## 3 Methods and Processing
More specifically, the displacement of the elastic wave radiated by a pulsating point source in an infinite homogeneous medium can be expressed by the following relationship:  
\[\varphi(R;\omega)=\frac{e^{i\cdot\omega(t-\frac{R}{a})}}{R} \tag{1}\]  
The parameter, \(\omega\), symbolizes the angular frequency; \(t\) is the time; the parameter, \(R\), expresses the distance between the source and the observation point; \(\alpha\) represents the speed of the P-waves. The above equation, in cylindrical system coordinates (i.e., \(r\), \(\vartheta\), and \(z\)), can be written using the Sommerfeld integral as shown in Equation (2):  
\[\varphi\ (r,z;\omega)=-i\cdot e^{i\omega\cdot t}\int_{0}^{\infty}\frac{k}{v}J_{0 }(kr)e^{-i\cdot v\cdot|z|}dk \tag{2}\]  
where \(v=\sqrt{k_{a}^{2}-k^{2}}\); Im (\(v\)) \(>0\); \(k_{a}=\frac{\omega}{a}\); and \(J_{0}\) are the Bessel functions of the first kind. The Sommerfeld integral is a typical partial differential solution equation (wave equation):  
\[\frac{\partial^{2}\varphi}{\partial r^{2}}+\frac{1}{r}\cdot\frac{\partial \varphi}{\partial r}+\frac{1}{r^{2}}\cdot\frac{\partial^{2}\varphi}{\partial \theta^{2}}+\frac{\partial^{2}\varphi}{\partial z^{2}}-\frac{1}{a^{2}}\cdot \frac{\partial^{2}\varphi}{\partial t^{2}}=0 \tag{3}\]  
the general solution of which is given by the equation presented below:  
\[\varphi(r,\vartheta,z;\omega)=e^{i\cdot\omega\cdot t}\sum_{n=-\infty}^{\infty} e^{i\cdot n\cdot\vartheta}\int_{0}^{\infty}g_{n}(\omega,k)J_{n}(kr)e^{-i\left|z \right|}dk \tag{4}\]where \(J_{n}\) represent the Bessel functions of the first kind of order \(n\). Considering a different solution for the above equation results in the function:  
\[\varphi_{s}(r,\vartheta,z;\omega)=-\frac{i}{\pi}\cdot e^{i\cdot\omega t}\sum_{n= 0}^{\infty}(-1)^{n}\cdot\varepsilon_{n}\cdot e^{i\cdot 2n\cdot(\theta-\theta_{s})} \int_{0}^{\infty}\frac{k}{v}J_{2n}(kr)e^{-iv|z|}dk \tag{5}\]  
where the term \(\varepsilon_{n}\) expresses [PERSON]'s factor and can take the values:  
\[\varepsilon_{n}= \begin{cases}2&n\
eq 0\\ 1&n=0\end{cases} \]  
where \(\varphi_{s}\) is the solution to Equation (3) and represents the field from a source located at a specific point. In a particular case where the source is positioned (i.e., \(L\), \(\vartheta_{s}\), and \(0\)) and the observation point is positioned (i.e., \(r,0\), and \(z\)), then Equation (5) is:  
\[\varphi_{s}(r,z;L,\vartheta_{s};\omega)=-\frac{i}{\pi}\cdot\sum_{n=0}^{\infty }(-1)^{n}\cdot\varepsilon_{n}e^{i2n\cdot\psi}\int_{0}^{\infty}\frac{k}{v}J_{2 n}\Big{(}k\sqrt{r^{2}+L^{2}-2 rL\cos\theta_{s}}\Big{)}e^{-iv|z|}dk \tag{6}\]  
where \(\psi\) represents the angular distance of the source between the source and observation point. Using the Bessel functions:  
\[e^{i\cdot 2n\cdot\psi}J_{2n}\Big{(}k\sqrt{r^{2}+L^{2}-2 rL\cos\theta_{s}}\Big{)} =\sum_{m=-\infty}^{\infty}J_{m}(kr)J_{2n+m}(kL)e^{im\theta_{s}} \tag{7}\]  
Equation (6) is transformed into the following:  
\[\varphi_{s}(r,z;L,\vartheta_{s};\omega)=-\frac{i}{\pi}\cdot\sum_{n=0}^{\infty }(-1)^{n}\cdot\varepsilon_{n}\int_{0}^{\infty}\frac{k}{v}\Bigg{(}\sum_{0}^{ \infty}J_{m}(kr)J_{2n+m}(kL)e^{im\theta_{s}}\Bigg{)}e^{-iv|z|}dk \tag{8}\]  
For the following, a set of elementary sources is considered as a given, and the radiation of which is symbolized by the parameters \(\varphi_{s}\) and \(\vartheta_{s}\).