Energy Eigenfunctions

A Gaussian pulse propagating on a Shive wave machine with 64 rods.

# Wave Machine

The Wave Machine model simulates the wave machine produced by John Shive at Bell Laboratories and made famous by the PSSC Simple Waves film.  The machine consists of n horizontal bars with moment of inertia In  welded to a torsion rod that is perpendicular to the bars.  The simulation allows the user to change the lengths of the bars, thereby simulating the effect of a wave propagating in a non-uniform medium. The default bar of length L=2 has a moment of inertia of one.  The maximum allowed bar length is 4 giving a moment of inertia of 4 and the minimum allowed length is 1/2 giving a moment of inertia of 1/16.

Twisting a bar about the torsion rod causes the bar to oscillate because the rod produces a restoring torque.  Because a bar twist acts on neighboring bars, the motions are coupled and a traveling wave results.  The speed of the wave depends on the torsional coupling between bars k and the moments of inertia of the bars.  A damping force can also be added using the model's damping parameter b.

The simulation allows various pulse shapes to be sent down the machine by twisting the first rod with the desired functional form or by dragging the first rod.  For example, applying a Gaussian twist produces a Gaussian traveling pulse but the width of this pulse depends on the wave speed.  The far end of the wave machine can be free or clamped and this changes the nature of the reflected wave.

Theoretical note:  The pulse shape will distort as the wave propagates on the wave machine because of dispersion effects.  This distortion is most apparent as the wavelength (or pulse width) approaches the rod separation.  Use the Driven Wave Machine model to explore dispersion these effects.

The Wave Machine model is distributed as a ready-to-run (compiled) Java archive.  Double clicking the ejs_mech_osc_chains_WaveMachine.jar file will run the program if Java is installed.  Other coupled oscillator models are available.  They can be found by searching the OSP Collection for coupled oscillations.

## References

• "Standing waves in a non-uniform medium," Paul Gluck, The Physics Teacher, (in press).

• "Making waves: A classroom torsional wave machine (Part I)," Kenneth D. Skeldon, Janet E. Milne, Alastair I. Grant, and David A. Palmer Phys. Teach. 36, 392 (1998)

• "Making waves: A classroom torsional wave machine (Part II)," Kenneth D. Skeldon, Janet E. Milne, Alastair I. Grant, and David A. Palmer Phys. Teach. 36, 466 (1998)

• University of Maryland Physics Lecture-Demonstration website section G3 http://www.physics.umd.edu/lecdem/services/demos/demosg3/demosg3.htm

• Similarities in wave behavior, John N. Shive, Bell Telephone Laboratories (1961). See also Am. J. of Physics 32, p572 (1964).

## Credits:

The Wave Machine model was created by Wolfgang Christian using the Easy Java Simulations (EJS) version 4.3 authoring and modeling tool created by Francisco Esquembre.

You can examine and modify this compiled EJS model if you run the model (double click on the model's jar file), right-click within a plot, and select "Open EJS Model" from the pop-up menu.  You must, of course, have EJS installed on your computer.  Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.

### Translations

Code Language Translator Run

### Software Requirements

SoftwareRequirements

 Android iOS Windows MacOS with best with Chrome Chrome Chrome Chrome support full-screen? Yes. Chrome/Opera No. Firefox/ Samsung Internet Not yet Yes Yes cannot work on some mobile browser that don't understand JavaScript such as..... cannot work on Internet Explorer 9 and below

### Credits

Wolfgang Christian - Davidson College; Wee Loo Kang (This email address is being protected from spambots. You need JavaScript enabled to view it.); Felix J. Garcia Clemente

### For Teachers

Legends of symbols used

Fixed End , means the far end of the wave machine is fixed, possibly for exploring reflecting wave formation etc.

Movable End, means the far end of the wave machine is it movable, like a ring on a sliding rod etc

A = Amplitude of Wave in Angle from centre of the rod, so A = 0 means no amplitude, A = π/2 is a 90 degree turn of the first rod on the wave machine

m = mass attached to the ends of the rods, to allow inquiry into the effects of adding extra point masses on the ends of the rods

L1 = near end of the wave machine variable from 2 to 4 m for inquiry into effects of length of rods and the effects of higher inertia to rotation.

L2 = far end of the wave machine variable from 2 to 4 m for inquiry into effects of length of rods and the effects of higher inertia to rotation.

k = torsional spring constant of the rope attaching to all the rods, low k means restoring moment lesser, high k means restoring moments higher

b = drag coefficient in a high viscous medium of air etc, to show effects of air drag

#### Teaching notes

1 As with the ripple tank, it helps if you start by demonstrating single pulses before going on to continuous waves. Deflect a section of the machine close to the end or you will get pulses running in opposite directions. Try to avoid setting up standing waves which can be confusing.
Show that a pulse/disturbance travels along the line. Point out that the mass (jelly babies) are displaced up and down (vertically) while the wave travels along horizontally.

