\kappa_{z}}e^{\kappa z_{0}}.\]  
Equation (11) becomes  
\[g_{z}(z;z_{0})=\left\{ \begin{array}{ll}-\frac{1}{2\kappa_{z}}e^{-\kappa_{z} (z_{0}-z)}&\quad 0<z<z_{0}<l\\ -\frac{1}{2\kappa_{z}}e^{-\kappa_{z}(z_{-}z_{0})}&\quad 0<z_{0}<z<l\end{array}  \right.. \tag{16}\]  
The solution of Equation (9) can be written as  
\[g_{\theta}(\theta;\theta_{0})=\left\{