Saturday, May 30, 2009

How to Get a Moron out of a Tree

[6/1: edited slightly. still opaque.]

Vaguely, this post is based on a stimulating comment at Unqualified Offerings that I read several weeks or months ago (on a post about physics education, which limits it to maybe 1 in 3, and by someone who's obviously a more enthusiastic sort of nerd than I am, which also doesn't narrow things down very much--naturally, I couldn't track down the original, so when the time comes, just assume I'm paraphrasing someone who knew what they were talking about), and the fact that this particular month, I'm busy feigning expertise in the gray areas between physical optics and acoustics. Now, if I were a better scientist, I might get worked up about some clever solution, or about some amazing counterintuitive result, or an impressive discovery, or I might keep myself up nights dreaming about my own rudimentary maths or Rube Goldberg apparati, and okay, sometimes I do those things, but what really tends to impress me are the things that good scientists have long since chalked up and left behind for the challenging problems--it's the remarkable fact that any of this shit works at all.

(And perhaps it's an appropriate moment to explore here the raison d'etre of this-here blog, if, in fact, there is one. I update much too rarely, and I'm never sufficiently on any particular point to grab readers (limited politics), and this particular sort of post has just got nothing for anyone. It's below the level of people who would know what they're doing in these fields, and it is yet too pointlessly geeky to entice any hypothetical passersby. It's as if I'm getting giddy over the overview chapter in my (company's) Optics book. But I admit that this is a triviality that I find satisfying and slightly amusing. And since I was motivated enough to draw a picture today, there could be no going back. Somebody please wave to that poor guy.)

The commenter in question stated that one potentially great thing to teach budding scientists would be the idea that not every goddamn term in the expansion is physically significant. "Yes," I thought at the time. "Damn straight." I had a vivid recollection of some group meeting back when where some solution of somethingorother resulted in several terms, and the authors went out of their way to make up meaning for each of them, independent of any theoretical justification. I had a hard time that day expressing just what it was that irritated me about the work, and here it was, years later, with the sort of pith I appreciate.

Mathematical series expansions are important tools, don't get me wrong. They come up in the physical sciences in an effort to simplify nonlinear problems, or else as solutions to the equations governing wave motion. Through them, you can generate functions represented as sums of other (ideally easier) functions, but is that a real physical picture? Are waves made up of actual little wavelets? Are those squared and cubed terms really physically significant in their own right?

I recall power series in various sorts of chemical engineering applications. For example, a useful equation of state (relating temperature, pressure and density of some fluid) generally varies from the simple ideal gas law, and one way to account for this is a power series expansion on one of the parameters, to get successively more accurate approximations with every term. [A general power series looks like A1x + A2x2 + A3x3 + ... on out forever. Generally there's a mathematical formula to derive the coefficients if you're trying to increase the accuracy of your function of x, but for physical reality, you can go with just fitting curves too, and in ideal cases terms will fall out of physical theory, possibly allowing for some fudge factors.] I don't know if it started with Van der Waals, but his corrections to the ideal gas law used the most intuitive ideas (molecules take up space), and it generated a few extra terms, much like a formal expansion would. Virial equations of state (as they came to be called) were developed based first on just the convenient mathematical expansion, but eventually the terms were assigned physical significance based on statistical mechanics. I like that they were made to mean something, but I don't really believe that any significant data could be made on for those high-order interactions with the tiny coefficients.

In science, Fourier series, which are sums of sines and cosines (and which add harmonics instead of powers, A1sin x + A2sin 2x + ...), get a lot more traction than power series do. There are a lot of reasons for this, most fundamentally because a lot of natural phenomena oscillate. Waves are everywhere (everywhere! thanks, Schmutzie), from sound to vibrating solids to light, and trigometric functions, which like to bob up and down, are an obvious way to represent them. Not that natural waves are always exactly sinusoidal, but most any periodic function can be represented by a Fourier series with enough terms. When you add periodic functions, you get new periodic functions, which is a handy thing when you want to describe how acoustic or electromagnetic waves interact with one another. The smaller waves superpose (i.e., add) to create some other wave, also represented by some Fourier series, whose properties it may be useful to consider separately. Conversely, some wave pattern can be imagined to be the sum of a whole lot of smaller wave patterns.

