Wavemaker Theory

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

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

Where [math]\displaystyle{ \Pi\, }[/math] is assumed known and real. 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, \Pi \, }[/math] and [math]\displaystyle{ H \, }[/math].

Other types of wavemaker modes may be treated similarly.

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

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

Where [math]\displaystyle{ \Pi(Z) \, }[/math] is a known function of [math]\displaystyle{ Z \, }[/math].

Let the total velocity potential be:

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

where

[math]\displaystyle{ \phi = \phi_\omega \ + \psi }[/math]

The first term is a velocity potential that represents a plane progressive regular wave down the tank with amplitude [math]\displaystyle{ A \, }[/math], yet unknown. Thus:

[math]\displaystyle{ \phi_\omega = \frac{igA}{\omega} \frac{\cosh K (Z+H)}{\cosh KH} e^{-iKX + i\omega t} }[/math]

with:

[math]\displaystyle{ \omega^2 = gK \tanh KH. \, }[/math]

The second component potential [math]\displaystyle{ \psi\, }[/math] is by definition a decaying disturbance as [math]\displaystyle{ X \to \infty \, }[/math] and otherwise satisfies the following boundary value problem:

[math]\displaystyle{ \begin{cases} \nabla^2 \psi = \psi_XX + \psi_ZZ = 0, -H \lt Z \lt 0 \\ \psi_Z - \frac{\omega^2}{g} \psi = 0, Z=0 \\ \psi_Z = 0, Z=-H \\ \psi \to 0, X \to \infty \end{cases} }[/math]

The condition on the wavemaker [math]\displaystyle{ (X=0) \, }[/math] is yet to be enforced.

Note that unlike [math]\displaystyle{ \phi_\omega, \psi \, }[/math] is not representing a propagating wave down the tank so it is called a non-wavelike mode. Such modes do exist as will be shown below. On the wavemaker [math]\displaystyle{ (X=0) \, }[/math] the horizontal velocity due to [math]\displaystyle{ \phi_\omega\, }[/math] and that due to [math]\displaystyle{ \psi\, }[/math] must sum to the forcing velocity due to [math]\displaystyle{ \xi(t) \, }[/math].

Noting that [math]\displaystyle{ \phi_\omega \sim e^{-iKX} \cosh K(Z+H) \, }[/math] we will try [math]\displaystyle{ \phi \sim e^{-\lambda x} \cos \lambda (Z+H) \, }[/math]. Its conjugate which satisfies the condition of vanishing value as [math]\displaystyle{ X \to \infty }[/math] for [math]\displaystyle{ \lambda \gt 0 \, }[/math].

Laplace : [math]\displaystyle{ \psi_XX + \psi_ZZ = 0, \, }[/math] verify for all [math]\displaystyle{ \lambda\, }[/math].

FS condition : [math]\displaystyle{ \psi_Z - \frac{\omega^2}{g} \psi = 0 \qquad \qquad \Longrightarrow \quad - \lambda \sin \lambda H - \frac{\omega^2}{g} \cos \lambda H = 0 }[/math]

[math]\displaystyle{ \Longrightarrow \quad \lambda \tan \lambda H = - \nu \equiv \frac{\omega^2}{g} }[/math]

Seafloor condition : [math]\displaystyle{ \psi_Z = 0, Z=-H \, }[/math]

So for the non-wavelike modes [math]\displaystyle{ \psi, \lambda \, }[/math] must satisfy the "dispersion" relation

[math]\displaystyle{ \lambda \tan \lambda H = - \nu = - \frac{\omega^2}{g} \lt 0 }[/math]

For positive values of [math]\displaystyle{ \lambda \, }[/math] so that [math]\displaystyle{ e^{-\lambda X} \to 0, X \to + \infty \, }[/math].

Values of [math]\displaystyle{ \lambda_i \, }[/math] satisfying the dispersion relation follow from the solution of the non-dimensional nolinear equation

[math]\displaystyle{ \tan \omega = - \frac{\nu}{\omega}, \omega = \lambda H \, }[/math]

Solutions [math]\displaystyle{ \omega_i, i = 1, 2, \cdots \, }[/math] exist as shown above with [math]\displaystyle{ \omega_i \sim i \pi \, }[/math] for large [math]\displaystyle{ i \, }[/math]. These values are known as the eigenvalues or eigen-wavenumbers of the non-wavelike modes. The eigen-wavenumber of the wavelike solution [math]\displaystyle{ K\, }[/math] is given by the dispersion relation:

[math]\displaystyle{ \frac{\omega^2 H}{g} = KH \tan KH. \, }[/math]