# Standard Linear Wave Scattering Problem

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\ltz\lt0[/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]. Variable Bottom Topography can also easily be included.

The equations are the following

(note that the last expression can be obtained from combining the expressions:

where [math]\alpha = \omega^2/g \,[/math])

where [math]\mathcal{L}[/math] is a linear operator which relates the normal and potential on the body surface through the physics of the body.

The simplest case is for a fixed body where the operator is [math]L=0[/math] but more complicated conditions are possible.

The equation is subject to some radiation conditions at infinity. We assume the following. [math]\phi^{\mathrm{I}}\,[/math] is a plane wave travelling in the [math]x[/math] direction,

where [math]A [/math] is the wave amplitude (in potential) [math]\mathrm{i} k [/math] is the positive imaginary solution of the Dispersion Relation for a Free Surface (note we are assuming that the time dependence is of the form [math]\exp(-\mathrm{i}\omega t) [/math]) and

In two-dimensions the Sommerfeld Radiation Condition is

where [math]\phi^{\mathrm{{I}}}[/math] is the incident potential.

In three-dimensions the Sommerfeld Radiation Condition is

where [math]\phi^{\mathrm{{I}}}[/math] is the incident potential.