Generalized Eigenfunction Expansion for Water Waves for a Fixed Body

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 (in some non-dimensional variables)

[math]\displaystyle{ \nabla^2 \phi = 0 }[/math]

[math]\displaystyle{ \phi_n - g \frac{\partial^2\phi}{\partial t^2} = 0, x\in F }[/math]

[math]\displaystyle{ \phi_n = 0, x\in \Gamma }[/math]

where [math]\displaystyle{ F }[/math] is the free-surface and [math]\displaystyle{ \Gamma }[/math] is the wetted body surface. We now have to transform this equation by introducing the Direchlet-Neuman map which gives

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