## 3 Mathematical Derivation
### Nonlocal Continuum Theory: Acoustic Waves in Piezoelectric Media
We are assuming the plane-wave solution in the form \(\overset{\rightarrow}{u}=u_{o}e^{i(\omega t-\overset{\rightarrow}{k}\cdot \overset{\rightarrow}{x})}\) for displacement and \(\phi=\phi_{0}e^{i(\omega t-\overset{\rightarrow}{k}\cdot\overset{ \rightarrow}{x})}\) for scalar electric potential. The electric field becomes  
\[E_{l}=(-\
abla\phi)_{l}=ik_{l}\phi_{0}e^{i(\omega t-\overset{\rightarrow}{k} \cdot\overset{\rightarrow}{x})} \tag{16}\]  
that is \(\overset{\rightarrow}{E}\bigg{|}\bigg{|}\overset{\rightarrow}{k}\). From the absence of free charges, the differential of Equation (8) yields:\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]0  
and, hence the amplitude of scalar potential function is obtained as  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]1  
Assuming the plane waves solution, the equation of motion (recall Equation (15)) is simplified as below  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]2  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]3  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]4  
The non-trivial solution from Equation (21) is given as  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]5  
After dividing by \(\rho\omega^{2}\) and introducing the Cartesian components \(n_{l}\) of the unit vector in the direction of \(k\) this leads to the modified Christoffel equation given in Equation (23)  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]6  
In Equation (23), the square brackets indicate the constitutive matrix considering mechanical and piezoelectric stiffness. In the absence of piezo-electric constants, the equation reduces to the classic Christoffel equation. The right-side term of the square bracket in Equation (23), indicates the piezo-electrically stiffened elastic constants. The stiffened piezo-electric constants are the function of both material properties and direction of propagation. Let, \(\Gamma_{il}=\frac{1}{\rho\omega^{2}}\Bigg{[}C_{ijlm}^{E}+\frac{ep_{ijr}n_{r}ep_ {lms}n_{s}}{\big{(}\varepsilon_{pq}^{S}n_{p}n_{q}\big{)}}\Bigg{]}n_{j}n_{m}\) and \(\big{|}k^{2}\big{|}=(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})\)  
The eigen values analysis of the biharmonic modified Christoffel equation gives  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]7  
The Equation (24) can be written in matrix form as below  
\[\left[ \begin{array}{cccc}\Gamma_{11}-\varepsilon_{ls}^{2}&\Gamma_{12}& \Gamma_{13}\\ \Gamma_{21}&\Gamma_{22}-\varepsilon_{ls}^{2}&\Gamma_{23}\\ \Gamma_{31}&\Gamma_{32}&\Gamma_{33}-\varepsilon_{ls}^{2}\end{array} \right]- \left[ \begin{array}{cccc}\frac{1}{\left|\mathbf{k}\right|^{2}}&0&0\\ 0&\frac{1}{\left|\mathbf{k}\right|^{2}}&0\\ 0&0&\frac{1}{\left|\mathbf{k}\right|^{2}}\end{array} \right]=0 \tag{25}\]  
Finally, Eigen value solution is performed on the following nonlocal Christoffel's equation  
\[\Bigg{[}\Big{(}\Gamma_{ij}-\varepsilon_{ls}^{2}\delta_{ij}\Big{)}-\frac{1}{ \left|\mathbf{k}\right|^{2}}\delta_{ij}\Bigg{]}=0 \tag{26}\]The solution yields solutions for three propagating modes: quasi-longitudinal, fast transversal and slow transversal waves. The main outcome of the theoretical model is the generation of directional group wave velocity in LiNbO\({}_{3}\) for slow and fast transverse wave velocity considering piezo-electric stiffening. Also, we develop directional group wave velocity for longitudinal wave velocity considering piezo stiffening. The crucial aspect of the theoretical model is an accurate generation of the dispersion relationship of the quasi-slow transversal and quasi-longitudinal wave mode resulting from intrinsic length scale varying from zero to unity. These developed dispersion curves will act as a calibrating chart on which the experimental results will be overlaid to quantify the nonlocal parameters representing a precursor to damage.