Difference between revisions of "Wavemaker Theory"

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= Introduction =
+
{{Ocean Wave Interaction with Ships and Offshore Structures
 +
| chapter title = Wavemaker Theory
 +
| next chapter = [[Ship Kelvin Wake]]
 +
| previous chapter =  [[Wave Momentum Flux]]
 +
}}
  
[[Image:Wavemaker.jpg|thumb|right|600px|Wavemaker]]
+
{{complete pages}}
 +
 
 +
== Introduction ==
 +
 
 +
[[Image:Wave_maker.png|600px|right|thumb|Wavemaker]]
  
 
We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]].
 
We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]].
 
A paddle is undergoing small amplitude horizontal oscillations with displacement
 
A paddle is undergoing small amplitude horizontal oscillations with displacement
<center><math> \xi (t) = \mathfrak{Re} \left \{\frac{1}{i\omega} f(z) e^{i\omega t} \right \} </math></center>
+
<center><math> \zeta (z,t) = \mathrm{Re} \left \{\frac{1}{-\mathrm{i}\omega} f(z) e^{-i\omega t} \right \} </math></center>
where <math> f(z) </math> is assumed known. Since the time <math>t=0</math> is arbitrary we can assume that
+
where <math> f(z) \, </math> is assumed known. Since the time <math>t=0 \,</math> is arbitrary we can assume that
<math>f(z)</math> is real but this is not necessary.  
+
<math>f(z)\,</math> is real but this is not necessary.  
 
Because the oscillations are small the [[Linear and Second-Order Wave Theory| linear equations]] apply (which will be given formally below).  
 
Because the oscillations are small the [[Linear and Second-Order Wave Theory| linear equations]] apply (which will be given formally below).  
 
This excitation creates plane progressive waves with amplitude <math> A \, </math> down the tank. The principal objective of wavemaker theory is to determine <math> A \, </math>  
 
This excitation creates plane progressive waves with amplitude <math> A \, </math> down the tank. The principal objective of wavemaker theory is to determine <math> A \, </math>  
 
as a function of <math> \omega, f(z) \, </math> and <math> h \, </math>. Time-dependent wavemaker theories can also be developed.
 
as a function of <math> \omega, f(z) \, </math> and <math> h \, </math>. Time-dependent wavemaker theories can also be developed.
  
= Expansion of the solution =  
+
== Expansion of the solution ==
 +
 
 +
{{frequency definition}}
 +
 
 +
 
 +
{{velocity potential in frequency domain}}
  
In general, the wavemaker displacement at <math> x=0\, </math> may be written in the form
+
The equations therefore become
<center><math> \xi(t) = \mathfrak{Re} \left \{ \frac{1}{i\omega} f (z) e^{i\omega t} \right \} </math></center>
+
{{standard linear wave scattering equations without body condition}}
where <math> f(z) \, </math> is a known function of <math> z \, </math>. The standard  
+
The boundary condition at the wavemaker is
[[Linear and Second-Order Wave Theory| linear equations]] apply.
 
Let the total velocity potential be
 
<center><math> \Phi(x,z,t) = \mathfrak{Re} \left \{ \phi(x,z) e^{i\omega t} \right \} </math></center>.
 
This gives us a [[Frequency Domain Problem]].
 
The water is assumed to have
 
constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
 
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
 
boundary value problem can therefore be expressed as
 
<center>
 
<math>
 
\Delta\phi=0, \,\, -h<z<0,
 
</math>
 
</center>
 
<center>
 
<math>
 
\phi_{z}=0, \,\, z=-h,
 
</math>
 
</center>
 
<center><math>
 
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<0,
 
</math></center>
 
 
<center>
 
<center>
 
<math>
 
<math>
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</math>
 
</math>
 
</center>
 
</center>
We
+
We must also apply the [[Sommerfeld Radiation Condition]]
must also apply the [[Sommerfeld Radiation Condition]]
 
 
as <math>x\rightarrow\infty</math>. This essentially implies
 
as <math>x\rightarrow\infty</math>. This essentially implies
 
that the only wave at infinity is propagating away.
 
that the only wave at infinity is propagating away.
  
