Difference between revisions of "Dispersion Relation for a Free Surface"

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== Separation of Variables ==
+
{{complete pages}}
  
The dispersion equation arises when separating
+
== Introduction ==
variables subject to the boundary conditions for a free surface.
 
The same equation arises when separating variables in two or three dimensions
 
and we present here the two-dimensional version.
 
We begin with the equations for the
 
[[Frequency Domain Problem]] with radial frequency <math>\,\omega</math> in terms of
 
the potential alone which are
 
  
<math>g \frac{\partial \phi}{\partial z} =
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The dispersion equation for a free surface is one of the most important equations in linear water wave theory. It arises when [http://en.wikipedia.org/wiki/Separation_of_Variables separating variables] subject to the boundary conditions for a free surface.
- \omega^2 \phi, \, z=0</math>
 
  
plus the equations within  the fluid
+
The same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. We denote the vertical coordinate by <math>z\,,</math> which points vertically upwards, and the free surface is at <math>z=0\,.</math>
  
<math>\nabla^2\phi =0 </math>
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{{frequency definition}}
  
<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math>
 
  
where <math> \,g </math> is the acceleration due to gravity,  <math> \,\rho_i </math> and <math> \,\rho </math>
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{{velocity potential in frequency domain}}
are the densities of the plate and the water respectively, <math> \,h </math> and <math> \,D </math> are
 
the thickness and flexural rigidity of the plate.
 
  
We then look for a separation of variables solution to [[Laplace's Equation]] and obtain the
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The equations therefore become
following expression for the velocity potential
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{{standard linear wave scattering equations without body condition}}
  
<math>\phi(x,z) = e^{ikx} \cosh k(z+H) \,</math>
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We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]
 +
{{separation of variables in two dimensions}}
  
If we then apply the condition at <math>z=0</math> we see that the constant <math>\,k</math>
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{{separation of variables for a free surface}}
(which corresponds to the wavenumber) is given by
 
  
<math>  - k \sinh(kH) =
 
- \omega^2 \cosh(kH)  \,\,\,(1) </math>
 
  
 +
== See Also ==
 +
* [http://en.wikipedia.org/wiki/Dispersion_(water_waves) Dispersion (water waves)]
  
This is the dispersion equation for a free surface.
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== Matlab Code ==
  
 
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A program to calculate solutions to the dispersion relation for a free surface
== Solution of the dispersion equation ==
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[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
 
+
[[Category:Linear Water-Wave Theory]]
The solution consists of one real and infinite
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[[Category:Pages with Matlab Code]]
number of imaginary roots with positive part plus their negatives. The vertical eigenfunctions
 
form complete set for <math>L_2[-H,0]\,</math> and they are orthogonal.
 

Latest revision as of 09:18, 20 October 2009


Introduction

The dispersion equation for a free surface is one of the most important equations in linear water wave theory. It arises when separating variables subject to the boundary conditions for a free surface.

The same equation arises when separating variables in two or three dimensions and we present here the two-dimensional version. We denote the vertical coordinate by [math]\displaystyle{ z\,, }[/math] which points vertically upwards, and the free surface is at [math]\displaystyle{ z=0\,. }[/math]

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 equations therefore become

[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])

We use separation of variables We express the potential as

[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]

and then Laplace's equation becomes

[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]

Separation of variables for a free surface

We use separation of variables

We express the potential as

[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]

and then Laplace's equation becomes

[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]

The separation of variables equation for deriving free surface eigenfunctions is as follows:

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime}(-h) = 0 }[/math]

and

[math]\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }[/math]

We can then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write

[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]

where we have chosen the value of the coefficent so we have unit value at [math]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha \, }[/math]

which is the Dispersion Relation for a Free Surface

The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by [math]\displaystyle{ k_{0}=\pm ik \, }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} \, }[/math], [math]\displaystyle{ m\geq1 }[/math]. The [math]\displaystyle{ k \, }[/math] of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

[math]\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }[/math]

to arrive at the dispersion relation

[math]\displaystyle{ \alpha = k\tanh kh. }[/math]

We note that for a specified frequency [math]\displaystyle{ \omega \, }[/math] the equation determines the wavenumber [math]\displaystyle{ k \, }[/math].

Finally we define the function [math]\displaystyle{ Z(z) \, }[/math] as

[math]\displaystyle{ \chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

[math]\displaystyle{ \int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right). }[/math]


See Also

Matlab Code

A program to calculate solutions to the dispersion relation for a free surface dispersion_free_surface.m