## 2 Basic Equation
### Dispersive Equation
#### 2.2.2 Dispersive Equation  
Based on the plane strain problem, in space, there is \(u_{z}=0\) and both \(u_{x}\) and \(u_{y}\) are functions of \((x,y)\) in space. The Helmholtz representation can be written as  
\[\mathbf{u}(x,y)=u_{x}(x,y)\mathbf{e}_{x}+u_{y}(x,y)\mathbf{e}_{y}=\
abla\varphi(x,y)+ \
abla\times\psi(x,y)\mathbf{e}_{z}, \tag{20a}\] \[\
abla\cdot\mathbf{u}(x,y)=\
abla^{2}\varphi(x,y),\] (20b) \[\
abla\times\mathbf{u}(x,y)=-\
abla^{2}\psi(x,y)\mathbf{e}_{z},\] (20c) \[\
abla\cdot(\
abla\times\psi)=0, \tag{20d}\]where \(\varphi(x,y)\) and \(\psi(x,y)\) are potential functions of displacement, substituting Equation (20) into Equation (15) yields:  
\[ \begin{array}{c}\
abla\Big{[}c(\lambda+2\mu)\
abla^{4}\varphi-\Big{(}\lambda+2 \mu+\mu_{0}H_{0}^{2}-\frac{\rho d^{2}\omega^{2}}{3}\Big{)}\
abla^{2}\varphi- \big{(}\omega^{2}\varepsilon_{0}\mu_{0}^{2}H_{0}^{2}+\rho\omega^{2}\big{)} \varphi\Big{]}\\ \qquad\qquad+\
abla\times\Big{[}c\mu\
abla^{4}\psi-\Big{(}\mu-\frac{\rho d^{2 }\omega^{2}}{3}\Big{)}\
abla^{2}\psi-\big{(}\omega^{2}\varepsilon_{0}\mu_{0}^ {2}H_{0}^{2}+\rho\omega^{2}\big{)}\psi\Big{]}=0,\end{array}  \tag{21}\]  
to ensure the validity of Equation (21), the values in the brackets must be equal to zero:  
\[c(\lambda+2\mu)\
abla^{4}\varphi-\bigg{(}\lambda+2\mu+\mu_{0}H_{0}^{2}-\frac{ \rho d^{2}\omega^{2}}{3}\bigg{)}\
abla^{2}\varphi-\Big{(}\omega^{2}\varepsilon _{0}\mu_{0}^{2}H_{0}^{2}+\rho\omega^{2}\Big{)}\varphi=0, \tag{22a}\] \[c\mu\
abla^{4}\psi-\bigg{(}\mu-\frac{\rho d^{2}\omega^{2}}{3}\bigg{)}\
abla^{2} \psi-\Big{(}\omega^{2}\varepsilon_{0}\mu_{0}^{2}H_{0}^{2}+\rho\omega^{2}\Big{)} \psi=0, \tag{22b}\]  
Equation (22) can be further expressed as:  
\[\
abla^{4}\varphi+\Bigg{(}\frac{1}{c}+\frac{\overline{H_{0}^{2}}}{c}-\frac{d^ {2}\omega^{2}}{3 cV_{P}^{2}}\Bigg{)}\
abla^{2}\varphi-\frac{\omega^{2}}{c} \Bigg{(}\frac{\overline{H_{0}^{2}}}{V_{D}^{2}}+\frac{1}{V_{P}^{2}}\Bigg{)} \varphi=0, \tag{23a}\] \[\
abla^{4}\psi+\Bigg{(}\frac{1}{c}-\frac{d^{2}\omega^{2}}{3 cV_{S}^{2}}\Bigg{)} \
abla^{2}\psi-\frac{\omega^{2}}{cV_{S}^{2}}\Bigg{(}\frac{\overline{H_{0}^{2 }}V_{P}^{2}}{V_{D}^{2}}+1\Bigg{)}\psi=0, \tag{23b}\]  
\(V_{P}=\sqrt{(\lambda+2\mu)/\rho}\) is the longitudinal wave speed in classical elastic solids, \(V_{S}=\sqrt{\mu/\rho}\) is the shear wave speed in classical elastic solids, \(V_{D}=1/\sqrt{\varepsilon_{0}\mu_{0}}\) is the electromagnetic wave speed, and \(\overline{H_{0}}=\frac{H_{0}}{V_{P}}\sqrt{\frac{\mu_{0}}{\rho}}\). From Equation (23), it is evident that the microstructural parameters and external magnetic field influence both the longitudinal and shear waves, albeit in different ways for each type of wave. Let the form of the solution to the wave equation be as follows:  
\[\varphi=A\exp[i(\mathbf{\sigma}_{P}\cdot\mathbf{r}-\omega t)], \tag{24a}\] \[\psi=B\exp[i(\mathbf{\sigma}_{S}\cdot\mathbf{r}-\omega t)], \tag{24b}\]  
where \(A\) and \(B\) represent the amplitudes of the P wave and SV waves, respectively, \(\mathbf{\sigma}\) is the wavenumber vector, and \(\mathbf{r}\) is the unit vector in the direction of wave propagation. Substituting Equation (24) into Equation (23) further simplifies this equation.