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
