Wavemaker Theory

Wave and Wave Body Interactions
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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

$\zeta (t) = \mathrm{Re} \left \{\frac{1}{\mathrm{i}\omega} f(z) e^{i\omega t} \right \}$

where $f(z) \,$ is assumed known. Since the time $t=0 \,$ is arbitrary we can assume that $f(z)\,$ 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 $A \,$ down the tank. The principal objective of wavemaker theory is to determine $A \,$ as a function of $\omega, f(z) \,$ and $h \,$. Time-dependent wavemaker theories can also be developed.

Expansion of the solution

We also assume that Frequency Domain Problem with frequency $\omega$ and we assume that all variables are proportional to $\exp(-\mathrm{i}\omega t)\,$

The water motion is represented by a velocity potential which is denoted by $\phi\,$ so that

$\Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}.$

The equations therefore become

$\begin{align} \Delta\phi &=0, &-h<z<0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align}$

(note that the last expression can be obtained from combining the expressions:

$\begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align}$

where $\alpha = \omega^2/g \,$) The boundary condition at the wavemaker is

$\left. \partial_x\phi \right|_{x=0} = \partial_t \xi = f(z).$

We must also apply the Sommerfeld Radiation Condition as $x\rightarrow\infty$. 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

$\phi(x,z) = X(x)Z(z)\,$

and then Laplace's equation becomes

$\frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2$

The separation of variables equation for deriving free surface eigenfunctions is as follows:

$Z^{\prime\prime} + k^2 Z =0.$

subject to the boundary conditions

$Z^{\prime}(-h) = 0$

and

$Z^{\prime}(0) = \alpha Z(0)$

We can then use the boundary condition at $z=-h \,$ to write

$Z = \frac{\cos k(z+h)}{\cos kh}$

where we have chosen the value of the coefficent so we have unit value at $z=0$. The boundary condition at the free surface ($z=0 \,$) gives rise to:

$k\tan\left( kh\right) =-\alpha \,$

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 $k_{0}=\pm ik \,$ and the positive real solutions by $k_{m} \,$, $m\geq1$. The $k \,$ of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

$\cos ix = \cosh x, \quad \sin ix = i\sinh x,$

to arrive at the dispersion relation

$\alpha = k\tanh kh.$

We note that for a specified frequency $\omega \,$ the equation determines the wavenumber $k \,$.

Finally we define the function $Z(z) \,$ as

$\chi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0$

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

$\int\nolimits_{-h}^{0}\chi_{m}(z)\chi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn}$

where

$A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right).$

Expansion in Eigenfunctions

The wavemaker velocity potential $\phi \,$ can be expressed simply in terms of eigenfunctions

$\phi = \sum_{n=0}^{\infty} a_n \phi_n (z) e^{-k_n x}$

and we can solve for the coefficients by matching at $x=0 \,$

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

It follows that

$a_n = -\frac{1}{k_n A_n} \int_{-h}^0 \phi_n(z) f(z)\mathrm{d}z$

Far Field Wave

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

$\lim_{x\to\infty} \phi = a_0 \phi_0(z) e^{-k_0 x} = a_0 \frac{\cos k_0(z+h)}{\cos k_0 h } e^{-k_0 x}$

Note that $k_0 \,$ 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 $f(z) \,$.

For what type of $f(z) \,$ are the non-wavelike modes zero? It is easy to verify by virtue of orthogonality that

$f(z) \ \sim \ \phi_0 (z)$

Unfortunately this is not a "practical" displacement since $\phi_0 (z) \,$ depends on $\omega\,$, 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

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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