Category:Linear Water-Wave Theory

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

[math] \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. [/math]

The coordinate system is the standard Cartesian coordinate system with the [math]z-[/math]axis pointing vertically up. The water surface is at [math]z=0[/math] and the region of interest is [math]-h\ltz\lt0[/math]. There is a body which occupies the region [math]\Omega[/math] and we denote the wetted surface of the body by [math]\partial\Omega[/math] We denote [math]\mathbf{r}=(x,y)[/math] as the horizontal coordinate in two or three dimensions respectively and the Cartesian system we denote by [math]\mathbf{x}[/math]. We assume that the bottom surface is of constant depth at [math]z=-h[/math]. Variable Bottom Topography can also easily be included but we do not consider this here.

The equations are the following

[math] \begin{align} \Delta\phi &=0, &-h\ltz\lt0,\,\,\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} [/math]

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

[math] \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} [/math]

where [math]\alpha = \omega^2/g \,[/math])

[math] \partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B, [/math]

where [math]\mathcal{L}[/math] 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 [math]L=0[/math] but more complicated conditions are possible.

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

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

where [math]A [/math] is the wave amplitude (in potential) [math]\mathrm{i} k [/math] 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 [math]\exp(-\mathrm{i}\omega t) [/math]) and

[math] \phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h} [/math]

In two-dimensions the Sommerfeld Radiation Condition is

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

where [math]\phi^{\mathrm{{I}}}[/math] is the incident potential.

In three-dimensions the Sommerfeld Radiation Condition is

[math] \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{.} [/math]

where [math]\phi^{\mathrm{{I}}}[/math] is the incident potential.