View source for 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>\omega</math>
The water motion is represented by a velocity potential which is
denoted by <math>\phi</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<z<0</math>. There is a body which occupies the region <math>\Omega</math>
and we denoted 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 but [[Variable Bottom Topography]]
can easily be included. 

The equations are the following
<center><math>
\nabla^{2}\phi=0, \, -\infty<z<0,\,\,\,\mathbf{x}\notin \Omega
</math></center>
<center><math>
\frac{\partial\phi}{\partial z}=0, \, z=h,
</math></center>
<center><math>
 \frac{\partial\phi}{\partial z} = \apha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
</math></center>
<center><math>
 \frac{\partial\phi}{\partial z} = L\phi, \, z\in\partial\Omega,
</math></center>

where <math>\aklpha</math> is the wavenumber in [[Infinite Depth]] which is given by 
<math>\aplha=\omega^2/g</math> where <math>g</math> is gravity. <math>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 usually assume that
there is an incident wave <math>\phi^{\mathrm{{In}}}\,</math>  
is a plane wave travelling in the <math>x</math> direction 
<center><math>
\phi^{\mathrm{{In}}}({r},z)=Ae^{{\rm i}kx}\cosh k(z+h)\,
</math></center>
where <math>A</math> is the wave amplitude and <math>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>\left|\mathbf{r}\right|\rightarrow
\infty</math>.

In two-dimensions the condition is 
<center><math>
\left(  \frac{\partial}{\partial|x|}-{i}k\right)
(\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
</math></center>
where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math>
is the wave number.

In three-dimensions the condition is 
<center><math>
\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></center>

[[Category:Linear Water-Wave Theory]]