Wavemaker Theory

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 with draft [math]\displaystyle{ D\, }[/math] is undergoing small amplitude horizontal oscillations with displacement

[math]\displaystyle{ \xi (t) = \mathfrak{Re} \left \{ 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

In general, the wavemaker displacement at [math]\displaystyle{ X=0\, }[/math] may be written in the form

[math]\displaystyle{ \xi(t) = \mathfrak{Re} \left \{ f (z) e^{i\omega t} \right \} }[/math]

where [math]\displaystyle{ f(z) \, }[/math] is a known function of [math]\displaystyle{ z \, }[/math]. The standard linear equations apply. Let the total velocity potential be

[math]\displaystyle{ \Phi(x,z,t) = \mathfrak{Re} \left \{ \phi(x,z) e^{i\omega t} \right \} }[/math]

.

This gives us a Frequency Domain Problem. The water is assumed to have constant finite depth [math]\displaystyle{ H }[/math] and the [math]\displaystyle{ z }[/math]-direction points vertically upward with the water surface at [math]\displaystyle{ z=0 }[/math] and the sea floor at [math]\displaystyle{ z=-H }[/math]. The boundary value problem can therefore be expressed as

[math]\displaystyle{ \Delta\phi=0, \,\, -H\lt z\lt 0, }[/math]

[math]\displaystyle{ \phi_{z}=0, \,\, z=-H, }[/math]

[math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt 0, }[/math]

[math]\displaystyle{ \partial_x\phi_{x}=f(z), \,\,x=0. }[/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

We now separate variables and write the potential as

[math]\displaystyle{ \phi(x,z)=\zeta(z)\rho(x) }[/math]

Applying Laplace's equation we obtain

[math]\displaystyle{ \zeta_{zz}+k^{2}\zeta=0. }[/math]

We then use the boundary condition at [math]\displaystyle{ z=-H }[/math] to write

[math]\displaystyle{ \zeta=\cos k(z+H) }[/math]

The boundary condition at the free surface ([math]\displaystyle{ z=0 }[/math]) is

[math]\displaystyle{ k\tan\left( kH\right) =-\alpha,\quad x\lt 0 }[/math]

which is the Dispersion Relation for a Free Surface We denote the positive imaginary solution of this equation by [math]\displaystyle{ k_{0} }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} }[/math], [math]\displaystyle{ m\geq1 }[/math]. We define

[math]\displaystyle{ \phi_{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 and

[math]\displaystyle{ \psi_{m}\left( z\right) = \cos\kappa_{m}(z+H),\quad m\geq 0 }[/math]

as the vertical eigenfunction of the potential in the dock covered region. For later reference, we note that:

[math]\displaystyle{ \int\nolimits_{-H}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}H\sin k_{m}H+k_{m}H}{k_{m}\cos ^{2}k_{m}H}\right) }[/math]

and

Orthogonal eigenfunctions

The solution of the for Dispersion Relation for a Free Surface leads to an infinite series of vertical eigenfunctions from Sturm-Liouville theory. This theory also shows that the eigenfunctions are orthogonal and we may define the following orthogonal eigenfunctions in the vertical direction [math]\displaystyle{ Z \, }[/math]:

[math]\displaystyle{ f_0 (Z) = \frac{\sqrt{2} \cosh K ( Z + H )}{{ (H + \frac{1}{v} \sinh^2 KH )}^{1/2}} }[/math] [math]\displaystyle{ f_n (Z) = \frac{\sqrt{2} \cosh \lambda_n ( Z + H )}{(H + \frac{1}{v} \sinh^2 \lambda_n H )}, \qquad n = 1, 2, \cdots }[/math]

Selected to satisfy:

[math]\displaystyle{ \begin{cases} \int_{-H}^0 f_0^2 (Z) dZ = \int_{-H}^0 f_n^2 (Z) dZ = 1 \\ \int_{-H}^0 f_m^2 (Z) f_n (Z) dZ = 0, \quad m \ne n \end{cases} }[/math]

