Poisson Equation on a Circular Membrane
This Demonstration shows the static response (Green's function) of a circular thin membrane to a force applied at a point on the membrane. The boundary is held to zero displacement (Dirichlet boundary condition). The source point is controlled by the (radius) and (angle) sliders. The scaling of the response is controlled by the "maximum " slider that controls the response value, which coincides with the top of the viewing box. If the response exceeds this value, it is clipped at the top.
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.