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 is assumed known. Since the time is arbitrary we can assume that 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 down the tank. The principal objective of wavemaker theory is to determine as a function of and . Time-dependent wavemaker theories can also be developed.
Expansion of the solution
We also assume that Frequency Domain Problem with frequency and we assume that all variables are proportional to
The water motion is represented by a velocity potential which is denoted by so that
The equations therefore become
(note that the last expression can be obtained from combining the expressions:
where ) The boundary condition at the wavemaker is
We must also apply the Sommerfeld Radiation Condition as . 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
and then Laplace's equation becomes
The separation of variables equation for deriving free surface eigenfunctions is as follows:
subject to the boundary conditions
We can then use the boundary condition at to write
where we have chosen the value of the coefficent so we have unit value at . The boundary condition at the free surface () gives rise to:
which is the Dispersion Relation for a Free Surface
The above equation is not really the dispersion relation for a free surface, it would be better to refer to it as 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 and the positive real solutions by , . The of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations
to arrive at the dispersion relation
We note that for a specified frequency the equation determines the wavenumber .
Finally we define the function as
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
Expansion in Eigenfunctions
The wavemaker velocity potential can be expressed simply in terms of eigenfunctions
and we can solve for the coefficients by matching at
It follows that
Far Field Wave
One of the primary objecives of wavemaker theory is to determine the far-field wave amplitude in terms of . The far-field wave component representing progagating waves is given by:
Note that 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 .
For what type of are the non-wavelike modes zero? It is easy to verify by virtue of orthogonality that
Unfortunately this is not a "practical" displacement since depends on , so one would need to build a flexible paddle.
A program to calculate the coefficients for the wave maker problems can be found here wavemaker.m
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