Template:Radiation condition for diffracted potential

[math]\displaystyle{ \phi^{\mathrm{D}} }[/math] satisfies the Sommerfeld Radiation Condition

[math]\displaystyle{ \frac{\partial}{\partial x} \left(\phi^{\mathrm{D}}-\phi^{\rm I} \right) \pm ik\left( \phi^{\mathrm{D}}-\phi^{\rm I}\right) = 0 ,\,\,\mathrm{as} \,\,x\rightarrow\pm\infty, }[/math]

where [math]\displaystyle{ k ,\, }[/math] is the wavenumber, which is the positive real solution of the Dispersion Relation for a Free Surface

[math]\displaystyle{ k\tanh(kh)=\omega^{2} \, }[/math]

and [math]\displaystyle{ \phi^{\rm I} }[/math] is the incident wave given by

[math]\displaystyle{ \phi^{\rm I} = \frac{\cosh(k(z+h))}{\cosh(kh)} e^{-i kx} \, }[/math]

(which has unit amplitude in potential) and is travelling towards positive infinity.