2 You can demonstrate the following:

• The basic principle of a wave: a vibrating source sends a disturbance through a medium. The wave travels, transferring energy, but the medium doesn’t move.
• A wave reflects when it meets a fixed end.
• Make bigger or smaller pulses – relate this to amplitude, and to energy, which is being transferred by the wave. Greater amplitude = greater energy.
• Make a faster disturbance – this makes a shorter pulse, but the speed is unchanged.
• Show that continuous disturbance by the source results in continuous waves.
• Emphasise the meanings of amplitude, wavelength, frequency and wave speed.
• Frequency and amplitude depend on the source; speed and wavelength depend on the medium.

3 You can adapt the machine to show that the speed of a wave changes when it moves into a different medium by adding or removing mass. A stopwatch is adequate for timing a wave to deduce its speed, but note that waves travel very quickly along a machine which has no jelly babies on its skewers.

4 Students could use the machine to investigate the relationship between wave speed, frequency and wavelength (speed = frequency ´ wavelength). They could also change various factors and find out how wave speed changes: separation of the skewers, mass/number of jelly babies, position of jelly babies on skewers, width of tape, thickness of tape, etc.
5 You could photograph the machine from the side to examine how the displacements of adjacent elements vary along the wave.

Big Picture

A wave is a disturbance that propagates through a medium, transferring energy. Water waves are familiar, but in science we extend the idea to other types of wave in other media.
A wave can only travel if the elements of the medium are connected in some way; in this case, each rod affects the next by twisting the tape that connects them.

Waves are characterised by their amplitude and frequency (determined by the source) and their speed (determined by the medium). The wavelength depends on the frequency and the speed.

### Wave Machine

The Wave Machine model simulates the wave machine produced by John Shive at Bell Laboratories and made famous by the PSSC Simple Waves film.  The machine consists of n horizontal bars with moment of inertia In  welded to a torsion rod that is perpendicular to the bars.  The simulation allows the user to change the lengths of the bars, thereby simulating the effect of a wave propagating in a non-uniform medium. The default bar of length L=2 has a moment of inertia of one.  The maximum allowed bar length is 4 giving a moment of inertia of 4 and the minimum allowed length is 1/2 giving a moment of inertia of 1/16. Twisting a bar about the torsion rod causes the bar to oscillate because the rod produces a restoring torque.  Because a bar twist acts on neighboring bars, the motions are coupled and a traveling wave results.  The speed of the wave depends on the torsional coupling between bars k and the moments of inertia of the bars.  A damping force can also be added using the model's damping parameter b.

The simulation allows various pulse shapes to be sent down the machine by twisting the first rod with the desired functional form or by dragging the first rod.  For example, applying a Gaussian twist produces a Gaussian traveling pulse but the width of this pulse depends on the wave speed.  The far end of the wave machine can be free or clamped and this changes the nature of the reflected wave.

Theoretical note:  The pulse shape will distort as the wave propagates on the wave machine because of dispersion effects.  This distortion is most apparent as the wavelength (or pulse width) approaches the rod separation.  Use the Driven Wave Machine model to explore dispersion these effects.

The Wave Machine model is distributed as a ready-to-run (compiled) Java archive but now available as JavaScript version.

### Demonstration and Building a Physical Demo wave machine

A wave machine is an entertaining way of introducing some basic ideas about wave motion.

Apparatus and material

For the demonstration

1. Duct tape
2. Clamps and stands, G clamps
3. Barbecue skewers
4. Jelly babies (jelly beans)
5. Metre rule
6. Stopwatch or stop clock

A 5 m wave machine will require about 100 skewers and 200 jelly babies.

Commercially-manufactured wave machines are available, but the attraction of this home-made version is that its construction and mechanism are clear to students.

Health & Safety and Technical notes

Students should be advised to take care when handling pointed skewers. They should be instructed not to eat the jelly babies, which should be disposed of safely after use (the jelly babies, not the students

### Video

Wave Machine Demonstration by National STEM Centre

### References

• "Standing waves in a non-uniform medium," Paul Gluck, The Physics Teacher, (in press).

• "Making waves: A classroom torsional wave machine (Part I)," Kenneth D. Skeldon, Janet E. Milne, Alastair I. Grant, and David A. Palmer Phys. Teach. 36, 392 (1998)

• "Making waves: A classroom torsional wave machine (Part II)," Kenneth D. Skeldon, Janet E. Milne, Alastair I. Grant, and David A. Palmer Phys. Teach. 36, 466 (1998)

• University of Maryland Physics Lecture-Demonstration website section G3 http://www.physics.umd.edu/lecdem/services/demos/demosg3/demosg3.htm

• Similarities in wave behavior, John N. Shive, Bell Telephone Laboratories (1961). See also Am. J. of Physics 32, p572 (1964).

### Credits:

The Wave Machine model was created by Wolfgang Christian and Loo Kang Wee (JavaScript version)  using the Easy Java Simulations (EJS) version 4.3 authoring and modeling tool created by Francisco Esquembre and Felix J. Garcia Clemente.