As the plate is floating on the surface, we can denote it as follows:
As the plate is floating on the surface, we can denote it as follows:
Line 92:
Line 23:
</math></center>
</math></center>
== Solution for Diffracted Potential ==
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:
{{Green's function equations for the diffracted potential}}
Details about this method can be found in [[Integral Equation for the Finite Depth Green Function at Surface]].
== Reflection and Transmission Coefficients ==
== Reflection and Transmission Coefficients ==
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> |R|^2+|T|^2=1\,</math> (and may often contain an imaginery element).
{{derivation of reflection and transmission in two dimensions}}
[[Image:Square_volume.png|600px|right|thumb|frame|A diagram depicting the area <math>\Omega\,</math> which is bounded by the rectangle <math>\partial \Omega \,</math>. The rectangle <math>\partial \Omega \,</math> is bounded by <math> -h \leq z \leq 0 \,</math> and <math>-\infty \leq x \leq \infty \,</math> or <math>-N \leq x \leq N\,</math>]]
We can calculate the Reflection and Transmission coefficients as follows:
Applying Green's theorem to <math>\phi\,</math> and <math>\phi^{\mathrm{I}}\,</math> gives:
- 2k_0 R \int_{-h}^{0} \left(\phi_0(z)\right)^2 \mathrm{d}z.
</math></center><br\>
where <math> k_0 \,</math> is the first imaginery root of the dispersion equation and the incident wave is of the form: <math> \phi^I=\phi_0(z)e^{-ikx} \,</math><br\><br\>
Therefore, in the case of a floating plate (where z=0):
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
