Integral Equation for the Finite Depth Green Function at Surface

We want to solve

[math]\displaystyle{ \phi(x) = \int_{-L}^{L}G(x,\xi) \left( \alpha\phi(\xi) - w(\xi) \right)d \xi }[/math]

where G(x,\xi) is the Free-Surface Green Function for two-dimensional waves, with singularity at the water surface. We break the surface into [math]\displaystyle{ N }[/math]evenly spaced point [math]\displaystyle{ x_n = -L = hn }[/math] where [math]\displaystyle{ h=2L/N }[/math] and [math]\displaystyle{ n=0,1,\dots,N }[/math]