## 4 Effect of Damping Control Parameters on Wave Energy Attenuation
### The Influence of Absorption Region Thickness \(L\)  
In exploring the effect of \(L\) on wave energy absorption, \(L\) values are determined at \(0.5\lambda_{min}\), \(\lambda_{min}\), \(2\lambda_{min}\), and \(3\lambda_{min}\) (i.e., 1 m, 2 m, 4 m, and 6 m, respectively), \(k\) = 1.08, \(\Delta l\) is the maximum mesh size (\(\Delta l\) = 20 cm), and \(n=L/\Delta l\). The model was calculated to have a size of 50 m \(\times\) 50 m, and the thicknesses of the unilateral absorbing region are 1 m and 4 m when \(L\) is \(0.5\lambda_{min}\) and \(2\lambda_{min}\). In this paper, the number of a single absorbing layer \(n\) and the corresponding increase in \(\alpha\) are responsible for different \(L\). As shown in Figure 15a, the rapid increase in \(\alpha\) and the  
Figure 14: \(\Delta l\) vs. wave energy absorption. (**a**) Number of absorbing layers vs. \(\alpha\) growth amplitude. (**b**) Wave energy attenuation gradient. Figure 13: Numerical model of uniform soil. relatively steep curves at \(L=0.5\lambda_{min}\) indicate that considerable damping is forced to be arranged in a reduced number of absorbing layers, leading to a surge in damping and a huge difference in wave impedance, which triggers interlayer reflections at the junction of the computational region and the absorbing layer and between the layer interiors. When \(L\) is equivalent to \(\lambda_{min}\), the increase in \(\alpha\) slightly slows down. In the case of \(L=2\lambda_{min}\) and \(3\lambda_{min}\), the curve increase is relatively smooth, indicating that the damping required to consume the wave energy matches the number of absorbing layers, and a larger \(L\) indicates a more continuous rise in \(\alpha\). As shown in Figure 15b, from Sensor 16 to Sensor t, the instantaneous energy amplitudes of four groups of models (\(L=0.5\lambda_{min}\), \(1\lambda_{min}\), \(2\lambda_{min}\), and \(3\lambda_{min}\)) were counted, and the wave energy under damping can be observed in the attenuation amplitude. When \(L\) is \(0.5\lambda_{min}\) and \(1\lambda_{min}\), the energy decay behavior is weak despite the rapid decrease in the wave energy, and cusps are generated, indicating that interlayer reflections may occur at this position due to excessive damping increase. For this reason, the snapshots of the wave fields at the two \(L\) values were specifically extracted, as shown in Figure 16. It can be seen that the outgoing wave is reflected in the absorbing layer when \(L\) is \(0.5\lambda_{min}\), and the MVDB works in an orderly manner when \(L\) is \(2\lambda_{min}\), verifying the conjecture that the interlayer reflection is generated at the cusp. In addition, the total damping values (Equation (5)) of the MVDB under different \(L\) and the residual wave energy of the pulse signal at the outermost absorbing layer were statistically investigated, as seen in Figure 17.  
As shown in Figure 17, the thickness of \(L\) determines the total damping values in the absorbing region. A larger \(L\) induces a greater total damping value (Figure 17a) and a better absorption effect (Figure 17b). When \(L\) is \(0.5\lambda_{min}\), the wave energy at the bottom boundary is attenuated by about one third; for \(L=\lambda_{min}\), the wave energy decays to about \(1/15\) of the original. It reduces to about \(1/120\) of the original in the case of \(L=2\lambda_{min}\), and when increasing \(L\) to \(3\lambda_{min}\), the wave energy decreases to about \(1/230\) of the original. Combined  
Figure 16: Ordered wave energy attenuation and interlayer reflection. (**a**) \(L\) is \(2\lambda_{min}\). (**b**) \(L\) is \(0.5\lambda_{min}\). Figure 15: Absorbing region thickness \(L\) vs. wave energy absorption. (**a**) \(\alpha\) growth amplitude. (**b**) Wave energy attenuation gradient. with Figures 15-17, \(L\) is positively correlated with the stability of the MVDB. A larger \(L\) indicates a higher accuracy of the MVDB. Notably, different materials have various physical properties, leading to varying wavelengths.