## 3 Displacement Fields Generated by a Point Source  
The NL equation governing a wave in an elastic and homogeneous medium subject to a local body force \(f\) can be written in vector form, as follows [27]  
\[(\lambda+2\mu)\vec{V}(\vec{V}\cdot\vec{u})-\mu\vec{V}\times(\vec{V}\times\vec {u})+f=\rho\frac{\partial^{2}\vec{u}}{\partial t^{2}}, \tag{43}\]  
where \(\vec{u}\) is the displacement vector, \(\lambda\) and \(\mu\) are the Lame constants, and \(\rho\) is the density of the media. In this paper, the displacement field \(\vec{u}\) in cylindrical coordinates proposed by [PERSON] and [PERSON] [23] is given as  
\[\vec{u}=\vec{V}\Phi+\vec{V}\times(X\hat{\vec{e}}_{z})+a\vec{V}\times\vec{V} \times(\vec{V}\hat{\vec{e}}_{z}), \tag{44}\]  
where \(\Phi\) is a scalar potential for the compressional wave (P), \(X\hat{\vec{e}}_{z}\) is a vector potential for the SH wave, \(\vec{V}\hat{\vec{e}}_{z}\) is a vector potential for the SV wave, and \(a\) is the radius of the cylinder. When the displacement field is generated by an intrinsic point defect, the three potentials are correlated with the force vector \(P\) (referred to as CFIP) as  
\[\Phi=\vec{V}\cdot\vec{P}\phi, \tag{45}\]  
\[X\hat{\vec{e}}_{z}=-\vec{V}\times P\chi, \tag{46}\]  
\[\Psi\hat{\vec{e}}_{z}=\vec{V}\times P\psi, \tag{47}\]  
where \(\phi,\chi\) and \(\psi\) are unknown scalar functions, and the negative sign for the SH potential is used for further applications. Factoring the spatial and temporal parts in the potentials leads to  
\[\phi(\vec{x},t;\;\vec{x}_{0})=\phi(\vec{x};\vec{x}_{0})e^{-i\omega t}, \tag{48}\]  
\[\chi(\vec{x},t;\vec{x}_{0})=\chi(\vec{x};\vec{x}_{0})e^{-i\omega t}, \tag{49}\]  
\[\psi(\vec{x},t;\vec{x}_{0})=\psi(\vec{x};\vec{x}_{0})e^{-i\omega t}, \tag{50}\]  
The displacement vector can be written as:  
\[ \begin{array}{c}\vec{u}(\vec{x},\vec{t};\vec{x}_{0},\vec{t}_{0})=\vec{u}( \vec{r},\vec{\theta},\vec{z},\vec{t};\vec{r}_{0},\vec{\theta}_{0},z_{0},\vec{t} _{0})\\ =[\vec{V}(\vec{V}\cdot P\phi)-\vec{V}\times(\vec{V}\times P\chi)-a\vec{V}\times \vec{V}\times(\vec{V}\times P\psi)]e^{-i\omega t},\end{array}  \tag{51}\]where \(\phi=\phi(x;x_{0})\), \(\chi=\chi(x;x_{0})\) and \(\psi=\psi(x;x_{0})\). By substituting Equation (51) into Equation (43) and applying some manipulation, such as \(\mathbf{\
abla}\cdot(\mathbf{\
abla}\phi)=\mathbf{\
abla}^{2}\phi\), \(\mathbf{\
abla}\times\phi=0\), \(\mathbf{\
abla}\cdot\psi=\mathbf{\
abla}\cdot\chi=0\), and \(\mathbf{\
abla}\times(\mathbf{\
abla}\cdot\mathbf{\
abla}\psi)=\mathbf{\
abla}\times(\mathbf{ \
abla}\cdot\mathbf{\
abla}\chi)=0\), Equation (43) can be rewritten as  
\[\mathbf{\
abla}\mathbf{\
abla}\cdot\left[(\lambda+2\mu)\left(\mathbf{\
abla}^{2}P\phi \right)+\rho\omega^{2}(P\phi)+Pg\right]\] \[+a\mathbf{\
abla}\times\left[\mu\mathbf{\
abla}^{4}(P\psi)+\rho\omega^{2} \mathbf{\
abla}^{2}(P\psi)\right] \tag{52}\] \[+\mathbf{\
abla}\times\mathbf{\
abla}\times\left[\mu\mathbf{\
abla}^{2}(\bm {\
abla}\chi)+\rho\omega^{2}(P\chi)+Pg\right]=0.\]  
In the above equation, the components in the gradient, the curl, and the curl of curl, can be independently zero  
\[(\lambda+2\mu)\mathbf{\
abla}^{2}\phi+\rho\omega^{2}\phi+g=0, \tag{53}\]  
\[\mu\mathbf{\
abla}^{4}\psi+\rho\omega^{2}\mathbf{\
abla}^{2}\psi=0, \tag{54}\]  
\[\mu\mathbf{\
abla}^{2}\chi+\rho\omega^{2}\chi+g=0. \tag{55}\]  
Introducing the longitudinal wave (P) speed \(\left(c_{P}=\sqrt{\lambda+2\mu/\rho}\right)\) and transverse wave (S) speed \(\left(c_{S}=\sqrt{\mu/\rho}\right)\) into the above equations leads to  
\[\mathbf{\
abla}^{2}\phi+k_{p}^{2}\phi+\frac{k_{p}^{2}}{\rho\omega^{2}}g=0, \tag{56}\]  
\[\mathbf{\
abla}^{2}\psi+k_{s}^{2}\psi=0, \tag{57}\]  
\[\mathbf{\
abla}^{2}\chi+k_{s}^{2}\chi+\frac{k_{s}^{2}}{\rho\omega^{2}}g=0, \tag{58}\]  
where \(k_{p}=\frac{\omega}{c_{P}}\) and \(k_{s}=\frac{\omega}{c_{S}}\), correspond to the angular wavenumbers of the P and the S waves, respectively.