Wavemaker Theory
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Introduction
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
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
(note that the last expression can be obtained from combining the expressions:
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:
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
and we can solve for the coefficients by matching at [math]\displaystyle{ x=0 \, }[/math]
It follows that
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:
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
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
