9853

Pendulum Waves

Set in motion 15 to 25 uncoupled pendulums with monotonically increasing lengths from the same initial angle, chosen between 10° and 60°. This produces a stunning visual display including traveling waves, standing waves, and simulated chaotic motion.
The lengths of the pendulums are adjusted so that their oscillation completes an integer number of cycles in 60 seconds. For small angles, the angular frequency and length are related by the approximate formula , but this Demonstration uses the exact relation , where denotes the elliptic integral of the first kind and numerical solutions are found for the classic pendulum differential equation .
Once set in motion, the pendulums quickly fall out of sync. But after 60 seconds (neglecting friction and air resistance), they have all undergone an integral number of cycles and return to their starting configuration. Another interesting configuration occurs at 30 seconds, when the pendulums are at alternating maximum phases. The states at 12, 15, 20, 24, 36, 40, 45, and 48 seconds also show some organization. The oscillations can be observed from the front, side, or top.
You can also choose to view a plot of all the pendulum angles against time. This shows the sinusoidal paths of the individual pendulums, color coded to match the corresponding pendulum balls. Note how the chaotic motion becomes ordered at certain times.

SNAPSHOTS

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

DETAILS

References:
R. E. Berg, "Pendulum Waves: A Demonstration of Wave Motion Using Pendula," American Journal of Physics, 59(2), 1991 pp. 186–187.
J. A. Flaten and K. A. Parendo, "Pendulum waves: A lesson in Aliasing," American Journal of Physics, 69(7), 2001 pp. 778–782.
Pendulum Waves: YouTube
Citadel Physics Department Wave Pendulum: YouTube

PERMANENT CITATION

    • 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+