## 4 Results and Analysis  
### Model Parameters  
Figures 4-6 show the effect of tunnel stiffness on tunnel responses. In comparison, the corresponding response of the single rigid tunnel model [6] is also plotted in each sub-figure. The parameters are as follows. The shear-velocity contrast between the tunnel and the half-space (we call it \"tunnel relative stiffness\") \(\beta_{0}/\beta\) = 2, 5, 10, and 100. The parameters for tunnel depth are \(D/a\) = 2 and tunnel distance \(s/a\) = 5, 10, and 20. The  
Figure 3: Tunnel displacement residuals convergence.  
wave incident angle \(\gamma=0^{\circ}\) (vertical incident), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) (horizontal incident). The dimensionless frequency of incident SH-wave \(\eta=0.25\), \(0.5\), \(1.0\), and \(2.0\).  
Figure 4: Typical spectral amplification of the tunnel displacement responses under different tunnel relative stiffness (\(\beta_{0}/\beta\)) and dimensionless frequency (\(\eta\)) of the incident wave. Tunnel burial depth \(D/a=2\), tunnel distance \(s/a=5\), incident angle \(\gamma=0^{\circ}\) (vertical incident). Left column: tunnel 1; right column: tunnel 2. Parts (**a–d**) correspond to the dimensionless frequency \(\eta=0.25\), \(0.5\), \(1.0\), and \(2.0\), respectively.  
Figure 5: Typical spectral amplification of the tunnel displacement responses under different incident angles (\(\gamma\)). Tunnel burial depth \(D/a=2\), tunnel distance \(s/a=5\), dimensionless frequency \(\eta=1.0\). Left column: tunnel 1; right column: tunnel 2. Parts (**a–d**) correspond to the incident angle \(\gamma=0^{\circ}\) (vertical incident), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) (horizontal incident), respectively.  
The analysis is arranged as follows. In Section 4.1, the model parameters are introduced. In Section 4.2, the influence of the tunnel relative stiffness (\(\beta_{0}/\beta\)) on the two tunnels' responses is investigated. For excluding the interference of other factors, only the results of the vertical incident (\(\gamma=0^{\circ}\)) are demonstrated. In Section 4.3, we further analyze the  
Figure 6: Typical spectral amplification of the tunnel displacement responses under different tunnel distance (\(s/a\)). Tunnel burial depth \(D/a=2\), dimensionless frequency \(\eta=1.0\), incident angle \(\gamma=0^{\circ}\) (vertical incident). Left column: tunnel 1; right column: tunnel 2. Parts (**a–c**) correspond to the tunnel distance \(s/a=5\), \(10\), and \(20\), respectively.  
influence of the wave incident angle on tunnel response. In Section 4.4, the effects of the dimensionless distance between the twin tunnels (\(s/a\)) are discussed.  
What needs to be pointed out is that, in this section, we only discuss the results that correspond to the tunnel buried depth \(D/a=2\). This is because the influence of the tunnel buried depth on the tunnel dynamic response has been reported in [6]. The conclusion in [6] shows that the shallower the tunnel buried depth (that is, the closer to the ground free surface), the greater the influence of the ground free surface and the stronger the interaction between the two tunnels. For deeper tunnels, the influence of the free surface is insignificant. In view of this, combining with some actual tunnel engineering data, we finally took the buried depth of the tunnel as \(D/a=2\) in this section, which is equivalent to studying the strongest interaction between the two tunnels.