# Ship-wake Module¶

Following Ertekin et al. (1986), Wu (1987) and Torsvik et al. (2008), the pressure disturbance with a center point at $$(x^*, y^*$$) is given by

(1)$p_a(\tilde{x},\tilde{y},t) = P f(\tilde{x},t) q(\tilde{y},t),$

where

(2)$\begin{split}f(\tilde{x},t) = \left \{ \begin{array}{rl} \cos^2\left [\frac{\pi(|\tilde{x}-x^*(t)|-\frac{1}{2}\alpha L)}{(1-\alpha) L} \right ], & \frac{1}{2}\alpha L < |\tilde{x} - x^*(t)| \le \frac{1}{2}L \\ 1, & |\tilde{x} - x^*(t)| \le \frac{1}{2}\alpha L \end{array} \right.\end{split}$
(3)$\begin{split}q(\tilde{y},t) = \left \{ \begin{array}{rl} \cos^2 \left [\frac{\pi(|\tilde{y}-y^*(t)|-\frac{1}{2}\beta R)}{(1-\beta) R} \right ], & \frac{1}{2}\beta R < |\tilde{y} - y^*(t)| \le \frac{1}{2}R \\ 1, & |\tilde{y} - y^*(t)| \le \frac{1}{2}\beta R \end{array} \right.\end{split}$

on the rectangle $$- L/2 \le \tilde{x} - x^*(t) \le L/2  and  - R/2 \le \tilde{y} - y^*(t) \le R/2$$, and zero outside this region, $$L$$ and $$R$$ represent the length and width of the pressure source, respectively. $$\alpha$$ and $$\beta$$ are parameters representing the shape of the draft region and $$0\le(\alpha,\beta)<1$$. They can be evaluated using the block coefficient of a watercraft as described below. ($$\tilde{x}, \tilde{y}$$) is the coordinate system for the pressure disturbance which may be rotated by an angle relative to the Boussinesq coordinate system ($$x,y$$). $$P$$ is a parameter controlling the surface displacement. In fact, $$p_a$$ is the static depression around the vessel.

In contrast to the formulation of the pressure distribution in the previous study such as Torsvik et al. (2008), $$P$$ has a unit of meters and can be interpreted as the inverse barometer effect corresponding to the static surface depression for a stationary vessel.

The values of $$\alpha$$ and $$\beta$$ are shape parameters and can be obtained by adjusting $$\alpha$$ and $$\beta$$ to get the displaced volume (static submerged volume of the vessel)

(4)$V_{\mbox{sub}} = \iint p_a d\tilde{x} d\tilde{y}$

which should match a given block coefficient $$C_B$$ defined by

(5)$C_B = \frac{V_{\mbox{sub}} }{L \cdot R \cdot D}$

in which $$D$$ represents draft of a vessel.

To set up vessels in the model, see Shipwakes Setup