# Integral Equation for the Finite Depth Green Function at Surface

We want to solve

$\phi(x) = \int_{-L}^{L}G(x,\xi) \left( \alpha\phi(\xi) - w(\xi) \right)\mathrm{d} \xi$

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 $N$ evenly spaced point $x_n = -L = hn$ where $h=2L/N$ and $n=0,1,\dots,N$

## Matlab Code

A program to calculate the coefficients for the semi-infinite dock problems can be found here matrix_G_surface.m