Difference between revisions of "Wavemaker Theory"

From WikiWaves
Jump to navigationJump to search
 
(78 intermediate revisions by 4 users not shown)
Line 1: Line 1:
A paddle with draft <math> D\, </math> is undergoing small amplitude horizontal oscillations with displacement
+
{{Ocean Wave Interaction with Ships and Offshore Structures
 +
| chapter title = Wavemaker Theory
 +
| next chapter = [[Ship Kelvin Wake]]
 +
| previous chapter =  [[Wave Momentum Flux]]
 +
}}
  
<center><math> \xi (t) = \mathfrak{Re} \left \{ \Pi e^{i\omega t} \right \} </math></center>
+
{{complete pages}}
  
Where <math> \Pi\, </math> is assumed known and real. 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, \Pi \, </math> and <math> H \, </math>.
+
== Introduction ==
  
Other types of wavemaker modes may be treated similarly.
+
[[Image:Wave_maker.png|600px|right|thumb|Wavemaker]]
  
In general, the wavemaker displacement at <math> X=0\, </math> may be written in the form
+
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
 +
<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
 +
<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).
 +
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.
  
<center><math> \xi(t) = \mathfrak{Re} \left \{ \Pi (Z) e^{i\omega t} \right \} </math></center>
+
== Expansion of the solution ==
  
Where <math> \Pi(Z) \, </math> is a known function of <math> Z \, </math>.
+
{{frequency definition}}
  
Let the total velocity potential be:
 
  
<center><math> \Phi = \mathfrak{Re} \left \{ \phi e^{i\omega t} \right \} </math></center>
+
{{velocity potential in frequency domain}}
  
where
+
The equations therefore become
 +
{{standard linear wave scattering equations without body condition}}
 +
The boundary condition at the wavemaker is
 +
<center>
 +
<math>
 +
\left. \partial_x\phi \right|_{x=0} = \partial_t \xi = f(z).
 +
</math>
 +
</center>
 +
We must also apply the [[Sommerfeld Radiation Condition]]
 +
as <math>x\rightarrow\infty</math>. This essentially implies
 +
that the only wave at infinity is propagating away.
  
<center><math> \phi = \phi_\omega \ + \psi </math></center>
+
{{separation of variables for a free surface}}
  
The first term is a velocity potential that represents a plane progressive regular wave down the tank with amplitude <math> A \, </math>, yet unknown. Thus:
+
== Expansion in Eigenfunctions ==
  
<center><math> \phi_\omega = \frac{igA}{\omega} \frac{\cosh K (Z+H)}{\cosh KH} e^{-iKX + i\omega t} </math></center>
+
The wavemaker velocity potential <math> \phi \,</math> can be expressed simply in terms of eigenfunctions
  
with:
+
<center><math> \phi = \sum_{n=0}^{\infty} a_n \phi_n (z) e^{-k_n x} </math></center>
  
<center><math> \omega^2 = gK \tanh KH. \,</math></center>
+
and we can solve for the coefficients by matching at <math>x=0 \,</math>
  
The second component potential <math>\psi\,</math> is by definition a decaying disturbance as <math> X \to \infty \, </math> and otherwise satisfies the following boundary value problem:
+
<center><math> \left. \phi_x \right|_{x=0} = \sum_{n=0}^{\infty} -k_n a_n  \phi_n (z) = f(z)
 +
</math></center>
 +
 
 +
It follows that
 +
 
 +
<center><math> a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z  </math></center>
  
<center><math> \begin{cases}
+
=== Far Field Wave ===
  \nabla^2 \psi = \psi_XX + \psi_ZZ = 0, -H < Z < 0 \\
+
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>.
  \psi_Z - \frac{\omega^2}{g} \psi = 0, Z=0 \\
+
The far-field wave component representing progagating waves is given by:
  \psi_Z = 0, Z=-H \\
+
 
  \psi \to 0, X \to \infty
+
<center><math> \lim_{x\to\infty} \phi = a_0 \phi_0(z) e^{-k_0 x} =
\end{cases}
+
a_0 \frac{\cos k_0(z+h)}{\cos k_0 h } e^{-k_0 x} </math></center>
</math></center>
+
 
 +
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
 +
 
 +
<center><math> f(z) \ \sim \ \phi_0 (z) </math></center>
 +
 
 +
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 ==
 +
 
 +
A program to calculate the coefficients for the wave maker problems can be found here
 +
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/wavemaker.m wavemaker.m]
 +
 
 +
=== Additional code ===
 +
 
 +
This program requires
 +
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
 +
to run
  
The condition on the wavemaker <math> (X=0) \, </math> is yet to be enforced.
+
-----
  
Note that unlike <math> \phi_\omega, \psi </math> is not representing a propagating wave down the tank so it is called a non-wavelike mode. Such modes do exist as will be shown below. On the wavemaker <math> (X=0) \, </math> the horizontal velocity due to <math> \phi_\omega\, </math> and that due to <math> \psi\,</math> must sum to the forcing velocity due to <math> \xi(t) \, </math>.
+
This article is based in part on the MIT open course notes and the original article can be found
 +
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/737E217E-0582-45D5-B1F9-B2ECF977C66E/0/lecture6.pdf here]
  
Noting that <math> \phi_\omega \approx e^{-iKX} \cosh K(Z+H) \, </math> we will try <math> \phi \approx e^{-\lambda x} \cos \lambda (Z+H} \,</math>. Its conjugate which satisfies the condition of vanishing value as <math> X \to \infty </math> for <math> \lambda > 0 </math>.
+
[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]
  
<u> Laplace </u>:
+
[[Category:Eigenfunction Matching Method]]
 +
[[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