So the wavemaker velocity potentials [math]\displaystyle{ \phi_w \, }[/math] and [math]\displaystyle{ \psi\, }[/math] can be expressed simply in terms of their respective eigen modes:

[math]\displaystyle{ \phi_w = a_0 f_0 (Z) e^{-iKX} }[/math] [math]\displaystyle{ \psi = \sum_{n=1}^{\infty} a_n f_n (Z) e^{-\lambda_n X} }[/math]

and:

[math]\displaystyle{ \Phi = \mathfrak{Re} \left \{ ( \phi_w + \psi)_ e^{i\omega t} \right \} }[/math]

On [math]\displaystyle{ X=0 \, }[/math]:

[math]\displaystyle{ \Phi_X = \mathfrak{Re} \left \{ \partial_X( \phi_W + \psi){X=0} e^{i\omega t} \right \} }[/math]

and

[math]\displaystyle{ \frac{d\xi}{dt} = \mathfrak{Re} \left \{ \Pi (Z) i \omega e^{i\omega t} \right \} }[/math]

Or:

[math]\displaystyle{ \frac{\partial}{\partial X} (\phi_W + \psi)_{X=0} = \Pi (Z) i \omega }[/math] [math]\displaystyle{ \left. \frac{\partial\phi_W}{\partial X} \right |_{X=0} = a_0 ( -iK) f_0 (Z) }[/math] [math]\displaystyle{ \left. \frac{\partial\psi}{\partial X} \right |_{X=0} = \sum_{n=1}^{\infty} a_n ( -\lambda_n) f_n (Z) }[/math]

It follows that:

[math]\displaystyle{ - i K a_0 f_0 (Z) + \sum_{n=1}^{\infty} a_n (- \lambda_n) f_n (Z) = i \omega \Pi (Z) }[/math]

Far Field Wave

One of the primary objecives of wavemaker theory is to determine [math]\displaystyle{ a_0 \, }[/math] (or the far-field wave amplitude [math]\displaystyle{ A \, }[/math] ) in terms of [math]\displaystyle{ \Pi (Z) \, }[/math]. Multiplying both sides by [math]\displaystyle{ f_0 (Z) \, }[/math], integrating from [math]\displaystyle{ - H \to 0 \, }[/math] and using orthogonality we obtain:

[math]\displaystyle{ - i K a_0 = i \omega \int_{-H}^0 dZ f_0 (Z) \Pi (Z) }[/math] [math]\displaystyle{ \Rightarrow \quad a_0 = - \frac{\omega}{K} \int_{-H}^0 dZ f_0 (Z) \Pi (Z) }[/math]

The far-field wave component representing progagating waves is given by:

[math]\displaystyle{ \phi_w = a_0 \frac{\sqrt{2} \cosh K (Z+H)}{{\left( H+\frac{1}{v} \sinh^2 KH \right)}^{1/2}} e^{-iKX} }[/math] [math]\displaystyle{ \equiv \frac{igA}{\omega} \frac{\cosh K (Z +H)}{\cosh KH} e^{-iKX} }[/math]

Plugging in [math]\displaystyle{ a_0\, }[/math] and solving for [math]\displaystyle{ A \, }[/math] we obtain the complex amplitude of the propagating wave at infinity, namely modulus and phase, in terms of the wave maker displacement [math]\displaystyle{ \Pi (Z) \, }[/math] and the other flow parameters.

For what type of [math]\displaystyle{ \Pi(Z) \, }[/math] are the non-wavelike modes [math]\displaystyle{ \psi \equiv 0 \, }[/math]? It is easy to verify by virtue of orthogonality that:

[math]\displaystyle{ \Pi(Z) \ \sim \ f_0 (Z) }[/math]

Unfortunately this is not a "practical" displacement since [math]\displaystyle{ f_0 (Z,K) \, }[/math] depends on [math]\displaystyle{ K\, }[/math], thus on [math]\displaystyle{ \omega\, }[/math]. So one would need to build a flexible paddle!

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This article is based on the MIT open course notes and the original article can be found here

Ocean Wave Interaction with Ships and Offshore Energy Systems