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.

Hadamard Finite Part Integrals

A hadamard finite part integral is an example of a hypersingular integral equation (an integral equation whose kernel has high order singularities), and must be treated with caution.

In order to evaluate our integral above, it is necessary to express the jump in [math]\displaystyle{ \phi \, }[/math] as

[math]\displaystyle{ [ \ \phi(x^{_{'}}) \ ] = (1-x^{_{'}2})^{1/2} \sum_{m=0}^{M}b_m U_m(x^{_{'}}) }[/math]

where [math]\displaystyle{ U_m(x^{_{'}}) \, }[/math] denotes Chebyshev Polynomials of the Second Kind.

It is also necessary to employ the powerful result discussed in Martin and Rizzo 1989:

[math]\displaystyle{ \oint_{-1}^{1} \frac{(1-v^2)^{1/2}U_m(v)}{(u-v)^2}dv = -\pi (m+1) U_m(u) }[/math]

Note that the structure has to be non-dimensionalised accordingly.

From this we can obtain

[math]\displaystyle{ -\frac{1}{2}\sum_{m=0}^{M} b_m (m+1)U_m(x) + \int_{\partial \Omega} \partial_{n^{_{'}}n} G \phi ds^{_{'}}- \int_{\partial \Omega}\partial_n G \partial_{n^{_{'}}} \phi ds^{_{'}} = 0 }[/math]

Multiplying this entire expression by [math]\displaystyle{ (1-x^2)^{1/2} U_n(x) \, }[/math] and integrating allows us to obtain the matrix representation

[math]\displaystyle{ \tilde{M} \vec{b} + \mathbb{Y} (\mathbb{G}^{(1)}\vec{\phi} - \mathbb{G}^{(2)} \vec{\phi}_{n^{_{'}}} ) = 0 }[/math]

where

[math]\displaystyle{ \begin{align} \tilde{M}_{nn} &= -\frac{\pi}{4}n \\ \mathbb{Y} &= \mathbb{U} \ \mathbb{W}^{(1)}\mathbb{W}^{(2)} \\ \mathbb{U} &= [U_1(x),U_2(x), ... , U_M(x)]\\ \mathbb{W}^{(1)}_{nn} &= (1-x_n^2)^{1/2} \end{align} }[/math]

Note that [math]\displaystyle{ \mathbb{W}^{(2)} }[/math] is a weights matrix for numerically integrating over [math]\displaystyle{ \Gamma \, }[/math], and the two [math]\displaystyle{ \mathbb{G} }[/math] matrices are obtained using boundary element method, which is discussed below.

Boundary Element Methods

For the purposes of this article, the webpages Boundary Element Method and Boundary Element Method for a Fixed Body in Finite Depth provide an excellent explanation to the Boundary Element method techniques used here.

Recall the expression

[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]

Using the Chebyshev series from above, we can represent this as

[math]\displaystyle{ \mathbb{G}^{(3)} \mathbb{W}^{(1)} \mathbb{U} \vec{b}+\mathbb{G}^{(4)}\vec{\phi} - \mathbb{G}^{(5)} \vec{\phi}_{n^{_{'}}} = \frac{1}{2} \vec{\phi} }[/math]

This leaves us with the following two equations

[math]\displaystyle{ \begin{align} \tilde{M} \vec{b} + \mathbb{Y} (\mathbb{G}^{(1)}\vec{\phi} - \mathbb{G}^{(2)} \vec{\phi}_{n^{_{'}}} ) &= 0 \\ \mathbb{G}^{(3)} \mathbb{W}^{(1)} \mathbb{U} \vec{b}+\mathbb{G}^{(4)}\vec{\phi} - \mathbb{G}^{(5)} \vec{\phi}_{n^{_{'}}} &= \frac{1}{2} \vec{\phi} \\ \end{align} }[/math]

Using the technique outlined in Boundary Element Method for a Fixed Body in Finite Depth we express

[math]\displaystyle{ \vec{\phi}_{n^{_{'}}} = A \vec{\phi} -\vec{f} }[/math]

Consequently we can represent our system in the following form

[math]\displaystyle{ \begin{bmatrix} \tilde{M} & \mathbb{Y}(\mathbb{G}^{(1)} - \mathbb{G}^{(2)}A) \\ \mathbb{G}^{(3)}\mathbb{W}^{(1)}\mathbb{U} & (\mathbb{G}^{(4)}- \mathbb{G}^{(5)}A - \frac{1}{2}\mathrm{I}) \end{bmatrix} \begin{bmatrix} \vec{b} \\ \vec{\phi} \end{bmatrix} = \begin{bmatrix} -\mathbb{Y}\mathbb{G}^{(2)}\vec{f} \\ -\mathbb{G}^{(5)}\vec{f} \end{bmatrix} \,\! }[/math]

From this we are able to compute the jump in [math]\displaystyle{ \phi \, }[/math], and the potential around the outer boundary [math]\displaystyle{ \partial \Omega }[/math]

Eigenfunction Matching

As an alternative to the the technique outlined on this webpage, Eigenfunction Matching for a Submerged Finite Dock can be used to determine the potential at any point in the region, as well as the jump in [math]\displaystyle{ \phi \, }[/math]

Matlab Code

Will be posted presently

References

Martin and Rizzo 1989

Linton and McIver 2001

Bibtex

The bibtex reference for this webpage is below:

@MISC{referenceID,

 author = {wikiwaves},
 title = { Wave Scattering by Submerged Thin Bodies },
 month = November,
 year = { 2024 },
 url = { https://wikiwaves.org/Wave_Scattering_by_Submerged_Thin_Bodies }

}