Template:Standard linear problem notation

We assume small amplitude so that we can linearise all the equations (see Linear and Second-Order Wave Theory). We also assume that Frequency Domain Problem with frequency $\omega$ and we assume that all variables are proportional to $\exp(-\mathrm{i}\omega t)\,$ The water motion is represented by a velocity potential which is denoted by $\phi\,$ so that
$\Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}.$
The coordinate system is the standard Cartesian coordinate system with the $z-$axis pointing vertically up. The water surface is at $z=0$ and the region of interest is $-h\lt z\lt 0$. There is a body which occupies the region $\Omega$ and we denote the wetted surface of the body by $\partial\Omega$ We denote $\mathbf{r}=(x,y)$ as the horizontal coordinate in two or three dimensions respectively and the Cartesian system we denote by $\mathbf{x}$. We assume that the bottom surface is of constant depth at $z=-h$.