Template:Standard linear problem notation

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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 [math]\omega[/math] and we assume that all variables are proportional to [math]\exp(-\mathrm{i}\omega t)\,[/math] The water motion is represented by a velocity potential which is denoted by [math]\phi\,[/math] so that

[math] \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. [/math]

The coordinate system is the standard Cartesian coordinate system with the [math]z-[/math]axis pointing vertically up. The water surface is at [math]z=0[/math] and the region of interest is [math]-h\lt z\lt 0[/math]. There is a body which occupies the region [math]\Omega[/math] and we denote the wetted surface of the body by [math]\partial\Omega[/math] We denote [math]\mathbf{r}=(x,y)[/math] as the horizontal coordinate in two or three dimensions respectively and the Cartesian system we denote by [math]\mathbf{x}[/math]. We assume that the bottom surface is of constant depth at [math]z=-h[/math].