Dispersion Relation for a Free Surface: Difference between revisions
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the potential alone which are | the potential alone which are | ||
<math> | <math>\frac{\partial \phi}{\partial z} + k_{infty} \phi, \, z=0</math> | ||
where <math>k_{\infty}</math> is the wavenumber in [[Infinite Depth]] which is given by | |||
<math>k_{\infty}=\omega^2/g</math> where <math>g</math> is gravity. We also have | |||
the equations within the fluid | |||
<math>\nabla^2\phi =0 </math> | <math>\nabla^2\phi =0 </math> | ||
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<math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math> | <math>\frac{\partial \phi}{\partial z} = 0, \, z=-H</math> | ||
where <math> \,g </math> is the acceleration due to gravity | where <math> \,g </math> is the acceleration due to gravity. | ||
We then look for a separation of variables solution to [[Laplace's Equation]] and obtain the | We then look for a separation of variables solution to [[Laplace's Equation]] and obtain the | ||
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This is the dispersion equation for a free surface. | This is the dispersion equation for a free surface. | ||
== Solution of the dispersion equation == | == Solution of the dispersion equation == | ||
Revision as of 22:31, 23 May 2006
Separation of Variables
The dispersion equation 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 begin with the equations for the Frequency Domain Problem with radial frequency [math]\displaystyle{ \,\omega }[/math] in terms of the potential alone which are
[math]\displaystyle{ \frac{\partial \phi}{\partial z} + k_{infty} \phi, \, z=0 }[/math]
where [math]\displaystyle{ k_{\infty} }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ k_{\infty}=\omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is gravity. We also have the equations within the fluid
[math]\displaystyle{ \nabla^2\phi =0 }[/math]
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \, z=-H }[/math]
where [math]\displaystyle{ \,g }[/math] is the acceleration due to gravity.
We then look for a separation of variables solution to Laplace's Equation and obtain the following expression for the velocity potential
[math]\displaystyle{ \phi(x,z) = e^{ikx} \cosh k(z+H) \, }[/math]
If we then apply the condition at [math]\displaystyle{ z=0 }[/math] we see that the constant [math]\displaystyle{ \,k }[/math] (which corresponds to the wavenumber) is given by
[math]\displaystyle{ - k \sinh(kH) = - \omega^2 \cosh(kH) \,\,\,(1) }[/math]
This is the dispersion equation for a free surface.
Solution of the dispersion equation
The solution consists of one real and infinite number of imaginary roots with positive part plus their negatives. The vertical eigenfunctions form complete set for [math]\displaystyle{ L_2[-H,0]\, }[/math] and they are orthogonal.
