Difference between revisions of "Template:Separation of variables for a free surface"

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=== Separation of variables for a free surface ===
 
=== Separation of variables for a free surface ===
  
The separation of variables equation for deriving free surface eigenfunctions is as follows:
+
{{separation of variable for a free surface first part}}
<center>
+
 
<math>
 
Z^{\prime\prime} + k^2 Z =0.
 
</math>
 
</center>
 
subject to the boundary conditions
 
<center>
 
<math>
 
Z^{\prime}(-h) = 0
 
</math>
 
</center>
 
and
 
<center>
 
<math>
 
Z^{\prime}(0) = \alpha Z(0)
 
</math>
 
</center>
 
We can then use the boundary condition at <math>z=-h \, </math> to write
 
<center>
 
<math>
 
Z = \frac{\cos k(z+h)}{\cos kh}
 
</math>
 
</center>
 
The boundary condition at the free surface (<math>z=0 \,</math>) gives rise to:
 
<center><math>
 
k\tan\left(  kh\right)  =-\alpha \,
 
</math></center>
 
which is the [[Dispersion Relation for a Free Surface]]
 
 
We denote the
 
We denote the
 
positive imaginary solution of this equation by <math>k_{0} \,</math> and
 
positive imaginary solution of this equation by <math>k_{0} \,</math> and

Revision as of 10:20, 24 August 2009

Separation of variables for a free surface

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

We denote the positive imaginary solution of this equation by [math]\displaystyle{ k_{0} \, }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} \, }[/math], [math]\displaystyle{ m\geq1 }[/math]. We define

[math]\displaystyle{ \phi_{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}\phi_{m}(z)\phi_{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]

.