Pendulum Wave: Seems like magic, but its physics!

Pendulum Wave: Seems like magic, but its physics!

Sometimes just witnessing a physics demonstration can inspire students to look more deeply into conceptual nature of the demonstration and strengthen their critical thinking skills. The Pendulum Wave is a device that provides such an opportunity. It consists of a series of pendulum with increasing periods that when simultaneously released, produce the effect of a changing transverse wave that cycles back to the beginning conditions. This issue of CoolStuff contributed by retired physics teacher Tom Senior, provides you with a creative method to demonstrate this with a standard overhead projector

From contributor Tom Senior: This device shows a hypnotic effect by having a series of pendula with increasing time periods so that when they are simultaneously released, they produce the effect of a wave.  The wave in this case diminishes in wavelength, then looks very chaotic, then comes back to the straight line from which it started.  The lengths of each pendulum is related to the one next to it in a way that after a certain number of swings, they are back in phase with each other.  When put to music,(see

the the brain of the beholder tries to put the audio and visual together, even though they do not match.  It can also be seen as a beat frequency demonstration.  I first saw this device from the video of the Arizona State University amazing automated hallway demo.

Arizona State University Display The Physics and Astronomy Instructional Resource Team unveiled a major display. Located with the Physics and Astronomy displays in the Physical Science F Wing lobby, the automated Pendulum Wave machine has been delighting and mesmerizing viewers ranging from faculty to school children. Wayne Easterling of PIRT was the project lead, with Tim Cook of PIRT providing some assistance. Gary Jarrette of the Physics & Astronomy Mechanical Instrument Shop fabricated the machine as well as engineering the drive mechanism.

To calculate the lengths of the pendula, start with the longest time period, T. Decide how many cycles of that pendulum you would want before all the pendula are back in phase, usually 12 to 20 cycles.  Let us use 16 cycles.   This means that the time period (Ta) of the next pendulum needs to be such that it takes 17 cycles to get back in phase with the first, the next will take 18 cycles, the next 19, etc… This leads to the relationships:  16T = 17 Ta,  16T = 18 Tb,  and 16T = 19 Tc, etc. To calculate the lengths, form the ratio of time periods as a function of lengths, then solve for the ratio of lengths as a function of time periods.  When you are finished this exercise for the student, the result shows that the length you are looking for is the length of the longest pendulum multiplied by the square of the ratio of the time periods squared. For example:  La = L (16/17)2,   Lb = L (16/18) 2,  Lc = L (16/19) 2,  etc….  where L is the length of the length of the longest pendulum. When you decide how long you want the longest pendulum to be, the lengths of the others can be calculated easily. (Spreadsheets are great for this!). To build the stand, calculate the difference in lengths for adjacent pendula.  Take the difference between adjacent pendula, this will form the steps on the stand so the swinging bobs are at the same level. The method of attachment is up to the builder, but the string supporting each pendulum needs to be pinched to make a definite length. The bob support string is bifilar, or two stringed, one on each side of the bob. The top ends need to be separated by enough distance to assure stability. The ability to make fine adjustments to the length is necessary so the device can come back to line after two or even three cycles.

One design for a 20 cm longest pendulum is to cut 5.0 cm long blocks of 1 by 2 pine and sand the sides so they will glue together firmly.  I then screw on a 1 cm wide strip of hard board across the bottom so that the edge of the strip lines up with the bottom of the block.  The strings to the bobs goes under the strip and is held in place by the screw.  Using the difference in length between the adjacent pendulums as the offset, glue the blocks together.

For the PTRA workshop at the 2009 Summer Meeting of the AAPT at the University of Michigan, I scaled down the Pendulum Wave to the over-head projector size.  When the pivot board is in the center of the stage, it does not affect the image on the screen, and the motion of the bobs is clearly seen. See the video on YouTube at:(

The Pendulum Kit Developed by retired physics teacher, Tom Senior, The Pendulum Wave Kit is designed to fit on an over-head projector. Discontinued!

NEW! Pendulum Wave now available.

Watch it in motion, see our video on the product page:

Acrylic Pendulum Wave

In Stock SKU: P4-1755

Overhead Projector Pendulum Wave Kit  (Discontinued!)

About the Contributor Tom Senior recently retired after teaching high school physics for 33 years at Radnor High School in Pennsylvania and at New Trier High School in Illinois.  He has participated with the Physics Instructional Resource Agent program of AAPT for the last 20 years, presenting workshops in make and take activities, the role of demonstrations in teaching physics among other activities.  He was awarded a Distinguished Service Citation at the 2008 Winter meeting of the AAPT.  He presently serves as Secretary to the Physics Instructional Resource Association (PIRA) and as a member of the Apparatus Committee.

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Comments (17)

  • flora Reply

    Hi. how can I rebuild it?
    Please help me
    Thanks alot

    April 12, 2012 at 9:22 am
    • Arbor Scientific Reply

      This product has been discontinued, it was a limited time offer. The included data sheet included more information. If that is not enough to help you rebuild your Pendulum Wave Kit let us know what you need.

      April 16, 2012 at 4:02 pm
  • khushi Reply

    very nice ,exellent
    i made this for science project and it was selected for school exhibition………..

    November 6, 2012 at 9:35 am
  • Astro Gremlin Reply

    One of the more beautiful demonstrations of harmony, wave interference, and beat frequencies!

    December 3, 2012 at 1:54 am
  • teacher Reply

    Are the lengths of the string only or to the center of the bobs?

    October 1, 2013 at 11:24 am
    • Arbor Scientific Reply

      The string would need to be twice the length of the desired pendulum.

      October 7, 2013 at 11:37 am
  • Keith Yager Reply

    Can you show the proof for the equations used for the lengths of the pendulua.

    December 1, 2013 at 5:49 pm
    • Arbor Scientific Reply

      Here is the equation.

      December 10, 2013 at 10:41 am
  • James Flaten Reply

    A discussion of the mathematical shapes exhibited by this demonstration — in particular the continuous function which is sampled at even intervals by the pendula resulting in aliasing — can be found in the American Journal of Physics. It turns out to be a neat little extension to the equation for traveling waves in one dimension.

    Pendulum waves: A lesson in aliasing, James A. Flaten and Kevin A. Parendo, Am. J. Phys. 69, 778 (2001)

    Available on-line at

    To see on-line simulations of the pendula and the underlying continuous function, contact James Flaten at [email protected]

    March 14, 2014 at 2:04 pm
  • person Reply

    what forces are in the penduum wave plz

    March 29, 2014 at 11:40 pm
  • Lisa Mills Reply

    Where the shadow is cast on the wall, what would a song sound like if the five lines of the treble clef were inscribed on the wall with the shadows being the notes?

    September 8, 2014 at 10:38 am
  • Noah Apa Reply

    What angle does the board that holds all the balls have to be?

    November 10, 2014 at 10:24 pm
  • Andy Reply

    What is the equation of the curve that maps the lengths of the strings? (Or in the case of your model, the curve of the top of the frame? )

    June 6, 2015 at 1:44 am
  • Matthew Reply

    The angle is irrelevant. It can be 90° or each pendulum can be in a 90° line or anything in between. Its the length of each pendulum that is the deciding factor of a succesful wave.

    June 10, 2015 at 3:02 am
  • Hesam Reply

    I have been trying to make pendulum waves for 8 months and i can’t make it yet.
    i adjust the lengths many times . i change the distance between the balls.
    but at last i can see just two or three good ocillation and after that it destroyed.
    I beg you please help me or introduce someone to help me. I’m confused. please. I don’t know what should i do.

    September 2, 2015 at 8:39 am

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