# Cnoidal Wave-Induced Residual Liquefaction in Loosely Deposited Seabed under Coastal Shallow Water
## 4 Result Analysis
### Displacement  
The cnoidal wave-induced displacement and the variation of the void ratio in the seabed floor at a typical position (\(x\) = 100 m, \(z\) = 28.32 m) are shown in Figure 3. It is observed that there is a significant horizontal lateral spreading to the right side and significant vertical subsidence occurring in the seabed floor under the continuous action of the cnoidal wave. At \(t\) = 500 s, the residual horizontal lateral spreading reaches up 45 cm, and the residual subsidence reaches up to about 7 cm (note: the initial subsidence under gravity at 5.5 cm was excluded). This significant lateral spreading is a type of the resultant phenomenon of the liquefaction or softening, which has occurred in the seabed floor caused by the pore pressure build-up. It is also interesting to find that the displacement amplitude horizontally is much less than that vertically.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Nevada Dense Sand** & **Unit** \\ \hline Kevo & 2000 & (kPa) \\ G\({}_{\rm eso}\) & 2600 & (kPa) \\ P\({}^{\prime}\)\({}_{0}\) & 4 & (kPa) \\ M\({}_{\rm g}\) & 1.32 & \\ M\({}_{\rm f}\) & 1.3 & \\ \(\alpha_{\rm f}\) & 0.45 & \\ \(\alpha_{\rm g}\) & 0.45 & \\ \(\beta_{0}\) & 4.2 & \\ \(\beta_{1}\) & 0.2 & \\ H\({}_{0}\) & 750 & \\ H\({}_{\rm U0}\) & 40,000 & (kPa) \\ \(\gamma_{\rm u}\) & 2.0 & \\ \(\gamma_{\rm DM}\) & 4.0 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Model parameters of the loose seabed soil for PZIII in analysis.  
Figure 3: Time history of the displacement and void ratio of the seabed at \(x\) = 100 m, \(z\) = 28.32 m. It is shown that the loose seabed soil has a significant lateral spreading, as well as a vertical residual subsidence and an oscillatory vibration driven by the cnoidal wave (note: \(x\) = 100 m is the symmetrical line of the seabed floor).  
The time history of the vertical displacement indicates that the loose seabed soil is continuously subsiding under the cnoidal wave continuous loading. In the meantime, the oscillatory amplitude becomes greater and greater. The phenomenon indicates that the stiffness of the soil skeleton is lost due to the reduction of effective stress between soil particles. This phenomenon is also a result of the liquefaction or softening of the seabed.  
The increasing residual subsidence of seabed soil is attributed to the process where some pore water is gradually drained out through the surface of the seabed floor driven by the upward pressure gradient in the seabed floor. It has been well known that the pore pressure will be accumulated in loosely deposited seabed floors under ocean wave loading. Meanwhile, the water pressure applied on the surface of the seabed floor does not accumulate. As a consequence, an upward pore pressure gradient is formed in the upper seabed floor. The pore water in the seabed floor will be drained out, driven by this upward pore pressure gradient. Once some pore water is drained out, the space previously occupied by this pore water will be occupied by the surrounding soil particles, leading to the reduction of the void ratio of soil, as illustrated in Figure 3. Overall, there are four types of simultaneous processes in the loosely deposited seabed floor under wave loading in total: (1) process of pore pressure buildup, (2) process of effective stress reduction, (3) process of pore water drained out, and (4) process of void ratio reduction. This mechanism is the reason why there are dense sand soils in nature. Dense sand could be evolved from loose sand in nature under dynamic loadings, such as ocean waves and seismic waves.