
\begin{table}
\begin{center}
\caption{Variables used in slide submarine mass failure model.}
\begin{tabular}{|l|l|}\hline
Variable name & Quantity \\ \hline
$b$ & length \\ \hline
$w$ & width \\ \hline
$T$ & thickness \\ \hline
$\gamma$ & density \\ \hline
$d$ & water depth to centre of mass \\ \hline
$\theta$ & bathymetric slope \\ \hline
$\psi$ & angular orientaion \\ \hline
$C_d$ & drag coefficient \\ \hline
$C_m$ & added mass coefficient \\ \hline
\end{tabular}
\end{center}
\end{table}


The following relationships are used to derive parameters describing a slide 
submarine mass failure:

characteristic time of motion

$$t_0 = \frac{u_t}{a_0}$$

initial acceleration

$$a_0 = g \sin \theta (\frac{\gamma-1}{\gamma + C_m}) (1 - \frac{\tan \psi}{\tan \theta})$$

theoretical terminal velocity

$$u_t = \sqrt{gd} \sqrt{ \frac{b \sin \theta}{d} \frac{\pi (\gamma-1)}{2 C_d}
(1 - \frac{\tan \psi}{\tan \theta}) }$$

characteristic distance of motion

$$s_0 = \frac{u_t^2}{a_0}$$

From these parameters, further parameters are derived which describe the water displacement produced by the slide:


characteristic tsunami wavelength

$$\lambda_0 = t_0 \sqrt{gd}$$

characteristic two dimensional amplitude

$$\eta_{0,2D} = s_0 (0.0574 - 0.0431 \sin \theta) (\frac{T}{b})
(\frac{b \sin \theta}{d})^{1.25} (1 - \exp(-2.2(\gamma-1) ) ) $$

characteristic three dimensional amplitude

$$\eta_{0,3D} = \frac{\eta_{0,2D}}{1 + 15.5 \sqrt{ \frac{d}{b \sin \theta}}} $$

Assuming a double Gaussian relationship in the $x$ direction (tsunami length) and a $\sech^2$ 
relationship in the $y$ direction (tsunami width), the initial water displacement for a slide may be represented by

$$\eta(x,y) = \eta_{0,3D} \frac{ (\exp(-(\frac{x-x_0}{\lambda_0})^2) - 
\kappa \exp(-(\frac{x-\delta x - x_0}{\lambda_0})^2))}{\cosh^2(\kappa\frac{y-y_0}{w+\lambda_0})}$$
