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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
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− | equations. We also assume that [[Frequency Domain Problem]] with frequency <math>\omega</math>
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− | The water motion is represented by a velocity potential which is
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− | denoted by <math>\phi</math>. The coordinate system is the standard Cartesian coordinate system
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− | with the <math>z</math> axis pointing vertically up. The water surface is at
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− | <math>z=0</math> and the region of interest is
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− | <math>-h<z<0</math>. There is a body which occupies the region <math>\Omega</math>
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− | and we denoted the wetted surface of the body by <math>\partial\Omega</math>
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− | We denote <math>\mathbf{r}=(x,y)</math> as the horizontal coordinate in two or three dimensions
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− | respectively and the cartesian system we denote by <math>\mathbf{x}</math>.
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− | We assume that the bottom surface is of constant depth but [[Variable Bottom Topography]]
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− | can easily be included.
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| | | |
− | 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>
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− | <center><math>
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− | \frac{\partial\phi}{\partial z}=0, \, z=h,
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− | </math></center>
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− | <center><math>
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− | \frac{\partial\phi}{\partial z} = k_{\infty}\phi,\,\mathbf{x}\in\partial\Omega,
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− | </math></center>
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− | <center><math>
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− | \frac{\partial\phi}{\partial z} = L\phi, \, z\in\Gamma_w.
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− | </math></center>
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| | | |
− | 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
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| 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. |
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− | 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>
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− | \phi^{\mathrm{{In}}}({r},z)=Ae^{{\rm i}kx}\cosh k(z+h)\,
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− | </math></center>
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− | where <math>A</math> is the wave amplitude and <math>k</math> is the wavenumber which is
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− | the positive real solution of the [[Dispersion Relation for a Free Surface]].
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− | We also have to apply the [[Sommerfeld Radiation Condition]] as <math>\left|\mathbf{r}\right|\rightarrow
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− | \infty</math>.
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| | | |
− | In two-dimensions the condition is
| + | {{sommerfeld radiation condition three dimensions}} |
− | <center><math>
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− | \left( \frac{\partial}{\partial|x|}-{i}k\right)
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− | (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}
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− | </math></center>
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− | where <math>\phi^{\mathrm{{In}}}</math> is the incident potential and <math>k</math>
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− | is the wave number.
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− | In three-dimensions the condition is
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− | <center><math>
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− | \sqrt{|\mathbf{r}|}\left( \frac{\partial}{\partial|\mathbf{r}|}-{i}k\right)
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− | (\phi-\phi^{\mathrm{{In}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.}
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− | </math></center>
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| [[Category:Linear Water-Wave Theory]] | | [[Category:Linear Water-Wave Theory]] |
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.