# Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures
## 2 Green's Function  
In Ref. [28], the CF as the internal PS located at \(x_{0}\) is formulated in terms of an oscillating impulse with natural frequencies of the material in a given geometry, as follows:  
\[\mathbf{f}=\mathbf{P}\delta(\mathbf{x}-\mathbf{x}_{0})e^{-i\omega t} \tag{1}\]  
where \(\mathbf{P}\) is the force vector and \(\omega\) is the predominant angular frequency \((\omega=2\pi\
u)\) of AE. [PERSON]'s delta function \(\delta(\mathbf{x}-\mathbf{x}_{0})\) provides a method for solving spatial problems dealing with a PS. [PERSON]'s function \(g(\mathbf{x};\mathbf{x}_{0})\), defined as  
\[\
abla^{2}g(\mathbf{x};\mathbf{x}_{0})=\delta(\mathbf{x}-\mathbf{x}_{0}) \tag{2}\]Is the solution for the delta function in elastodynamics, and provides the spatial distribution of the CF at a given time. In cylindrical coordinates, Equation (2) is expressed as  
\[\
abla^{2}[g_{r}(r;r_{0})g_{\theta}(\theta;\theta_{0})g_{z}(z;z_{0})]=\frac{ \delta(r-r_{0})\delta(\theta-\theta_{0})\delta(z-z_{0})}{\mathbf{r}} \tag{3}\]  
where \(\
abla^{2}\) is the Laplacian in cylindrical coordinates, and the CF is located at \(r_{0},\theta_{0}\), and \(z_{0}\). At any point in the cylindrical domain except the CF locating point, Equation (3) becomes zero,  
\[\
abla^{2}[g_{r}(r;r_{0})g_{\theta}(\theta;\theta_{0})g_{z}(z;z_{0})]=0. \tag{4}\]  
In cylindrical \((r,\theta,z)\) coordinates, Equation (4) can be rewritten as  
\[\frac{r^{2}}{g_{r}}\frac{\partial^{2}g_{r}}{\partial r^{2}}+\frac{r}{g_{r}} \frac{\partial g_{r}}{\partial r}+\frac{1}{g_{\theta}}\frac{\partial^{2}g_{ \theta}}{\partial\theta^{2}}+\frac{r^{2}}{g_{z}}\frac{\partial^{2}g_{z}}{ \partial z^{2}}=0.\]  
Substituting the following relations  
\[\frac{1}{g_{z}}\frac{\partial^{2}g_{z}}{\partial z^{2}}=\kappa_{z}^{2},\]  
\[\frac{1}{g_{\theta}}\frac{\partial^{2}g_{\theta}}{\partial\theta^{2}}=-v^{2},\]  
Into the above PDE results, we obtain  
\[r^{2}\frac{\partial^{2}g_{r}}{\partial r^{2}}+r\frac{\partial g_{r}}{\partial r }+\Big{(}\kappa_{z}^{2}r^{2}-v^{2}\Big{)}g_{r}=0\]  
Note that we selected exponential, rather than oscillating, solutions in the z-direction. This implies that the radial solutions are appropriate for the particular set of boundary conditions under consideration. In ref. [28], we obtained the axial, tangential, and radial components of Green's function under certain conditions, such as the continuity and discontinuity principles, angular symmetry for the cylindrical domain with length \(l\) and a radius of \(a\). The inner radius \(b\) is introduced to the shell structure as an additional radial boundary condition. \[g_{z}(z;z_{0})= \begin{cases}-\frac{1}{2\kappa_{z}}e^{-\kappa_{z}(z_{0}-z)}\ 0<z<z_{0}<l\\ -\frac{1}{2\kappa_{z}}e^{-\kappa_{z}(z-z_{0})}\ 0<z_{0}<z<l\end{cases}  \tag{5}\]  
\[g_{\theta}(\theta;\theta_{0})=\cos[v(\theta-\theta_{0})]\ (v=0,\pm 1,\ \pm 2,\ \cdots), \tag{6}\]  
\[g_{r}(r;r_{0})=J_{v}(\kappa_{z}r_{0})J_{v}(\kappa_{z}r)\qquad(b<r<a) \tag{7}\]  
In Equation (7), \(J_{v}\) is a Bessel function of the first kind of \(v\)-th order. The Bessel function of the second kind is excluded because there is no singularity in the cylindrical domain. The value of \(\kappa_{z}\) can be obtained by applying the boundary conditions at the outer or inner surfaces of the cylindrical shell to Equation (7):  
\[\left.J_{v}(\kappa_{z}r)\right|_{r=a-r_{0}}=0\ \ (b<r_{0}<r<a),\]  
\[\left.J_{v}(\kappa_{z}r)\right|_{r=r_{0}-b}=0\ \ (b<r<r_{0}<a).\]  
As an alternative, the boundary condition \(\left.g_{r}(r;r_{0})\right|_{r=a}=0\) was applied [30]. However, this condition resulted in a discontinuous and asymmetric Green's function. Selecting the first root (\(r_{v1}\)) of the Bessel function, \(r_{v1}=\kappa_{z}(a-r_{0})\) or \(r_{v1}=\kappa_{z}(r_{0}-b)\). These relations give  
\[\kappa_{z}= \begin{cases}\frac{r_{v1}}{(a-r_{0})}\ \ (a+b>2r_{0})\\ \frac{r_{v1}}{(r_{0}-b)}\ \ (a+b<2r_{0})\.\end{cases}  \tag{8}\]For the outer surface, \(\kappa_{z}=\frac{r_{v1}}{(a-r_{0})}\). Introducing a parameter, \(A_{v1}\), we obtain [PERSON]'s function:  
\[