## 2 Computational Simulation for Oblique Incidence of SV Waves
### Free-Field Ground Motion  
As shown in Figure 1, the plane SV wave with an incident angle \(\alpha\) is reflected on the free surface, forming a plane SV wave and a plane P wave with the reflection angle of \(\alpha\) and \(\beta\), respectively. The particle vibration direction of the medium propagating SV wave is perpendicular to the forward direction of the SV wave, and shear deformation of the medium occurs when the SV wave passes through. However, the particle vibration direction of the medium propagating P wave is consistent with the forward direction of the P wave, and the passing of the P wave causes the volume expansion and contraction of the medium. According to the stress balance and displacement continuity, the angle and amplitude amplification factor of reflected wave can be expressed as shown in Equation (1). \[\left\{ \begin{array}{l}\beta=\arcsin(\frac{c_{p}\sin\alpha}{c_{s}})\\ A_{1}=\frac{c_{s}^{2}\sin 2\alpha\sin 2\beta-c_{p}^{2}\cos^{2}2\alpha}{c_{s}^{2} \sin 2\alpha\sin 2\beta+c_{p}^{2}\cos^{2}2\alpha}\\ A_{2}=\frac{c_{p}c_{s}\sin 2\alpha\cos 2\alpha}{c_{s}^{2}\sin 2\alpha\sin 2\beta+c_{p}^{2 }\cos^{2}2\alpha}\end{array} \right., \tag{1}\]  
where \(A_{1}\) and \(A_{2}\) are the amplitude amplification factors of the reflected SV wave and reflected P wave, respectively. The reflection angle \(\beta\) is smaller than \(\pi/2\), such that there is a critical value for the incident angle of the plane SV wave, i.e.,  
\[\alpha<\arcsin(c_{s}/c_{p}). \tag{2}\]  
A cube soil region is intercepted, which is subjected to a plane SV wave with an oblique incidence angle \(\alpha\). Assuming that the wave front of the incident SV wave at time zero passes through the \(z\)-axis and forms an angle \(\alpha\) with the ground (Figure 2a). The length, width, and height of the truncated region are \(Lx\), \(Ly\), and \(Lz\), respectively, while the coordinates of a node \(l\) in the truncated region are (\(x\), \(y\), \(z\)). Figure 2b is the corresponding 2D model, which shows the wave field acting on each truncated boundary. The wave field of the truncated boundary consists of the directly incident SV wave, the SV wave reflected by the ground surface and the P wave reflected by the ground surface generated by wave type conversion. Considering the time delay of the wave in the propagation process, the time of the incident SV wave, reflected SV wave, and reflected P wave propagation to the node \(l\) are set as \(\Delta t1\), \(\Delta t2\), and \(\Delta t3\), respectively; the expressions for \(\Delta t1\), \(\Delta t2\), and \(\Delta t3\) are as follows:  
\[\left\{ \begin{array}{l}\Delta t_{1}=\frac{x\sin\alpha+y\cos\alpha}{c_{s}}\\ \Delta t_{2}=\frac{x\sin\alpha+(2L_{y}-y)\cos\alpha}{c_{s}}\\ \Delta t_{3}=\frac{x\sin\alpha+y\cos\alpha}{c_{s}\cos\beta}+\frac{(L_{y}-y) \cos(\alpha+\beta)}{c_{s}\cos\beta}+\frac{L_{y}-y}{c_{p}\cos\beta}\end{array}  \right., \tag{3}\]  
where \(c_{s}\) and \(c_{p}\) are the shear and compression wave velocities of the medium, respectively. Assuming that the displacement time history of input seismic SV wave is \(u_{0}(t)\), then the wave field on each truncated boundary is composed of an incident SV wave \(u_{0}(t-\Delta t_{1})\), a reflected SV wave \(u_{0}(t-\Delta t_{2})\), and a reflected P wave \(u_{0}(t-\Delta t_{3})\). The displacement of the wave field can be written as  
\[\left\{ \begin{array}{l}u^{f}_{l1}(t)=u_{0}(t-\Delta t_{1})\cos\alpha-A_{1}u_ {0}(t-\Delta t_{2})\cos\alpha+A_{2}u_{0}(t-\Delta t_{3})\sin\beta\\ u^{f}_{l2}(t)=-u_{0}(t-\Delta t_{1})\sin\alpha-A_{1}u_{0}(t-\Delta t_{2})\sin \alpha-A_{2}u_{0}(t-\Delta t_{3})\cos\beta\end{array} \right.. \tag{4}\]  
The displacement \(u^{f}_{l3}(t)\) in the \(z\)-direction on each truncated boundary surface is zero, where \(u^{f}_{l1}(t)\), \(u^{f}_{l2}(t)\), and \(u^{f}_{l3}(t)\) are the displacement of the wave field of node \(l\) in the \(x\)-, \(y\)-, and \(z\)-directions respectively.