## 2 Governing Equations
\begin{array}{l}\frac{\partial}{\partial x}=\frac{\partial}{ \partial x}\mathrm{e}^{\mathrm{i}\alpha}+\frac{\partial}{\partial\phi}\mathrm{ e}^{-\mathrm{i}\alpha}\\ \frac{\partial}{\partial\alpha}=\mathrm{i}\frac{\partial}{\partial x}-\mathrm{i} \overline{z}\frac{\partial}{\partial\overline{z}}\end{array} \right.,\; \mathrm{e}^{\mathrm{i}\alpha}=\left.\frac{h^{\prime}(\zeta)}{\left|h^{\prime}( \zeta)\right|}\mathrm{e}^{\mathrm{i}\theta},\mathrm{d}z=h^{\prime}(\zeta) \mathrm{d}\zeta.\end{array}\right. \tag{9}\]  
Figure 1: The outer domain of the non-circular boundary of the \(z\)-plane is mapped to the outer domain of the unit circle of the \(\zeta\)-plane. Equation (7) can be transformed as  
\[ \begin{array}{l}\tau_{yz}=\frac{e^{*}}{\left|\bar{h}^{\prime}(\zeta)\right|} \Big{(}\frac{\partial w}{\partial\zeta}\mathrm{e}^{\mathrm{i}\theta}+\frac{ \partial w}{\partial\zeta}\mathrm{e}^{-\mathrm{i}\theta}\Big{)}+\frac{h_{15}^{ 2}/\mu_{11}}{\left|\bar{h}^{\prime}(\zeta)\right|}\left(\frac{\partial f}{ \partial\zeta}\mathrm{e}^{\mathrm{i}\theta}+\frac{\partial f}{\partial\zeta} \mathrm{e}^{-\mathrm{i}\theta}\right),\\ \tau_{\theta z}=\frac{\mathrm{i}e^{*}}{\left|\bar{h}^{\prime}(\zeta)\right|} \Big{(}\frac{\partial w}{\partial\zeta}\mathrm{e}^{\mathrm{i}\theta}-\frac{ \partial w}{\partial\zeta}\mathrm{e}^{-\mathrm{i}\theta}\Big{)}+\frac{\mathrm{ i}h_{15}^{2}/\mu_{11}}{\left|\bar{h}^{\prime}(\zeta)\right|}\left(\frac{ \partial f}{\partial\zeta}\mathrm{e}^{\mathrm{i}\theta}-\frac{\partial f}{ \partial\zeta}\mathrm{e}^{-\mathrm{i}\theta}\right),\\ B_{\eta}=-\frac{h_{15}}{\left|\bar{h}^{\prime}(\zeta)\right|} \Big{(}\frac{\partial f}{\partial\zeta}\mathrm{e}^{\mathrm{i}\theta}+\frac{ \partial f}{\partial\zeta}\mathrm{e}^{-\mathrm{i}\theta}\Big{)},\\ B_{\theta}=-\frac{\mathrm{i}h_{15}}{\left|\bar{h}^{\prime}(\zeta)\right|} \Big{(}\frac{\partial f}{\partial\zeta}\mathrm{e}^{\mathrm{i}\theta}-\frac{ \partial f}{\partial\zeta}\mathrm{e}^{-\mathrm{i}\theta}\Big{)}.\end{array}  \tag{10}\]  
In the original plane, the angle between the SH wave vector and the positive \(x\)-axis vector is denoted as \(\varphi\). In the mapping plane, we can express the elastic field and magnetic potential as follows  
\[ \begin{array}{l}w^{(i)}=w_{0}\mathrm{e}^{\frac{\mathrm{i}\theta}{2}\left[ \bar{h}(\zeta)\mathrm{e}^{-\mathrm{i}\varphi}+\overline{h(\zeta)}\mathrm{e}^{ \mathrm{i}\varphi}\right]},\\ \phi^{(i)}=\frac{h_{15}}{\mu_{11}}w_{0}\mathrm{e}^{\frac{\mathrm{i}\theta}{2} \left[\bar{h}(\zeta)\mathrm{e}^{-\mathrm{i}\varphi}+\overline{h(\zeta)} \mathrm{e}^{\mathrm{i}\varphi}\right]},\end{array}  \tag{11}\]  
where the \(w_{0}\) is the amplitude of the incident wave. The above equation omits the time factor \(\mathrm{e}^{-\mathrm{i}\omega t}\). By employing the wave function expansion method, we can express Equation (11) as follows  
\[ \begin{array}{l}w^{(i)}=w_{0}\sum\limits_{n=-\infty}^{\infty}i^{n}J_{n}(k|h( \zeta)|)\left\{\frac{\hbar(\zeta)}{\left|\bar{h}(\zeta)\right|}\right\}^{n} \mathrm{e}^{-\mathrm{i}n\varphi},\\ \phi^{(i)}=\frac{h_{15}}{\mu_{11}}w_{0}\sum\limits_{n=-\infty}^{\infty}i^{n}J_{n }(k|h(\zeta)|)\left\{\frac{\hbar(\zeta)}{\left|\bar{h}(\zeta)\right|}\right\}^ {n}\mathrm{e}^{-\mathrm{i}n\varphi}.\end{array}  \tag{12}\]  
Simultaneously, the scattered wave generated by an irregular opening can be expressed as  
\[ \begin{array}{l}w^{(s)}=w_{0}\sum\limits_{n=-\infty}^{\infty}A_{n}H_{n}^{(1 )}(k|h(\zeta)|)\left\{\frac{\hbar(\zeta)}{\left|\bar{h}(\zeta)\right|}\right\}^ {n},\\ f^{(s)}=B_{0}+w_{0}\sum\limits_{n=1}^{\infty}\Big{(}B_{n}\zeta^{-n}+C_{n} \overline{\zeta}^{-n}\Big{)},\\ \phi^{(s)}=\frac{h_{15}}{\mu_{11}}\Big{(}w^{(s)}+f^{(s)}\Big{)},\end{array}  \tag{13}\]  
where the \(H_{n}^{(1)}\) represents the first-type Hankle function, and \(A_{n}\), \(B_{n}\), \(C_{n}\), and \(B_{0}\) are undetermined constants. For piezomagnetic materials, we can express the total elastic field and total magnetic potential as follows  
\[