We now consider the scattered potentials <math>\phi^{\mathrm{S}}</math>. The relationship between scattered potentials, diffracted potentials and the incident wave are as follows:
The problem of a two-dimensional finite dock is solved using a green function.
The same problem is solved using eigenfunction matching in Eigenfunction Matching for a Finite Dock.
Equations for a Dock in the Frequency Domain
Wave scattering by a finite dock
We consider here the Frequency Domain Problem for a finite dock which occupies
the region [math]\displaystyle{ -L\lt x\lt L }[/math] (we assume [math]\displaystyle{ e^{i\omega t} }[/math] time dependence).
The water is assumed to have
constant finite depth [math]\displaystyle{ h }[/math] and the [math]\displaystyle{ z }[/math]-direction points vertically
upward with the water surface at [math]\displaystyle{ z=0 }[/math] and the sea floor at [math]\displaystyle{ z=-h }[/math]. The
boundary value problem can therefore be expressed as
We
must also apply the Sommerfeld Radiation Condition
as [math]\displaystyle{ |x|\rightarrow\infty }[/math]. This essentially implies
that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave
and a wave propagating away.
We begin with the diffraction potential [math]\displaystyle{ \phi^{\mathrm{D}} }[/math] which
satisfies the following equations
[math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math]
is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,
[math]\displaystyle{
\phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \,
}[/math]
where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \mathrm{i} k }[/math] is
the positive imaginary solution of the Dispersion Relation for a Free Surface
(note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math])
and
[math]\displaystyle{
\phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h}
}[/math]
As the plate is floating on the surface, we can denote it as follows:
The Reflection and Transmission Coefficients represent the ratio of the amplitude of the reflected or transmitted wave to the amplitude of the incident wave. Conservation of energy means that [math]\displaystyle{ |R|^2+|T|^2=1\, }[/math].
A diagram depicting the area [math]\displaystyle{ \Omega\, }[/math] which is bounded by the rectangle [math]\displaystyle{ \partial \Omega \, }[/math]. The rectangle [math]\displaystyle{ \partial \Omega \, }[/math] is bounded by [math]\displaystyle{ -h \leq z \leq 0 \, }[/math] and [math]\displaystyle{ -\infty \leq x \leq \infty \, }[/math] or [math]\displaystyle{ -N \leq x \leq N\, }[/math]
We can calculate the Reflection and Transmission coefficients by
applying Green's theorem to [math]\displaystyle{ \phi\, }[/math] and [math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math][math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math]
is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction,
[math]\displaystyle{
\phi^{\mathrm{I}}(x,z)=A \phi_0(z) e^{\mathrm{i} k x} \,
}[/math]
where [math]\displaystyle{ A }[/math] is the wave amplitude (in potential) [math]\displaystyle{ \mathrm{i} k }[/math] is
the positive imaginary solution of the Dispersion Relation for a Free Surface
(note we are assuming that the time dependence is of the form [math]\displaystyle{ \exp(-\mathrm{i}\omega t) }[/math])
and
[math]\displaystyle{
\phi_0(z) =\frac{\cosh k(z+h)}{\cosh k h}
}[/math]
We assume that [math]\displaystyle{ A=1 }[/math]. This gives us
We now consider the scattered potentials [math]\displaystyle{ \phi^{\mathrm{S}} }[/math]. The relationship between scattered potentials, diffracted potentials and the incident wave are as follows:
and they are referred to as the added mass and damping coefficients (see Added-Mass, Damping Coefficients And Exciting Forces
for the equivalent definition for a rigid body).
respectively.
This equation is solved by truncating the number of modes.
Solution for the Radiation and Diffracted Potential
We use the Free-Surface Green Function for two-dimensional waves, with singularity at
the water surface since we are only
interested in its value at [math]\displaystyle{ z=0. }[/math] Using this we can transform the system of equations to
The Reflection and Transmission Coefficients represent the ratio of the amplitude of the reflected or transmitted wave to the amplitude of the incident wave. They hold the property that [math]\displaystyle{ |R|^2+|T|^2=1\, }[/math] (and may often contain an imaginery element).
A diagram depicting the area [math]\displaystyle{ \Omega\, }[/math] which is bounded by the rectangle [math]\displaystyle{ \partial \Omega \, }[/math]. The rectangle [math]\displaystyle{ \partial \Omega \, }[/math] is bounded by [math]\displaystyle{ -h \leq z \leq 0 \, }[/math] and [math]\displaystyle{ -\infty \leq x \leq \infty \, }[/math] or [math]\displaystyle{ -N \leq x \leq N\, }[/math]
We can calculate the Reflection and Transmission coefficients as follows:
Applying Green's theorem to [math]\displaystyle{ \phi\, }[/math] and [math]\displaystyle{ \phi^{\mathrm{I}}\, }[/math] gives:
where [math]\displaystyle{ k_0 \, }[/math] is the first imaginery root of the dispersion equation and the incident wave is of the form: [math]\displaystyle{ \phi^I=\phi_0(z)e^{-ikx} \, }[/math]<br\><br\>
Therefore, in the case of a floating plate (where z=0):
