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 occupies the region [math]\displaystyle{ -h\lt z\lt 0. }[/math] We denote the free surface by [math]\displaystyle{ \Gamma_s }[/math] (located at [math]\displaystyle{ z=0 }[/math]) and the wetted surface of the ice floe by [math]\displaystyle{ \Gamma_w }[/math]. We denote [math]\displaystyle{ \mathbf{r}= x }[/math] or [math]\displaystyle{ \mathbf{r}=(x,y) }[/math] as the horizontal coordinate in two or three dimensions respectively.

The equations are the following

[math]\displaystyle{ \nabla^{2}\phi=0, \, -\infty\lt z\lt 0 }[/math]

[math]\displaystyle{ \frac{\partial\phi}{\partial z}=0, \, z=h, }[/math]

[math]\displaystyle{ \frac{\partial\phi}{\partial z} = k_{\infty}\phi,\,z\in\Gamma_s, }[/math] [math]\displaystyle{ \frac{\partial\phi}{\partial z} = L\phi, \, z\in\Gamma_w. }[/math]

where [math]\displaystyle{ k_{\infty} }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ k_{\infty}=\omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is gravity. [math]\displaystyle{ L }[/math] is a linear operator which relates the normal and potential on the body surface through the physics of the body. For a fixed body the operator [math]\displaystyle{ L=0 }[/math].

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 condition is

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

where [math]\displaystyle{ \phi^{\mathrm{{In}}} }[/math] is the incident potential and [math]\displaystyle{ k }[/math] is the wave number.

In three-dimensions the 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]