# Category:Linear Water-Wave Theory

## 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 $\displaystyle{ \omega }$ and we assume that all variables are proportional to $\displaystyle{ \exp(-\mathrm{i}\omega t)\, }$ The water motion is represented by a velocity potential which is denoted by $\displaystyle{ \phi\, }$ so that

$\displaystyle{ \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. }$

The coordinate system is the standard Cartesian coordinate system with the $\displaystyle{ z- }$axis pointing vertically up. The water surface is at $\displaystyle{ z=0 }$ and the region of interest is $\displaystyle{ -h\lt z\lt 0 }$. There is a body which occupies the region $\displaystyle{ \Omega }$ and we denote the wetted surface of the body by $\displaystyle{ \partial\Omega }$ We denote $\displaystyle{ \mathbf{r}=(x,y) }$ as the horizontal coordinate in two or three dimensions respectively and the Cartesian system we denote by $\displaystyle{ \mathbf{x} }$. We assume that the bottom surface is of constant depth at $\displaystyle{ z=-h }$. Variable Bottom Topography can also easily be included but we do not consider this here.

The equations are the following

\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 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:

\displaystyle{ \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 $\displaystyle{ \alpha = \omega^2/g \, }$)

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

where $\displaystyle{ \mathcal{L} }$ 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 $\displaystyle{ L=0 }$ but more complicated conditions are possible.

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

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

where $\displaystyle{ A }$ is the wave amplitude (in potential) $\displaystyle{ \mathrm{i} k }$ 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 $\displaystyle{ \exp(-\mathrm{i}\omega t) }$) and

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

In two-dimensions the Sommerfeld Radiation Condition is

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

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

In three-dimensions the Sommerfeld Radiation Condition is

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

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

## Subcategories

This category has the following 10 subcategories, out of 10 total.

## Pages in category "Linear Water-Wave Theory"

The following 23 pages are in this category, out of 23 total.