Difference between revisions of "Standard Linear Wave Scattering Problem"

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The standard linear wave scattering problem is based on assuming irrotational and inviscid fluid motion and the wave
+
{{complete pages}}
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
+
{{standard linear problem notation}}
<center><math>
+
[[Variable Bottom Topography]]
\nabla^{2}\phi=0, \, -\infty<z<0,\,\,\,\mathbf{x}\notin \Omega
+
can also easily be included.
</math></center>
 
<center><math>
 
\frac{\partial\phi}{\partial z}=0, \, z=h,
 
</math></center>
 
<center><math>
 
\frac{\partial\phi}{\partial z} = k_{\infty}\phi,\,\mathbf{x}\in\partial\Omega,
 
</math></center>
 
<center><math>
 
\frac{\partial\phi}{\partial z} = L\phi, \, z\in\Gamma_w.
 
</math></center>
 
  
where <math>k_{\infty}</math> is the wavenumber in [[Infinite Depth]] which is given by
+
{{standard linear wave scattering equations}}
<math>k_{\infty}=\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
+
The simplest case is for a fixed body  
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.
 
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
+
{{incident plane wave}}
there is an incident wave <math>\phi^{\mathrm{{In}}}\,</math> 
+
 
is a plane wave travelling in the <math>x</math> direction
+
{{sommerfeld radiation condition two dimensions}}
<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
+
{{sommerfeld radiation condition three dimensions}}
<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]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 19:14, 8 February 2010


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

[math]\displaystyle{ \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]\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 denote 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 at [math]\displaystyle{ z=-h }[/math]. Variable Bottom Topography can also 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 assume the following. [math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,

[math]\displaystyle{ \phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \, }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \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]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math]) and

[math]\displaystyle{ \phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h} }[/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.