Difference between revisions of "Removing the Depth Dependence"

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{{complete pages}}
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We are considering the [[Frequency Domain Problem]] for linear wave waves.  
 
We are considering the [[Frequency Domain Problem]] for linear wave waves.  
{{fixed body finite depth equations in two dimensions}}
 
  
If we have a problem in which all the scatterers are of constant cross sections so
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{{standard linear problem notation}}
that
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<center>
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{{boundary value problem for a fixed body}}
<math>\partial\Omega = \partial\hat{\Omega} \times z\in[-h,0]
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</math>
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{{incident plane wave}}
</center>
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where <math>\partial\hat{\Omega} </math> is a function only of <math>x,y</math>
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{{sommerfeld radiation condition two dimensions}}
i.e. the boundary of the scattering bodies is uniform with respect to depth.
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We can remove the depth dependence [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]
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{{removing the depth dependence}}
and obtain that the dependence on depth is given by
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<center>
 
<math>
 
\phi(x,y,z) = \frac{\cos \big( k_0 (z+h) \big)}{\cos(k_0 h)} \Phi(x,y)
 
</math>
 
</center>
 
Since <math>\phi</math> satisfies [[Laplace's Equation]], then <math>\Phi</math> satisfies [[Helmholtz's Equation]]
 
<center>
 
<math>\nabla^2 \phi + k_0^2 \phi = 0 </math>
 
</center>
 
in the region not occupied by the scatterers. Not that this is not the standard why to write [[Helmholtz's Equation]]
 
because <math>k_0</math> is the pure imaginary, and it is more normal to write
 
<center>
 
<math>\nabla^2 \phi - k*2 \phi = 0 </math>
 
</center>
 
where <math>k=-ik_0.</math>
 
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 19:15, 8 February 2010


We are considering the Frequency Domain Problem for linear wave waves.

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].

The Standard Linear Wave Scattering Problem in Finite Depth for a fixed body is

[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]) The body boundary condition for a rigid body is just

[math]\displaystyle{ \partial_{n}\phi=0,\ \ \mathbf{x}\in\partial\Omega_{\mathrm{B}}, }[/math]

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.

If we have a problem in which all the scatterers are of constant cross sections so that

[math]\displaystyle{ \partial\Omega = \partial\hat{\Omega} \times z\in[-h,0] }[/math]

where [math]\displaystyle{ \partial\hat{\Omega} }[/math] is a function only of [math]\displaystyle{ x,y }[/math] i.e. the boundary of the scattering bodies is uniform with respect to depth. We can remove the depth dependence separation of variables and obtain that the dependence on depth is given by

[math]\displaystyle{ \phi(x,y,z) = \frac{\cosh \big( k (z+h) \big)}{\cosh(k h)} \bar{\phi}(x,y) }[/math]

Since [math]\displaystyle{ \phi }[/math] satisfies Laplace's Equation, then [math]\displaystyle{ \Phi }[/math] satisfies Helmholtz's Equation

[math]\displaystyle{ \nabla^2 \bar{\phi} + k^2 \bar{\phi} = 0 }[/math]

in the region not occupied by the scatterers.