A note on the reflection coefficient in a water wave scattering problem (Q2639262)

We are concerned with the following boundary value problem \[ \phi_{xx}+\phi_{yy}=0\text{ in } y>0,\quad K\phi +\phi_ y=0\text{ on } y=0, \] \[ \phi_ x=0\text{ on } x=0,\quad a<y<\infty,\quad \phi,\nabla \phi \to 0\quad as\quad x\to \infty, \] \(r^{1/2}\nabla \phi\) remains bounded as \(r=\{x^ 2+(y-a)^ 2\}^{1/2}\to 0\), and \[ \phi \sim \exp (-Ky+iKx)\quad as\quad x\to \infty, \] \[ \phi \sim \exp (-Ky+iKx)+R \exp (-Ky-iKx)\quad as\quad x\to -\infty. \] A method based on variational principle is used to evaluate the reflection coefficient R.