Standard Linear Wave Scattering Problem

The standard linear wave scattering problem is based on assuming irrotational and inviscid fluid motion and the wave sufficiently small amplitude so that we can linearise all the equations. We also assume that Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] The water motion is represented by a velocity potential which is denoted by [math]\displaystyle{ \phi }[/math]. The coordinate system is the standard Cartesian coordinate system with the [math]\displaystyle{ z }[/math] axis pointing vertically up. The water surface is at [math]\displaystyle{ z=0 }[/math] and the region of interest is [math]\displaystyle{ -h\lt z\lt 0 }[/math]. There is a body which occupies the region [math]\displaystyle{ \Omega }[/math] and we denoted the wetted surface of the body by [math]\displaystyle{ \partial\Omega }[/math] We denote [math]\displaystyle{ \mathbf{r}=(x,y) }[/math] as the horizontal coordinate in two or three dimensions respectively and the cartesian system we denote by [math]\displaystyle{ \mathbf{x} }[/math]. We assume that the bottom surface is of constant depth but Variable Bottom Topography can easily be included.

The equations are the following

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

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

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

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

[math]\displaystyle{ \partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B, }[/math]

where [math]\displaystyle{ \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]\displaystyle{ L=0 }[/math] but more complicated conditions are possible.

The equation is subject to some radiation conditions at infinity. We usually assume that there is an incident wave [math]\displaystyle{ \phi^{\mathrm{{In}}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction

[math]\displaystyle{ \phi^{\mathrm{{In}}}({r},z)=Ae^{{\rm i}kx}\cosh k(z+h)\, }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude and [math]\displaystyle{ k }[/math] is the wavenumber which is the positive real solution of the Dispersion Relation for a Free Surface. We also have to apply the Sommerfeld Radiation Condition as [math]\displaystyle{ \left|\mathbf{r}\right|\rightarrow \infty }[/math].

In two-dimensions the Sommerfeld Radiation Condition is

[math]\displaystyle{ \left( \frac{\partial}{\partial|x|} - \mathrm{i} k \right) (\phi-\phi^{\mathrm{{I}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.} }[/math]

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

In three-dimensions the Sommerfeld Radiation Condition is

[math]\displaystyle{ \sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|} - \mathrm{i} k \right) (\phi-\phi^{\mathrm{{I}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} }[/math]

where [math]\displaystyle{ \phi^{\mathrm{{I}}} }[/math] is the incident potential. In three-dimensions the Sommerfeld Radiation Condition is

[math]\displaystyle{ \sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right) (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.} }[/math]