Visualizing solitary waves

As I’ve mentioned before, people in physics love to talk about waves.  Seemingly everything in nature gets described as a wave at some point or another.  And for the most part, our wave descriptions are done using sines and cosines (frequently disguised as complex exponentials).  Sines and cosines are familiar, easy to visualize, and have lots of helpful mathematics built around them.

But in some respects, a sine wave is an extremely unrealistic object.  For one thing, it fills all of space.  A sine wave goes on and on, oscillating forever without any easily-defined “extent”.  A real wave would never fills all of space; a real wave is a disturbance over a particular region.  You can think of it this way: a wave is composed of many individual objects moving up and down or back and forth.  If you had a wave that filled all of space, it would require infinitely many of these little objects to be moving.  As such, it would have infinite energy.  Clearly, a real wave must have a finite size.  What’s more, the size of a wave tends to be small, since larger waves require more energy, which is hard to come by.

This problem alone could be a deal-breaker, but there is another issue with sines and cosines.  A sine wave cannot carry momentum.  Imagine, for example, a rightward-moving sine wave.  The wave is called “transverse” if the objects that comprise it are moving up and down while the wave is moving to the right.  Like this:

Even though the wave (black line) is moving to the right, the particles that make up the wave (blue dot) are moving up and down.  Clearly, if this wave hit you, it wouldn’t push you to the right.  It would push you upward, and then drop you down again.

A sine wave can also be “longitudinal”, which means it is composed of particles moving back and forth along the direction of the wave.  Like this:

There is certainly a lot of rightward motion in this wave, so it looks like it has momentum.  But focus on any one particle and you’ll see that it spends as much time moving left as it does moving right.  A person standing in the middle of this wave pattern might be alternatingly pushed right and left, but would not be given any net momentum.

A real wave, on the other hand, can have momentum.  A strong wave at the beach can knock you over (or, if it’s really strong, it can knock your house over).  So apparently there are waves that cannot be described by any combination of sines and cosines.

There is another kind of wave, called a soliton, that does not suffer from either of these problems.  A soliton is a single wave front that can propagate without decaying.  It carries momentum, has a definite energy, and occupies a definite range of space.

A soliton can appear whenever the “particles” that comprise the wave can have more than one equilibrium position.  In the animation above, each particle has a definite center of motion from which it never drifts.  As a consequence, the particles must do as much backward movement as forward movement, and the wave cannot carry momentum.  This is the limitation of sine and cosine waves: they describe waves where each constituent particle is “stuck” around a single position.

In soliton waves, this is not the case.  Imagine stretching out a slinky and fixing the far right end to the wall.  If you oscillate the left end back and forth around a fixed position, you get a sine wave.  Each coil in the slinky will move back and forth but will never stray far from its original position.  If you take the left end and suddenly move it forward a few inches, you get a soliton wave.  Each coil will move forward a few inches, and in doing so will push the coils in front of it.  The wave will appear as a single high-density region propagating forward, and it will have definite forward momentum.

Since I recently discovered that making low-quality movies can be fun, you can check out these visualizations of a soliton wave:

Here is an actual soliton made in a wave tank:

and you can check out this link to see one propagating on a stretched string.

For the small subset of readers who would like to know the mathematics behind solitons, I’ll give a brief summary here.  Sine and cosine waves are solutions to the wave equation.  Solitons result from a modified wave equation called the Sine-Gordon equation.  This equation allows for stable 360-degree “flips” in the phase (see the pendulum illustration in the first youtube video above) which the normal wave equation does not allow.  The phase propagates through space and time as

,

where v is the speed of the wave and is its size.

Footnotes

• The longitudinal wave animation is taking from Dr. Dan Russell’s web page at Kettering University: http://www.kettering.edu/~drussell/demos.html.  I highly recommend looking through it; there are lots of great wave illustrations.
• There may be other mathematical formulations of a soliton wave other than through the Sine-Gordon equation.  I just used the one I am most familiar with.