Category:Linear Water-Wave Theory

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Introduction

Linear water waves are small amplitude waves for which we can linearise the equations of motion (Linear and Second-Order Wave Theory). It is also standard to consider the problem when waves of a single frequency are incident so that only a single frequency needs to be considered, leading to the Frequency Domain Problem. The linear theory is applicable until the wave steepness becomes sufficiently large that non-linear effects become important.

Equations in the Frequency Domain

We assume small amplitude so that we can linearise all the equations (see Linear and Second-Order Wave Theory). We also assume that Frequency Domain Problem with frequency \omega 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 coordinate system is the standard Cartesian coordinate system with the z-axis pointing vertically up. The water surface is at z=0 and the region of interest is -h<z<0. There is a body which occupies the region \Omega and we denote the wetted surface of the body by \partial\Omega We denote \mathbf{r}=(x,y) as the horizontal coordinate in two or three dimensions respectively and the Cartesian system we denote by \mathbf{x}. We assume that the bottom surface is of constant depth at z=-h. Variable Bottom Topography can also easily be included but we do not consider this here.

The equations are the following


\begin{align}
\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}},
\end{align}


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


\begin{align}
 \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}},
\end{align}

where \alpha = \omega^2/g \,)


\partial_n\phi  = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B,

where \mathcal{L} is a linear operator which relates the normal and potential on the body surface through the physics of the body.

The simplest case is for a fixed body where the operator is L=0 but more complicated conditions are possible.

The equation is subject to some radiation conditions at infinity. We assume the following. \phi^{\mathrm{I}}\, is a plane wave travelling in the x direction,


\phi^{\mathrm{I}}(x,z)=A  \phi_0(z) e^{\mathrm{i} k x} \,

where A is the wave amplitude (in potential) \mathrm{i} k is the positive imaginary solution of the Dispersion Relation for a Free Surface (note we are assuming that the time dependence is of the form \exp(-\mathrm{i}\omega t) ) and


 \phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h}

In two-dimensions the Sommerfeld Radiation Condition is


\left(  \frac{\partial}{\partial|x|} - \mathrm{i} k \right)
(\phi-\phi^{\mathrm{{I}}})=0,\;\mathrm{{as\;}}|x|\rightarrow\infty\mathrm{.}

where \phi^{\mathrm{{I}}} is the incident potential.

In three-dimensions the Sommerfeld Radiation Condition is


\sqrt{|\mathbf{r}|}\left(  \frac{\partial}{\partial|\mathbf{r}|} - \mathrm{i} k \right)
(\phi-\phi^{\mathrm{{I}}})=0,\;\mathrm{{as\;}}|\mathbf{r}|\rightarrow\infty\mathrm{.}

where \phi^{\mathrm{{I}}} is the incident potential.