Standard Linear Wave Scattering Problem

This is a complete page

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.

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.