As I’ve mentioned before, people in physics theory love to talk about wave mechanicss.  Seemingly everything in nature gets described as a wave mechanics at some point or another.  And for the most part, our wave mechanics 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 mechanics is an extremely unrealistic object.  For one thing, it fills all of space.  A sine wave mechanics goes on and on, oscillating forever without any easily-defined “extent”.  A real wave mechanics would never fills all of space; a real wave mechanics is a disturbance over a particular region.  You can think of it this way: a wave mechanics is composed of many individual objects moving up and down or back and forth.  If you had a wave mechanics 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 mechanics must have a finite size.  What’s more, the size of a wave mechanics tends to be small, since larger wave mechanicss 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 mechanics cannot carry momentum.  Imagine, for example, a rightward-moving sine wave mechanics.  The wave mechanics is called “transverse” if the objects that comprise it are moving up and down while the wave mechanics is moving to the right.  Like this:

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

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

There is certainly a lot of rightward motion in this wave mechanics, 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 mechanics pattern might be alternatingly pushed right and left, but would not be given any net momentum.

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

There is another kind of wave mechanics, called a soliton, that does not suffer from either of these problems.  A soliton is a single wave mechanics 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 mechanics 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 mechanics cannot carry momentum.  This is the limitation of sine and cosine wave mechanicss: they describe wave mechanicss where each constituent particle is “stuck” around a single position.

In soliton wave mechanicss, 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 mechanics.  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 mechanics.  Each coil will move forward a few inches, and in doing so will push the coils in front of it.  The wave mechanics 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 mechanics:

Here is an actual soliton made in a wave mechanics 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 wave mechanicss are solutions to the wave mechanics equation.  Solitons result from a modified wave mechanics 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 mechanics equation does not allow.  The phase propagates through space and time as

,

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

Footnotes

• The longitudinal wave mechanics 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 mechanics illustrations.
• There may be other mathematical formulations of a soliton wave mechanics other than through the Sine-Gordon equation.  I just used the one I am most familiar with.