== Separation of variables ==
+
{{separation of variables for a free surface}}
 
 
We now separate variables and write the potential as
 
<center>
 
<math>
 
\phi(x,z)=\zeta(z)\rho(x)
 
</math>
 
</center>
 
Applying Laplace's equation we obtain
 
<center>
 
<math>
 
\zeta_{zz}+k^{2}\zeta=0.
 
</math>
 
</center>
 
We then use the boundary condition at <math>z=-h</math> to write
 
<center>
 
<math>
 
\zeta=\cos k(z+h)
 
</math>
 
</center>
 
The boundary condition at the free surface (<math>z=0</math>) is
 
<center><math>
 
k\tan\left(  kh\right)  =-\alpha,\quad x<0
 
</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>
 
</center>
 
as the vertical eigenfunction of the potential in the open
 
water region. From [http://en.wikipedia.org/wiki/Sturm-Liouville_theory 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
 
<center>
 
<math>
 
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{n}\delta_{mn}
 
</math>
 
</center>
 
where
 
<center>
 
<math>
 
A_{n}=\frac{1}{2}\left(  \frac{\cos k_{n}h\sin k_{m}h+k_{n}h}{k_{n}\cos
 
^{2}k_{n}h}\right)
 
</math>
 
</center>
 
and
 
  
= Expansion in Eigenfunctions =
+
== Expansion in Eigenfunctions ==
  
The wavemaker velocity potential <math> \phi </math> can be expressed simply in terms of eigenfunctions
+
The wavemaker velocity potential <math> \phi \,</math> can be expressed simply in terms of eigenfunctions
  
<center><math> \phi = \sum_{n=0}^{\infty} a_n \phi_n (x) e^{-k_n x} </math></center>
+
<center><math> \phi = \sum_{n=0}^{\infty} a_n \phi_n (z) e^{-k_n x} </math></center>
  
and we can solve for the coefficients by matching at <math>x=0</math>
+
and we can solve for the coefficients by matching at <math>x=0 \,</math>
  
<center><math> \left. \phi_x(z) \right|_{x=0} = \sum_{n=0}^{\infty} -k_n a_n  \phi_n (x) e^{-k_n x} = f(z)
+
<center><math> \left. \phi_x \right|_{x=0} = \sum_{n=0}^{\infty} -k_n a_n  \phi_n (z) = f(z)
 
</math></center>
 
</math></center>
  
 
It follows that  
 
It follows that  
  
<center><math>  a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)dz </math></center>
+
<center><math>  a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z </math></center>
  
== Far Field Wave ==
+
=== Far Field Wave ===
One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude <math> A \, </math> ) in terms of <math> f(z) </math>.  
+
One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude <math> A \, </math> in terms of <math> f(z) \, </math>.  
 
The far-field wave component representing progagating waves is given by:
 
The far-field wave component representing progagating waves is given by:
  
<center><math> \phi_w = a_0 \frac{\cos k_0(z-h)}{\cos k_0 h } e^{-ik_0 x} </math></center>
+
<center><math> \lim_{x\to\infty} \phi = a_0 \phi_0(z) e^{-k_0 x} =
 +
a_0 \frac{\cos k_0(z+h)}{\cos k_0 h } e^{-k_0 x} </math></center>
  
We therefore obtain the complex amplitude of the propagating wave at infinity, namely modulus and phase, in terms of the wave maker displacement <math> f(z) </math>.
+
Note that  <math> k_0 \, </math> is imaginery. We therefore obtain the complex amplitude of the propagating wave at infinity, namely modulus and phase, in terms of the wave maker displacement <math> f(z) \, </math>.
  
For what type of <math> f(z) </math> are the non-wavelike modes zero? It is easy to verify by virtue of orthogonality that
+
For what type of <math> f(z) \, </math> are the non-wavelike modes zero? It is easy to verify by virtue of orthogonality that
  
 
<center><math> f(z) \ \sim \ \phi_0 (z) </math></center>
 
<center><math> f(z) \ \sim \ \phi_0 (z) </math></center>
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Unfortunately this is not a "practical" displacement since <math> \phi_0 (z) \, </math> depends on <math> \omega\, </math>, so one would need to build a flexible paddle.
 
Unfortunately this is not a "practical" displacement since <math> \phi_0 (z) \, </math> depends on <math> \omega\, </math>, so one would need to build a flexible paddle.
  
