Difference between revisions of "Generalized Eigenfunction Expansion for Water Waves for a Fixed Body"
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= Outline of the theory = | = Outline of the theory = | ||
− | We consider the case of a rigid body in water of finite depth. We can solve | + | We consider the case of a rigid body in water of finite depth. The time-dependent equations |
+ | of motion are | ||
+ | <math>\nabla^2 \phi = 0</math> | ||
+ | <math>\phi_n + \frac{\omega^2}{g} \phi = 0, x\in F</math> | ||
+ | <math>\phi_n = 0, x\in F</math> | ||
+ | |||
+ | |||
+ | We can solve | ||
the single frequency problem with frequency <math>\omega</math> for an incident wave | the single frequency problem with frequency <math>\omega</math> for an incident wave | ||
travelling from <math>-\infty</math> | travelling from <math>-\infty</math> | ||
Line 16: | Line 23: | ||
to obtain a solution <math>\phi(\omega,x)</math>. This looks like an incident plus reflected | to obtain a solution <math>\phi(\omega,x)</math>. This looks like an incident plus reflected | ||
− | wave as <math>x\to\ | + | wave as <math> x\to -\infty</math> and like a transmitted wave as <math> x\to -\infty</math>. |
+ | We call these solutions generalised eigenfunctions. | ||
+ | We then try and expand the solution in terms of this solutions (which we call the generalised | ||
+ | eigenfunctions) because they solve for |
Revision as of 05:08, 18 April 2006
This page is under construction.
The theory of generalised eigenfunction is described in Hazard and Lenoir 2002
and Meylan 2002 for the case of a rigid body in water of infinite depth and for
a thin plate on water of shallow draft respectively. We will present here the theory for a rigid body
in water of finite depth.
Outline of the theory
We consider the case of a rigid body in water of finite depth. The time-dependent equations of motion are [math]\displaystyle{ \nabla^2 \phi = 0 }[/math] [math]\displaystyle{ \phi_n + \frac{\omega^2}{g} \phi = 0, x\in F }[/math] [math]\displaystyle{ \phi_n = 0, x\in F }[/math]
We can solve
the single frequency problem with frequency [math]\displaystyle{ \omega }[/math] for an incident wave
travelling from [math]\displaystyle{ -\infty }[/math]
[math]\displaystyle{ \phi^i = e^{i \omega x} }[/math]
to obtain a solution [math]\displaystyle{ \phi(\omega,x) }[/math]. This looks like an incident plus reflected wave as [math]\displaystyle{ x\to -\infty }[/math] and like a transmitted wave as [math]\displaystyle{ x\to -\infty }[/math]. We call these solutions generalised eigenfunctions. We then try and expand the solution in terms of this solutions (which we call the generalised eigenfunctions) because they solve for