Template:Radiation potential equations for a dock like structure

[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]