Details

The static solution (Green's function) for the Poisson equation for a single force applied on a circular thin membrane at the point (, ) is called the Green's function for the circular membrane, which is parameterized by (, ). Poisson's equation for the Green's function is

and the Green's function for the Dirichlet boundary condition, when the circular boundary at radius is held to zero displacement, is given by (for )

.

This solution is, with some manipulation, from Duffy, eq. 5.2.36. Because the solution has no series summation, like many Green's functions for the Poisson equation, it can be computed rather rapidly. The solution depends on and on the angular difference in such a way that there is little value in enabling the viewer to manipulate , but this is provided anyway. The user can cut off at any arbitrary positive value.

Reference: D. G. Duffy, Green's Functions with Applications, Boca Raton, FL: Chapman & Hall/CRC, 2001.