Wave Scattering by Submerged Thin Bodies

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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:

[math]\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} }[/math]

Plus radiation conditions:

[math]\displaystyle{ \partial_x \phi \pm i k \phi = 0, \qquad as \ x \rightarrow \pm \infty \, }[/math]

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 [math]\displaystyle{ x= a_1 \, }[/math] and [math]\displaystyle{ x= a_2 \, }[/math]. We denote these conditions by

[math]\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} }[/math]

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 [math]\displaystyle{ -d }[/math]):

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

We denote this segment of the boundary by [math]\displaystyle{ \Gamma \, }[/math].

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

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

Observe that the normal derivatives are related in the following manner

[math]\displaystyle{ \begin{align} & \partial_{n^{-}} = - \partial_{n^{+}} & \, \end{align} }[/math]

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

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

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

Integral Equation formulation

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

[math]\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^{_{'}} }[/math]

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

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

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

[math]\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 }[/math]

Note that in our case, [math]\displaystyle{ \oint \, }[/math] 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?