**5. Wave-Induced Liquefaction**

The stability of the coastal structures and its seabed foundation is one of the main concerns for engineering design procedure. The wave-induced liquefaction in a porous seabed is one of the most significant unstable factors. Zen and Yamazaki [55] pointed out that the liquefaction of the porous seabed is responding to the variation of the ocean wave, which is actually caused by the periodic upward seepage force. Thus, we proposed to estimate the liquefaction state in two-dimensions, i.e.,

$$
\sigma\_0' + (P\_b - p\_s) \le 0 \tag{35}
$$

where *σ* 0 0 represents the initial effective stress, *P<sup>b</sup>* is the wave pressure acting on seabed, while *p<sup>s</sup>* is the wave-induced transient pore pressure. The value of *P<sup>b</sup>* − *p<sup>s</sup>* is equal to the excess pore pressure generated by the wave loading (|*u<sup>e</sup>* |).

Figure 13 shows the wave-induced transient liquefaction area around the immersed tunnel at three typical times (*t* = 12 s, *t* = 14 s and *t* = 15.5 s) separately. As shown in the figure, the transient liquefaction area moves along the direction of the wave propagation. The previously liquefied area is able to recover as the wave trough go away. This process is repeated periodically under the cyclic wave loading. The maximum liquefaction depth in this case is 0.8 m below the seabed surface, as seen in Figure 13a, while the soil that covered the tunnel is fully liquefied during one wave period which illustrated in Figure 13b. Thus, the back filling soil above the tunnel can not protect the immersed tunnel any more in this circumstance. Moreover, the maximum liquefaction depth of the rightward seabed of the tunnel is 0.6 m, which is slightly shallow than that in leftwards of 0.8 m.

**Figure 13.** The liquefaction area surrounding immersed tunnel for (**a**) *t* = 12 s; (**b**) *t* = 14 s and (**c**) *t* = 15.5 s respectively.
