View source for Wavemaker Theory

A paddle with draft <math> D\, </math> is undergoing small amplitude horizontal oscillations with displacement

<center><math> \xi (t) = \mathfrak{Re} \left \{ \Pi e^{i\omega t} \right \} </math></center>

Where <math> \Pi\, </math> is assumed known and real. This excitation creates plane progressive waves with amplitude <math> A \, </math> down the tank. The principal objective of wavemaker theory is to determine <math> A \, </math> as a function of <math> \omega, \Pi \, </math> and <math> H \, </math>.

Other types of wavemaker modes may be treated similarly.

In general, the wavemaker displacement at <math> X=0\, </math> may be written in the form

<center><math> \xi(t) = \mathfrak{Re} \left \{ \Pi (Z) e^{i\omega t} \right \} </math></center>

Where <math> \Pi(Z) \, </math> is a known function of <math> Z \, </math>.

Let the total velocity potential be:

<center><math> \Phi = \mathfrak{Re} \left \{ \phi e^{i\omega t} \right \} </math></center>

where

<center><math> \phi = \phi_\omega \ + \psi </math></center>

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

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

with:

<center><math> \omega^2 = gK \tanh KH </math></center>.

The second component potential <math>\psi\,</math> is by definition a decaying disturbance as <math> X \to \infty \, </math> and otherwise astisfies the following boundary value problem:

<center><math> \begin{cases}
  \nabla^2 \psi = \psi_XX + \psi_ZZ = 0, -H < Z < 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></center>