= Matlab Code =
+
== Matlab Code ==
  
 
A program to calculate the coefficients for the wave maker problems can be found here
 
A program to calculate the coefficients for the wave maker problems can be found here
[http://www.math.auckland.ac.nz/~meylan/eigenfunction_matching/code/wavemaker.m wavemaker.m]
+
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/wavemaker.m wavemaker.m]
  
== Additional code ==
+
=== Additional code ===
  
 
This program requires
 
This program requires
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[[Category:Eigenfunction Matching Method]]
 
[[Category:Eigenfunction Matching Method]]
 
[[Category:Pages with Matlab Code]]
 
[[Category:Pages with Matlab Code]]
 +
[[Category:Complete Pages]]

Latest revision as of 01:13, 5 May 2023

Wave and Wave Body Interactions
Current Chapter Wavemaker Theory
Next Chapter Ship Kelvin Wake
Previous Chapter Wave Momentum Flux



Introduction

Wavemaker

We will derive the potential in a two-dimensional wavetank due the motion of the wavemaker. The method is based on the Eigenfunction Matching Method. A paddle is undergoing small amplitude horizontal oscillations with displacement

[math]\displaystyle{ \zeta (z,t) = \mathrm{Re} \left \{\frac{1}{-\mathrm{i}\omega} f(z) e^{-i\omega t} \right \} }[/math]

where [math]\displaystyle{ f(z) \, }[/math] is assumed known. Since the time [math]\displaystyle{ t=0 \, }[/math] is arbitrary we can assume that [math]\displaystyle{ f(z)\, }[/math] is real but this is not necessary. Because the oscillations are small the linear equations apply (which will be given formally below). This excitation creates plane progressive waves with amplitude [math]\displaystyle{ A \, }[/math] down the tank. The principal objective of wavemaker theory is to determine [math]\displaystyle{ A \, }[/math] as a function of [math]\displaystyle{ \omega, f(z) \, }[/math] and [math]\displaystyle{ h \, }[/math]. Time-dependent wavemaker theories can also be developed.

Expansion of the solution

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]) The boundary condition at the wavemaker is

[math]\displaystyle{ \left. \partial_x\phi \right|_{x=0} = \partial_t \xi = f(z). }[/math]

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ x\rightarrow\infty }[/math]. This essentially implies that the only wave at infinity is propagating away.

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]

Expansion in Eigenfunctions

The wavemaker velocity potential [math]\displaystyle{ \phi \, }[/math] can be expressed simply in terms of eigenfunctions

[math]\displaystyle{ \phi = \sum_{n=0}^{\infty} a_n \phi_n (z) e^{-k_n x} }[/math]

and we can solve for the coefficients by matching at [math]\displaystyle{ x=0 \, }[/math]

[math]\displaystyle{ \left. \phi_x \right|_{x=0} = \sum_{n=0}^{\infty} -k_n a_n \phi_n (z) = f(z) }[/math]

It follows that

[math]\displaystyle{ a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z }[/math]

Far Field Wave

One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude [math]\displaystyle{ A \, }[/math] in terms of [math]\displaystyle{ f(z) \, }[/math]. The far-field wave component representing progagating waves is given by:

[math]\displaystyle{ \lim_{x\to\infty} \phi = a_0 \phi_0(z) e^{-k_0 x} = a_0 \frac{\cos k_0(z+h)}{\cos k_0 h } e^{-k_0 x} }[/math]

Note that [math]\displaystyle{ k_0 \, }[/math] is imaginery. We therefore obtain the complex amplitude of the propagating wave at infinity, namely modulus and phase, in terms of the wave maker displacement [math]\displaystyle{ f(z) \, }[/math].

For what type of [math]\displaystyle{ f(z) \, }[/math] are the non-wavelike modes zero? It is easy to verify by virtue of orthogonality that

[math]\displaystyle{ f(z) \ \sim \ \phi_0 (z) }[/math]

Unfortunately this is not a "practical" displacement since [math]\displaystyle{ \phi_0 (z) \, }[/math] depends on [math]\displaystyle{ \omega\, }[/math], so one would need to build a flexible paddle.

Matlab Code

A program to calculate the coefficients for the wave maker problems can be found here wavemaker.m

Additional code

This program requires dispersion_free_surface.m to run


This article is based in part on the MIT open course notes and the original article can be found here

Ocean Wave Interaction with Ships and Offshore Energy Systems