Wavemaker Theory

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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 ω 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

\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}},


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

\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}},

where α = ω2 / 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

We denote the negative imaginary solution of this equation by k_{0} \, and the positive real solutions by k_{m} \,, m\geq1. We define

\phi_{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}\phi_{m}(z)\phi_{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

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

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