[math]\displaystyle{
\Delta\phi_n^{\mathrm{R}} =0,\,\,-h\lt z\lt 0,
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} =0,\,\,z=-h,
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} =\alpha\phi_n^{\mathrm{R}},\,\,x\notin(-L,L),\, \,
z=0
}[/math]
[math]\displaystyle{
\partial_{z}\phi_n^{\mathrm{R}} = i\omega X_{n},\,\,x\in(-L,L),\,\,z=0.
}[/math]
The radiation condition for the radiation potential is
[math]\displaystyle{
\frac{\partial\phi_n^{\mathrm{R}}}{\partial x}\pm ik\phi_n^{\mathrm{R}}=0,\,\,\mathrm{as}
\,\,x\rightarrow\pm\infty.
}[/math]
