## 5 Results and Discussions  
In this section, we combine the understanding from Sections 2-4. Directional group velocities are calculated for Lithium Niobate piezoelectric crystal considering piezoelectric stiffening using Equation (26) [37]. The constitutive material properties of LiNbO\({}_{3}\) is given in Table 1 ([PERSON] et al., 1999) [49].  
Figures 8a, 9a and 10a show directional slow transversal, fast transversal and longitudinal group wave velocity for LiNbO\({}_{3}\) without piezo stiffening. Similarly,Figures 8b, 9b and 10b show directional slow transversal, fast transversal and longitudinal group wave velocity for LiNbO\({}_{3}\) with piezo stiffening. The angular dispersions shown in Figures 8-10 are calculated for a nominal elastic and piezoelectric properties at zero intrinsic length scale parameter \(\varepsilon_{ls}=0\) and referred to as undamaged or reference state. At zero-degree orientation of wave propagation, the group wave velocity for a slow transversal wave in non-piezo stiffened and piezo stiffened materials are 3571 m/s and 3592 m/s, respectively. However, for longitudinal waves, the computed group velocities are 7696 m/s and 7873 m/s in non-piezo and piezo stiffened materials. We assume that any changes in the group wave velocity of the transversal wave mode are due to the surface damage, which is the manifestation of the perturbation in elastic material constants alone but independent of piezoelectric constants. The same cannot be argued for the longitudinal wave velocity, as piezo stiffening plays a significant role in governing the group wave velocity. Therefore, any alteration in group wave velocity of the longitudinal wave is due to a combined effect from the degradation of mechanical and piezoelectric constants.  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Elastic Constant (\(\times\)10\({}^{11}\) N/m\({}^{2}\))** & **Piezoelectric Constant (C/m\({}^{2}\))** \\ \hline C\({}_{\text{1}}^{E}\) & 1.9886 & C\({}_{15}\) & 3.655 \\ C\({}_{\text{1}}^{E}\) & 0.5467 & C\({}_{22}\) & 2.407 \\ C\({}_{\text{1}}^{E}\) & 0.6799 & C\({}_{31}\) & 0.328 \\ C\({}_{\text{1}}^{E}\) & 0.0783 & C\({}_{33}\) & 1.894 \\ C\({}_{\text{3}}^{E}\) & 2.3418 & & Dielectric constant \\ C\({}_{\text{4}}^{E}\) & 0.5985 & \(\varepsilon_{11}^{S}/\epsilon_{O}\) & 44.9 \\ C\({}_{\text{6}}^{E}\) & 0.7209 & \(\varepsilon_{33}^{S}/\epsilon_{O}\) & 26.7 \\ \hline \end{tabular}
\end{table}
Table 1: Acoustic physical constant of LiNbO\({}_{3}\) of single crystal.  
In order to quantify the relative magnitude of the damage on the sample, dispersion relationship (group velocity vs frequency) are plotted for LiNbO\({}_{3}\) for non-piezo and piezoelectric stiffened (refer Section 2) specimens. Figures 11 and 12 represent the dispersion relation of the LiNbO\({}_{3}\) considering intrinsic length scale nonlocal parameter \(\varepsilon_{ls}\) ranging from 0 to 1, where \(\varepsilon_{ls}=0\) is an  
Figure 8: Directional group wave velocity in LiNbO\({}_{3}\) for slow transverse wave velocity (**a**) without piezo stiffening (**b**) with piezo stiffening mode.  
Figure 10: The directional group wave velocity plot in LiNbO\({}_{3}\) for longitudinal wave velocity (**a**) without piezo stiffening (**b**) with piezo stiffening mode.  
Figure 9: The directional group wave velocity plot in LiNbO\({}_{3}\) for fast transverse wave velocity (**a**) without piezo stiffening (**b**) with piezo stiffening mode.  
undamaged state and \(\varepsilon_{ls}\) = 1 is relative complete damage state. Figure 11a,b and Figure 12a,b show the dispersion relationship considering piezo and non-piezo stiffening for slow transversal and longitudinal waves velocity, respectively. The dispersion relationship shows lowering in group velocity with an increase in frequency for the given wave mode. Also, the group wave velocity decreases with an increase in nonlocal parameter magnitude. It is observed that the surface and subsurface defects will cause scattering, reflection, and attenuation of guided waves leading to change in time of flight (TOF).