## 4 Effect of Damping Control Parameters on Wave Energy Attenuation
### The Influence of Absorption Region Thickness \(L\)
In the present work, the determination of the MVDB action range depends on the wavelength. In this sense, the applicability of the MVDB viscous damping approximation (Equation (11)) to other materials should be considered. For a clear interpretation, other materials were introduced for arithmetic examples, and the meaning of \(L\) is explained through numerical results. \(L\) refers to the extent of the entire absorbing region, and its core value lies in the design of the number of absorbing material layers when determining the thickness of a single absorbing material. Due to the different physical properties of materials, a research gap remains on the rigorous definition of the number of absorbing layers for studied media and the desirable calculations. Seismic waves at a consistent frequency exhibit varying wavelengths when propagating in different media, and wavelength is only related to the wave velocity and source frequency. For this consideration, it was selected as a medium to design a suitable absorbing layer scheme for different materials according to Equation (11). The given \(\Delta l\) and the source frequency were used to convert the wavelength into the thickness of the absorbing region \(L\), which is material specific, and then the number of suitable absorbing layers \(n\) was derived. In this sense, the significance of \(L\) is determining a reasonable number of absorbing layers \(n\). Due to limited space, this paper only cites the numerical case of concrete materials for explanation. C30 concrete has [PERSON]'s modulus \(E=20\) GPa, [PERSON]'s ratio \(\
u=0.2\), and density \(\rho=2400\) kg/m\({}^{3}\), and the effective computational size of the model is 80 cm \(\times\) 80 cm. The thickness of the unilateral absorbing region is 10 cm for Model 1 and 8 cm for Model 2. The mesh of Model 2 was refined (\(\Delta l=0.4\) cm), and the number of absorbing layers is kept consistent with that of Model 1, where \(k=1.08\), \(n=L/\Delta l\), calculation time = 1 ms, and time step = 5 \(\times\) 10\({}^{-4}\) ms. MVDB parameters were obtained from Equations (11) and (12), as shown in Table 1, and the model is schematically shown in Figure 18. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Model** & \(f\) & \(\lambda_{min}\) & \(L\) & \(\Delta I\) & \(n\) & \(t_{z}\) & \(\Delta t_{max}\) \\ \hline
1 & 40 kHz & 5 cm & \(2\lambda_{min}\) & 0.5 cm & 20 & 1 ms & \(5\times 10^{-4}\) ms \\
2 & 40 kHz & 5 cm & \(1.6\lambda_{min}\) & 0.4 cm & 20 & 1 ms & \(5\times 10^{-4}\) ms \\ \hline \end{tabular}
\end{table} 
Table 1: Model and partial discrete parameters. Figure 17: Total damping values and residual wave energy for the \(L\) of the absorbing region. (**a**) Total damping values. (**b**) Residual wave energy amplitudes of the outermost absorbing layer. Figure 19 shows the snapshots of the wave fields of Models 1 and 2. It can be seen that the wave energy attenuation of the two models is highly consistent. Despite a small difference in \(L\), the approximate results can still be obtained due to the consistent numbers of absorbing layers, and such a slight difference may be due to the attenuation properties of the remaining thick concrete medium (\(0.4\lambda_{min}\)). Based on the above analysis, the conclusion of the \(0.5\)-\(3\lambda_{min}\) calculation is only feasible for the current mesh accuracy (Equation (12)). Nonetheless, it still reflects that the MVDB is equally applicable to other materials.