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 astisfies 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]