Wednesday, June 4, 2008

Wednesday Math, Vol. 26: The Soliton

How many great stories of mathematical physics involve a young man on horseback chasing a strange natural phenomenon? Well, there's this one! If any of my math-y or science-y friends know another, I'd be glad to hear it.

In 1834, 24 year old Scottish mathematician John Scott Russell was at the Union Canal. A barge was being drawn by two horses and when the barge stopped suddenly, the wave off the bow continued. Most waves have a crest and a trough and crash quickly enough, but not this one. It was like a bump of water, about 30 feet long and well over a foot high, traveling at a constant rate and not slowing down or shrinking noticeably. Russell knew enough physics to know this was unusual, so he got on his horse and chased the wave. He and the horse caught up to the wave, which traveled at about 8 or 9 mph and Russell observed its size, shape and speed, which changed very little though Russell pursued it for over a mile until the winding of the canal and the divergence of the trail made pursuit impossible. Russell worked on studying this phenomenon, off and on, for the rest of his career. He called the phenomenon The Wave Of Translation.

We no longer call it a Wave of Translation, but a solitary wave, often shortened to soliton. They are odd ducks in the world of waves. Most waves have crests and troughs, but a soliton is all crest and no trough. Most waves interact with each other by canceling each other out, what with crests and troughs coming together, but not solitons. Big solitons travel faster than small solitons, and if a big one catches up with a small one, it simply passes the smaller with very little transference of energy. For the time they inhabit the same space, the little soliton looks like a bump on top of the big soliton.

Solitons are not just of academic interest. Tsunamis are almost always solitons. As I said, bigger solitons move faster, and you didn't have to get an A in physics to realize that Big + Fast = really Destructive when a soliton finally hits something. A small blessing from the physical world is that solitons are difficult to create. Some ships bow waves create solitons, but most don't. An underwater earthquake can create a soliton, but not always. If a landslide hits a body of water just right, it can create a soliton. All these things are possible, but the conditions have to be just so.

For the math-y part of this mostly physics-y post, the differential equation that describes a soliton is also an odd duck. In physics, the scientist's best friend is symmetry. As I was taught it by Stu Smith, and I don't argue physics with Stu Smith, every other differential equation that has been completely mathematically solved by humans (which is different from being numerically solved) has some underlying symmetry. The soliton is the only asymmetrical differential equation that we figured out by doing the math instead of getting lots of readings of a physical object and building an approximate differential equation about the object.

So if you hear the word soliton in everyday conversation, an unlikely occurrence I grant you, you will know what it is, and you might remember young John Scott Russell chasing a wave saying "Hoot, man! Aye, laddie! Now that's a fookin' wave! If it's not a wave like that, it's crap!" and other Scottish things.*

*(Quotes from Russell entirely made up by Matty Boy.)

dguzman said...

Fookin' deddly fascinatin', Matty Boy!

I remember enough calculus to remember differential equations, but an asymmetrical one? Say it isn't so.

Karlacita! said...

I love the soliton! If I was still in the new age, I'm sure I'd be thinking about how this was a proof of something or the other.

But now that I'm all skeptical-like, I can just admire it!

jolie said...

thanks for the terrific post, matty boy. very cool.