## 2 Governing Equations  
For a steady-state elastic field and electric field under the action of a periodic load, the field variables can be expressed in the following separated form with time-harmonic factors and spatial variables  
\[L^{*}(x,y,t)=L(x,y)\mathrm{e}^{-\mathrm{i}\omega t}, \tag{1}\]  
where \(L^{*}\) is a complex function and its real part is the solution of the problem. The spatial function \(L\) is the main subject of study. Considering the case of SH wave incident on the \(xy\)-plane, assuming that the \(z\)-axis is the polarization direction, the control equations of the steady-state anti-plane dynamic problem are  
\[ \begin{array}{c}\frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz }}{\partial y}+\rho\omega^{2}w=0,\\ \frac{\partial B_{y}}{\partial x}+\frac{\partial B_{y}}{\partial y}=0,\end{array}  \tag{2}\]  
where the components \(\tau_{xz}\) and \(\tau_{yz}\) are the anti-plane shear stress components; \(B_{x}\) and \(B_{y}\) are the magnetic displacement components. \(w\), \(\rho\), and \(\omega\), respectively, represent the displacement on the \(z\)-axis, the density, and the circular frequency of the steady-state waves. Thus, the constitutive relation of piezomagnetic materials can be expressed as  
\[ \begin{array}{l}\tau_{xz}=c_{44}\frac{\partial w}{\partial x}+h_{15}\frac{ \partial\phi}{\partial x},\tau_{yz}=c_{44}\frac{\partial w}{\partial y}+h_{15} \frac{\partial\phi}{\partial y},\\ B_{x}=h_{15}\frac{\partial w}{\partial x}-\mu_{11}\frac{\partial\phi}{ \partial x},B_{y}=h_{15}\frac{\partial w}{\partial y}-\mu_{11}\frac{\partial \phi}{\partial y},\end{array}  \tag{3}\]  
where the \(c_{44}\), \(h_{15}\), and \(\mu_{11}\) represent the elastic constants, piezomagnetic stress constants, and magnetic constants, respectively. \(\phi\) represents the magnetic potential within the medium. Substituting Equation (3) into Equation (2), we can obtain the following form  
\[ \begin{array}{l}\
abla^{2}w+k^{2}w=0,\\ \
abla^{2}f=0,\end{array}  \tag{4}\]  
where the \(k=\sqrt{\rho\omega^{2}/c^{*}}\) represents the wave number and \(c^{*}=c_{44}+h_{15}^{2}/\mu_{11}\) represents the modulus of magnetic hardening elastic constant. The magnetic potential can be determined by  
\[\phi=\frac{h_{15}}{\mu_{11}}(w+f). \tag{5}\]  
In a cylindrical coordinate system, Equation (4) can be written in the following form  
\[ \begin{array}{l}\frac{\partial^{2}w}{\partial x^{2}}+\frac{1}{r}\frac{ \partial w}{\partial x}+\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial x^{2}}+ k^{2}w=0,\\ \frac{\partial^{2}f}{\partial x^{2}}+\frac{1}{r}\frac{\partial f}{\partial x}+ \frac{1}{r^{2}}\frac{\partial^{2}f}{\partial x^{2}}=0.\end{array}  \tag{6}\]  
Meanwhile, Equation (3) can be transformed into the following form  
\[ \begin{array}{l}\tau_{xz}=c_{44}\frac{\partial w}{\partial x}+h_{15}\frac{ \partial\phi}{\partial x},\tau_{xz}=c_{44}\frac{\partial w}{\partial x}+h_{15} \frac{\partial\phi}{\partial x},\\ B_{r}=h_{15}\frac{\partial w}{\partial x}-\mu_{11}\frac{\partial\phi}{ \partial x},B_{\alpha}=h_{15}\frac{1}{r}\frac{\partial w}{\partial\alpha}-\mu_ {11}\frac{1}{r}\frac{\partial\phi}{\partial\alpha}.\end{array}  \tag{7}\]  
The method of complex function and conformal mapping can be employed to solve the boundary value problem of non-circular openings, as shown in Figure 1. After introducing complex variables \(z=r\mathrm{e}^{\mathrm{i}\alpha}\) and \(\overline{z}=r\mathrm{e}^{-\mathrm{i}\alpha}\) and using conformal mapping, Equation (4) can be expressed as  
\[ \begin{array}{l}\frac{\partial^{2}w}{\partial x\partial x}=\left(\frac{ik}{2 }\right)^{2}h^{\prime}(\zeta)\overline{h^{\prime}(\zeta)}w(\zeta,\overline{ \zeta}),\\ \frac{\partial^{2}f}{\partial x\partial x}=0.\end{array}  \tag{8}\]  
On the mapping plane (\(\eta\), \(\theta\)), we can utilize the following equation  
\[\left\{