# Difference between revisions of "Wave Scattering by Submerged Thin Bodies"

## Introduction

We consider the problem of a thin submerged structure assumed to be rigid, submerged in a semi-infinite domain of constant depth. This submerged structure is then subject to some incident wave forcing, and the resulting scattering problem is investigated.

## Equations

The governing equations for our problem are as follows:

\displaystyle{ \begin{align} & \Delta \phi=0, &\qquad in \ \Omega\, \\ & \partial_z \phi=0, &\qquad at\ z=-h\, \\ & \partial_z \phi=\alpha \phi, &\qquad at\ z=0\, \\ \end{align} }

$\displaystyle{ \partial_x \phi \pm i k \phi = 0, \qquad as \ x \rightarrow \pm \infty \, }$

For the purposes of using Boundary Element Methods later in this article, we restrict our domain to a finite region by imposing artificial boundary conditions at $\displaystyle{ x= a_1 \, }$ and $\displaystyle{ x= a_2 \, }$. We denote these conditions by

\displaystyle{ \begin{align} & \phi=\tilde{\phi}_1(z), &\qquad at \ x=a_1\, \\ & \phi=\tilde{\phi}_2(z), &\qquad at \ x=a_2\, \\ \end{align} }

and will be considered in more detail later on.

## Thin bodies

Additionally, we have a condition for the response of our structure (which is submerged at constant finite depth $\displaystyle{ -d }$):

\displaystyle{ \begin{align} & \partial_n \phi = f(x,t), &\qquad x \in (-L,L), \ z=-d \, \end{align} }

We denote this segment of the boundary by $\displaystyle{ \Gamma \, }$.

There are additional considerations to be made when dealing with thin obstacles - we need to split $\displaystyle{ \Gamma \, }$ into two regions ($\displaystyle{ \Gamma^{\pm} \, }$) and distinguish between their respective normal derivatives ($\displaystyle{ \partial_n \rightarrow \partial_{n^{\pm}}\, }$). This allows us to express the boundary condition in the form

\displaystyle{ \begin{align} & \partial_{n^{\pm}} \phi = \pm f(x,t), &\qquad x \in (-L,L), \ z=-d \, \end{align} }

Observe that the normal derivatives are related in the following manner

\displaystyle{ \begin{align} & \partial_{n^{-}} = - \partial_{n^{+}} & \, \end{align} }

From this, we can express our problem involving the submerged structure entirely in terms of $\displaystyle{ \Gamma^{+} \, }$, and omit the notation in the future (ie $\displaystyle{ \partial_{n^{+}} = \partial_n }$).

For the case of a rigid structure, we define the associated boundary condition to be

\displaystyle{ \begin{align} & \partial_n \phi = 0, &\qquad on \ \Gamma \, \end{align} }

## Integral Equation formulation

Using Green's second identity, in tandem with the discussion of thin bodies (above), we can obtain an expression for $\displaystyle{ \phi \, }$ around the outer boundary ($\displaystyle{ \partial \Omega \, }$) of our domain:

$\displaystyle{ \frac{1}{2}\phi(\textbf{x})=\int_{\Gamma} [ \ \phi \ ] \partial_{n^{_{'}}} G ds^{_{'}} +\int_{\partial \Omega} (\phi \partial_{n^{_{'}}} G - G \partial_{n^{_{'}}} \phi )ds^{_{'}} }$

where $\displaystyle{ [ \ \phi \ ]=\phi(x^{+})-\phi(x^{-}) }$, and is typically referred to as the 'jump in $\displaystyle{ \phi \, }$'.

Note carefully that $\displaystyle{ \phi \, }$ relates to points on the outer boundary, $\displaystyle{ \partial \Omega \, }$, whereas $\displaystyle{ [ \ \phi \ ] \, }$ relates to points on the inner boundary, $\displaystyle{ \Gamma \, }$.

In order to compute $\displaystyle{ [ \ \phi \ ] \, }$, we will restrict all x points in the expression above to be on $\displaystyle{ \Gamma \, }$, and then take the normal derivative of the expression to obtain

$\displaystyle{ \oint_{\Gamma} [ \ \phi \ ] \partial_{n^{_{'}} n} G ds^{_{'}} +\int_{\partial \Omega} (\phi \partial_{n^{_{'}}n} G - \partial_n G \partial_{n^{_{'}}} \phi )ds^{_{'}} =0 }$

Note that in our case, $\displaystyle{ \oint \, }$ refers to a Hadamard finite-part integral, as opposed to a contour integral.

## Boundary Element Methods

refer to the wikiwaves page etc

## Eigenfunction Matching

refer to the wikiwaves page etc

## References

Article by Yang? Linton McIvor?