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

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== Separation of variables for a free surface ==  
=== Separation of variables for a free surface ===


The equation  
We use [http://en.wikipedia.org/wiki/Separation_of_Variables separation of variables]
 
{{separation of variables in two dimensions}}
 
{{separation of variables for a free surface first part}}
 
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>k_{0}=\pm ik \,</math> and
the positive real solutions by <math>k_{m} \,</math>, <math>m\geq1</math>. The <math>k \,</math> of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations
<center>
<center>
<math>
<math>
- \frac{1}{Z(z)}
\cos ix = \cosh x, \quad \sin ix = i\sinh x,
\frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = \eta^2.
</math>
</math>
</center>
</center>
subject to the boundary conditions
to arrive at the dispersion relation
<center>
<center>
<math>
<math>
\frac{dZ}{dz}(-h) = 0
\alpha = k\tanh kh.
</math>
</math>
</center>
</center>
and
We note that for a specified frequency <math>\omega \,</math> the equation determines the wavenumber <math>k \,</math>.
<center>
 
<math>
Finally we define the function <math>Z(z) \,</math> as
\frac{dZ}{dz}(0) = \alpha Z(0)
</math>
</center>
We then use the boundary condition at <math>z=-h</math> to write
<center>
<center>
<math>
<math>
Z = \frac{\cos k(z+h)}{\cos kh}
\chi_{m}\left(  z\right)  =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
</math>
</center>
The boundary condition at the free surface (<math>z=0</math>) is
<center><math>
k\tan\left(  kh\right)  =-\alpha,
</math></center>
which is the [[Dispersion Relation for a Free Surface]]
We denote the
positive imaginary solution of this equation by <math>k_{0}</math> and
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>.  We define
<center>
<math>
\phi_{m}\left(  z\right)  =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
</math>
</math>
</center>
</center>
Line 45: Line 34:
<center>
<center>
<math>
<math>
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{n}\delta_{mn}
\int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn}
</math>
</math>
</center>
</center>
Line 51: Line 40:
<center>
<center>
<math>
<math>
A_{n}=\frac{1}{2}\left(  \frac{\cos k_{n}h\sin k_{m}h+k_{n}h}{k_{n}\cos
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)
^{2}k_{n}h}\right).
</math>
</center>.
 
The equation
<center>
<math>
- \frac{1}{Z(z)}
\frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = \eta^2.
</math>
</center>
is the equation for separation of variables for a free surface.
In the setting of water of finite depth, the general solution 
<math>Z(z)</math> can be written as
<center>
<math>
Z(z) = F \cos \big( \eta (z+d) \big) + G \sin \big( \eta (z+d) \big),
\quad \eta \in \mathbb{C} \backslash \{ 0 \},
</math>
</center>
since <math>\eta = 0</math> is not an eigenvalue.
To satisfy the bed condition, <math>G</math> must be <math>0</math>.
<math>Z(z)</math> satisfies the free surface condition, provided the separation
constants <math>\eta</math> are roots of the equation
<center>
<math>
- F \eta \sin \big( \eta (z+d) \big) - \alpha F \cos \big( \eta (z+d)
  \big) = 0, \quad z=0,
</math>
</center>
or, equivalently, if they satisfy the [[Dispersion Relation for a Free Surface]]
<center><math>
\alpha + \eta \tan \eta d = 0\,.
</math></center>
This equation has an
infinite number of real roots, denoted by <math>k_m</math> and <math>-k_m</math> (<math>m \geq
1</math>), but the negative roots produce the same eigenfunctions as the
positive ones and will therefore not be considered. It also has a pair of purely imaginary roots which
will be denoted by <math>k_0</math>. Writing <math>k_0 = - \mathrm{i} k</math>, <math>k</math> is the
(positive) root of the [[Dispersion Relation for a Free Surface]]
<center><math>
\alpha = k \tanh k d,\,
</math></center>
again it suffices to consider only the positive root of this equation. The solutions can
therefore be written as
<center>
<math>
Z_m(z) = F_m \cos \big( k_m (z+d) \big), \quad m \geq 0.
</math>
</math>
</center>
</center>
It follows that <math>k</math> is the previously introduced wavenumber and the [[Dispersion Relation for a Free Surface]]
gives the required relation between the radian frequency and the wavenumber.

Latest revision as of 04:40, 16 June 2015

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]

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