Category:Linear Water-Wave Theory

From WikiWaves
Jump to: navigation, search


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.