## 3 Mathematical Derivation
### Nonlocal Continuum Theory: Acoustic Waves in Piezoelectric Media  
In this section, we derive the angular dispersion relationship of ultrasonic wave propagation in the piezoelectric crystal by incorporating a nonlocal elastic kernel function in [PERSON]'s solution. Einstein index notations have been used throughout this paper with indices taking values 1, 2 and 3. Let us consider a body \(\Omega\) with boundary \(\Gamma\). At any point \(\mathbf{y}\) in \(\Omega\), the constitutive relationship is written as  
\[\sigma_{ij}=C_{ijlm}^{E}S_{lm}-ep_{ijr}E_{r} \tag{7}\]  
\[D_{r}=ep_{ijr}S_{ij}+\varepsilon_{rl}^{S}E_{l} \tag{8}\]  
\(\sigma\) denotes the elastic stress tensor, \(S\) as elastic strain tensor, the electric field \(\overset{\rightarrow}{E}_{r}\), \(\overset{\rightarrow}{D}_{r}\) the electric displacement, \(C^{E}\) the elastic modulus tensor at constant electric field, \(ep\) the piezoelectric stress tensor, and \(\varepsilon^{S}\) the permittivity tensor at constant strain. Equation (7) represents the experimental stress strain behavior developed in the anisotropic crystal depending on the constraints imposed in the electric field. Equation (8) shows that the resultant stress tensor depends on whether the electric field is allowed to develop in the crystal, if strain \(S\) is applied. The piezoelectric effect in the sensor depends on the crystal symmetry. The displacement of microscopic charge in crystal lattice results in the generation of electric displacement in the piezo-electric crystal. The equations of motion can be written as [38]. \[\rho\ddot{u}_{i}(\mathbf{x},\mathbf{t})\delta_{il}=\frac{\partial\dot{\alpha} _{ij}}{\partial x_{j}} \tag{9}\]  
According to the nonlocal elasticity, the stress at a point of interest \(\mathbf{x}(x_{n})\) can be written as  
\[t_{ij}(\mathbf{x})=\int\limits_{\Omega}C_{ijkl}(\mathbf{y})\kappa(|\mathbf{y} -\mathbf{x}|)S_{kl}(\mathbf{y})d\Omega \tag{10}\]  
where, \(\kappa(|\mathbf{y}-\mathbf{x}|)\) is a nonlocal kernel or nonlocal modulus. Various researchers investigated the suitable use of the nonlocal kernel for specific problem formulation. A list of different types of kernels is presented in the literature [41]. A functional form of nonlocal elasticity kernel is presented by [PERSON] [44] and the potential is developed from a molecular dynamic study. From these past studies, it is concluded that the kernel function can also be considered as the probability density function. The nonlocal kernels have a few specific properties [41]. The integro-differential equation of motion at \(\mathbf{x}\) can be written as  
\[\int\limits_{\Omega}\frac{\partial}{\partial x_{j}}\big{(}S_{ij}(\mathbf{y}) \kappa(|\mathbf{y}-\mathbf{x}|)\big{)}d\Omega(\mathbf{y})+F_{i}(\mathbf{x})= \rho(\mathbf{x})\ddot{u}_{k}\delta_{ik} \tag{11}\]  
The above equation can be manipulated in a different way by operating it with the operator \(L\). Thus Equation (11) can be written as  
\[C_{ijkl}\int\limits_{\Omega}L\kappa(|\mathbf{y}-\mathbf{x}|)S_{kl,j}(\mathbf{y })d\Omega(\mathbf{y})+LF_{i}(\mathbf{x})=L\big{(}\rho(\mathbf{x})\ddot{u}_{k} \big{)}\delta_{ik} \tag{12}\]  
\[C_{ijkl}\int\limits_{\Omega}\delta(|\mathbf{y}-\mathbf{x}|)S_{kl,j}(\mathbf{y })d\Omega(\mathbf{y})+LF_{i}(\mathbf{x})=L\big{(}\rho(\mathbf{x})\ddot{u}_{k} \big{)}\delta_{ik} \tag{13}\]  
\[C_{ijkl}S_{kl,j}(\mathbf{x})-L\big{(}\rho(\mathbf{x})\ddot{u}_{k}\big{)}\delta _{ik}=-LF_{i}(\mathbf{x}) \tag{14}\]  
The equation of motion for vibration in anisotropic piezoelectric solid is given as (the dummy index are interchanged, without loss of generality)  
\[\rho L(\ddot{u}_{i})\delta_{il}=C_{ijml}^{E}\frac{\partial^{2}u_{l}}{\partial x _{j}\partial x_{m}}-ep_{ijr}\frac{\partial E_{r}}{\partial x_{j}} \tag{15}\]  
The operator \(L\) is given as \(L=1-\varepsilon_{sl}^{2}\
abla^{2};\varepsilon_{sl}=e_{o}l\) is the intrinsic length scale factor.