# Cnoidal Wave-Induced Residual Liquefaction in Loosely Deposited Seabed under Coastal Shallow Water
## 3 Boundary Conditions and Hydrodynamic Loading  
The length and thickness of the seabed floor used in the computation are 200 and 30 m, respectively, as illustrated in Figure 1. In total, 12,190 four-node elements are utilized to discretize the computation domain. The maximum size of the elements is 0.87 m horizontally and 0.6 m vertically. At the bottom of the seabed floor, the horizontal and vertical displacements are all fixed as 0, and only the horizontal displacement is fixed along the two lateral sides of the computational domain. On the surface of the seabed floor, the hydrodynamic loading and the hydrostatic pressure are jointly applied. However, it is worthy to note that the effective stresses on the surface of the seabed floor must always be kept zero all the time in the computation.  
According to [PERSON] and [PERSON] (1983) [26], the profile of the cnoidal wave is expressed as  
\[\eta=d+H[\frac{1}{m^{2}}-1-\frac{E(m)}{m^{2}K(m)}]+Hcn^{2}[2K(m)\cdot(\frac{x} {L}-\frac{t}{T})] \tag{6}\]  
where \(d\) is the water depth, \(H\) is the wave height, and \(T\) is the wave period. \(cn(\cdot)\) is the Jacobi elliptic function. \(m\in(0,1)\) is the modulus of the cnoidal wave. \(K(m)\) and \(E(m)\) are the complete ellipse integral of the first kind and the second kind, respectively:  
\[K(m)=\int_{0}^{\pi/2}\frac{1}{\sqrt{1-m^{2}sin^{2}\varphi}}d\varphi \tag{7}\]  
\[E(m)=\int_{0}^{\pi/2}\sqrt{1-m^{2}sin^{2}\varphi}\;d\varphi \tag{8}\]  
\(L\) is the length of the cnoidal wave, formulated as  
\[L=mK(m)\sqrt{\frac{16d^{3}}{3H}} \tag{9}\]  
The cnoidal wave-induced hydrodynamic loading on the surface of the seabed floor is expressed as  
\[p=\rho_{f}gd+\rho_{f}gH\{[\frac{1}{m^{2}}-1-\frac{E(m)}{m^{2}K(m)}]+cn^{2}[2K (m)\cdot(\frac{x}{L}-\frac{t}{T})]\} \tag{10}\]  
Figure 1: Diagram of the computational domain for the seabed floor.  
where \(\rho_{f}\) is the density of seawater and \(g\) is the gravity. It has been well known that when the Ursell parameter \(U=HL^{2}/d^{3}>25\), the hydrodynamic loading determined by cnoidal wave theory will be much more accurate than that by Stokes wave theory [27].  
In fact, it is inconvenient for us to compute the complete ellipse integral of the first kind and second kind \(K(m)\) and \(E(m)\), as formulated in Equations (7) and (8). To solve this problem, [PERSON] and [PERSON] [28] proposed the following formulations containing an infinite series to approach the real value of \(K(m)\) and \(E(m)\).  
\[K(m)=\frac{\pi}{2}+\frac{\pi}{2}\sum_{n=1}^{\infty}\left[\frac{1}{2^{2n-1}} \frac{(2n-1)!}{(n-1)!n!}\right]^{2}\times m^{n} \tag{11}\]  
\[E(m)=\frac{\pi}{2}-\frac{\pi}{2}\sum_{n=1}^{\infty}\big{[}\frac{1}{2n-1}|[ \frac{1}{2^{2n-1}}\frac{(2n-1)!}{(n-1)!n!}]^{2}\times m^{n} \tag{12}\]  
Since it is also not easy to get the value of the Jacobi elliptic function \(cn^{2}\big{[}2K(m)\big{(}\frac{\pi}{L}-\frac{t}{T}\big{)}\big{]}\) in Equation (10), [PERSON] (1977) [29] proposed the following Fourier series approximation to reliably estimate the value of \(cn^{2}\big{[}2K(m)\big{(}\frac{\pi}{L}-\frac{t}{T}\big{)}\big{]}\).  
\[cn^{2}[2K(m)(\frac{x}{L}-\frac{t}{T})]=1+\frac{\frac{E(m)}{K(m)}-1}{m}+\sum_{ j=1}^{N}A_{j}\cos[j\frac{\pi}{K(m)}2K(m)(\frac{x}{L}-\frac{t}{T})] \tag{13}\]  
\[A_{j}=\frac{j\pi^{2}}{mK^{2}(m)\times\sin h[\frac{j\pi K(1-m)}{K(m)}]} \tag{14}\]  
It is found that the modulus of the cnoidal wave \(m\in\) (0, 1) is the key parameter determining the waveform of the wave-induced pressure applied on the surface of the seabed floor. This effect is clearly illustrated in Figure 2. It is observed that the waveforms of the cnoidal wave-induced hydrodynamic loading on the surface of the seabed floor are highly dependent on the value of \(m\). When \(m\leq\) 0.1, it is a little approach to the linear Stokes wave. When 1 \(>m>\) 0.9, the nonlinearity is very significant. In this study, \(m=\) 0.99 is used in the computation.