Template:Incident plane wave

The equation is subject to some radiation conditions at infinity. We usually assume that there is an incident wave [math]\displaystyle{ \phi^{\mathrm{{In}}}\, }[/math] is a plane wave travelling in the [math]\displaystyle{ x }[/math] direction

[math]\displaystyle{ \phi^{\mathrm{{In}}}({r},z)=Ae^{k_0 x}\frac{\cos k_0(z+h)}{\cos k_0 h} }[/math]

where [math]\displaystyle{ A }[/math] is the wave amplitude and [math]\displaystyle{ k_0 }[/math] is the positive imaginary solution of the Dispersion Relation for a Free Surface. We also have to apply the Sommerfeld Radiation Condition as [math]\displaystyle{ \left|\mathbf{r}\right|\rightarrow \infty }[/math].