# Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures
## 2 Green's Function
\begin{array}{c}g(r,\theta,z;r_{0},\theta_{0},z_{0})=A_{v1}\ J_{v}(\kappa_{z}r _{0})J_{v}(\kappa_{z}r)\cos[v(\theta-\theta_{0})]\\ \times\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}.\end{array}  \tag{9}\]  
In ref. [28], the parameter \(A_{v1}\) was determined by integrating the delta function in the cylindrical domain. The integrations of \(\delta(\theta-\theta_{0})\delta(z-\kappa_{0})\) for the cylindrical shell are the same as those for the cylinder, because these two delta functions are independent of \(r\). The remaining problem is to integrate \(\frac{\delta(r-r_{0})}{r}\),  
\[\pi A_{v1}J_{v}(\kappa_{z}r_{0})J_{v}(\kappa_{z}r)\left[e^{-\kappa_{z}z_{0}}+ e^{-\kappa_{z}(l-z_{0})}-2\right]=\frac{\delta(r-r_{0})}{r}. \tag{10}\]  
Multiplying both sides of Equation (10) by \(J_{p}(\kappa_{z}r)\ rdr\) and integrating over \((b,a)\) gives the following:  
\[ \begin{array}{c}\pi A_{v1}\Big{[}e^{-\kappa_{z}z_{0}}+e^{-\kappa_{z}(l-z_{0 })}-2\Big{]}\ J_{v}(\kappa_{z}r_{0})\int_{b}^{a}J_{v}(\kappa_{z}r)J_{p}(\kappa _{z}r)rdr\\ =\int_{b}^{a}\delta(r-r_{0})J_{p}(\kappa_{z}r)\ dr.\end{array}  \tag{11}\]  
The integrations on the left side of the above equation can be divided into two parts:  
\[\int_{b}^{a}J_{v}(\kappa_{z}r)J_{p}(\kappa_{z}r)rdr=\left\{\int_{r_{0}}^{a}[J _{v}(\kappa_{z}r)]^{2}rdr+\int_{b}^{r_{0}}[J_{v}(\kappa_{z}r)]^{2}rdr\right\} \delta_{pv}.\]  
Applying the normalization and orthogonality principles of Bessel functions,  
\[\int_{0}^{c}J_{p}\left(\alpha\frac{r}{c}\right)J_{q}\left(\alpha\frac{r}{c} \right)rdr=\frac{c^{2}}{2}\left[J_{p+1}(\alpha)\right]^{2}\delta_{pq},\]  
And the following concepts of the delta function,  
\[\int_{b}^{a}J_{v}(\kappa_{z}r)J_{p}(\kappa_{z}r)=\frac{a^{2}}{2}[J_{v+1}( \kappa_{z}a)]^{2}-\frac{b^{2}}{2}[J_{v+1}(\kappa_{z}b)]^{2}\]  
\[\int_{b}^{a}\delta(r-r_{0})J_{v}(\kappa_{z}r)\ dr=J_{v}(\kappa_{z}r_{0}),\]  
To Equation (11), we obtain the constant  
\[A_{v1}=\frac{1}{\pi[e^{-\kappa_{z}z_{0}}+e^{-\kappa_{z}(l-z_{0})}-2]}\ \times\ \frac{2}{\left\{a^{2}[J_{v+1}(\kappa_{z}a)]^{2}-b^{2}[J_{v+1}(\kappa_{z}b)]^{2 }\right\}}. \tag{12}\]  
For the cylindrical problem, the corrected value of \(A_{v1}\) is given in (A1). Introducing Equation (12) to Equation (9) gives the Green's function for the Kronecker delta function:  
\[g(r,\theta,z;r_{0},\theta_{0},z_{0})=G_{v1}\ J_{v}(\kappa_{z}r_{0})J_{v}( \kappa_{z}r)\{\cos[v(\theta-\theta_{0})]\}, \tag{13}\]  
where  
\[G_{v1}=-\frac{1}{2\kappa_{z}}A_{v1}J_{v}(\kappa_{z}r_{0})\times \begin{cases}e ^{-\kappa_{z}(z_{0}-z)}\ (0<z<z_{0}<l)\\ e^{-\kappa_{z}(z-z_{0})}\ (0<z_{0}<z<l)\end{cases} . \tag{14}\]  
Since the CF direction is determined by the \(\mathbf{P}\) vector, [PERSON]'s function is assumed to be azimuthally independent (\(v=0\)). In addition, it is very convenient to take the location of the PS as the new origin and introduce relative coordinates to the location of the PS, defined as \(\xi=r-r_{0}\), \(\theta=\theta-\theta_{0}\), and \(\eta=z-z_{0}\), where \(\xi_{0}=0\), \(\theta_{0}=0\), and \(\eta_{0}=0\) (hereafter,referred to as the PS-oriented \((\xi,\vartheta,\eta)\) coordinate system), Equations (13) and (14) can be rewritten as  
\[g(\xi,\ \eta)=G_{01}J_{0}(\kappa_{z}\xi), \tag{15}\]  
\[G_{01}=-\frac{1}{2\kappa_{z}}A_{01}J_{0}(\kappa_{z}\xi_{0})\times \begin{cases}e^ {\kappa_{z}\eta}\ (z_{0}-l<\eta<0)\\ e^{-\kappa_{z}\eta}\ (0<\eta<l-z_{0})\end{cases} . \tag{16}\]  
In Equation (15), the value of \(\xi\) at a given point is the shortest distance from the PS. In cylindrical geometry,  
\[\xi=\sqrt{\left(x_{i}-x_{0i}\right)^{2}+\left(x_{j}-x_{0j}\right)^{2}}\ \rightarrow\ \left(r_{0}=\sqrt{x_{0i}^{2}+x_{0j}^{2}}\right). \tag{17}\]  
For the shell geometry, the calculation of \(\xi\) is somewhat complicated due to its hollow interior. No linear distance exists between the two points across the hollow interior. In Figure 1b, an arc connecting the PS and a given point is introduced.