9860

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.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+