One of the basic ideas in optics--and one of the student's first what-the-fuck moments--is Huygens' principle, which he imagined as a physical explanation. Huygens averred that a propagating wave--whether in matter or (as he thought of it) the luminiferous aether--was in fact the sum of a whole bunch tiny spherical wavelets, which could be thought to exist at every point in the advancing phase front, which all superposed to form the wave as it was observed. As the wave progressed, new wavelets would be generated and propagated forward too. It went far (and goes far) to describe phenomena like diffraction of light and other waves when it passes through an aperture.

This was in the seventeenth century, by the way, contending some of Newton's famous points, and it's interesting that the physical interpretation apparently came first. (Maybe I should check this? People were monkeying with infinite series by then and maybe it informed the idea?) Auguste Fresnel (remember him?) fixed one of the big basic hangups before too long (spherical wavelets needed some justification for not radiating backwards as well as forwards), but it took quite awhile before Huygens' principle was shown to satisfy a general wave equation for propagation (with, I believe I've read, some dubious boundary conditions). And yeah, that's the other great thing about Fourier series: harmonic functions turn out to be solutions for any number of differential equations that you can derive for mechanical motion or electromagnetism. These predict the oscillatory phenomena that appear throughout classical physics, and as you might imagine, a solution that can be represented by a Fourier series, can be represented by more than one Fourier series, or by sums of Fourier series. But that's math. As a physical matter, this Huygens thing remains pretty brazen: sure you can add up the cosines, but is a real wave one oscillation and a zillion wavelets both at the same time?

Huygens' principle has had its doubters, a quiet dissent that's evidently (with a little background reading) persisted. Einstein dismissed any significance of wavelets beyond a mathematical convenience. And Feynman thought that it wasn't the right physical picture either, although by then he was advancing the quantum mechanical nature of light: not wavelets, he argued, but probability distributions, which then sum up to physical observation. (Quantum mechanics can probably be taken as a more "real" picture for the action of fundamental particles, but this idea of wavelets helps out in diffraction of mechanical waves too, which don't need quantum. Quantum mechanics, at least as it applies to electrons in matter, likes to take on (sums of) harmonic functions too, and the huge conceptual leap there is that it evolves probability distributions instead of any clear picture of motion. Which isn't so bizarre until you try to wrap your head around the idea that these fundamental particles are odd physical things. I won't pretend to any useful understanding of QED, however.)

When waves travel through some medium, they cause sympathetic disturbances within it, and any amount of localized funny business ensues, propagating waves interacting with the vibrations they excite in the first place. (It occurred to me recently that I've spent years now, off and on trying to artificially create physical media that capture the quantum or the classical dynamics explicitly, on larger-than-usual scales.) Classical wave theory can get you pretty far even in these modern times. Generally speaking, it gets more challenging the more the waves interact with their media, and there are any number of degrees of description between, say, Maxwell's equations in free space and an entirely quantum picture. Acoustics generally works fine only considering matter as a continuum (but quantum vibrations exist, which matter in some fields).

When waves in media get intense, they can alter the properties matter as they pass through it, which can make the (classical) physics a lot harder. I don't know much about nonlinear optics, but when I've dabbled with nonlinear acoustics, second-order terms in various power series could be developed to explain those high-amplitude phenomena--you take another expansion to bring those second-order effects into reality. (Coincidentally enough, adding up powers of cosines, which you might do for that nonlinear acoustics problem, is basically the same thing as adding up harmonics of cosines as in a Fourier series, by virtue of some basic trigonometric identities. "Second order" often means the same thing in a conversational sense.) So a second harmonic can be teased out of the power series, a double-fast wave which the superposition principle tells us could have really been there all along. Sure, it's the next-best mathematical formula, but damned if the frequency double that evolves in a real measurement doesn't look a pure thing all by itself. Was it there lurking under there under the primary wave all along?

(How to get a moron out of a tree? Wave to him. Get it?)

20 comments:

LentenStuffe said...

OK K,

I did read this, and I did try to understand what-the-fuck you were on about, but I'm too stupid. But I persisted because it was you writing.

I'll be honest, I usually only sift through this kind of material to see if I can harvest it for a metaphor. So far I'm liking the Huygens' Principle and could use that.

Your formatting and lay-out is very user-friendly. Should mention that too.

Keifus said...

I hope I didn't abuse your trust too much there, but I only have so much room in my head, and this was getting in the way. I wrestled with whether I should post it. (I.e., sure it's my blog, but I like it when people occasionally read it.)

Maybe I'll edit it (if I can uncover some goddamn time to). Basically the idea was that mathematical series expansions come up in the physical sciences in an effort to simplify nonlinear problems, or else as solutions to the equations governing wave motion. You get functions represented as sums of other little functions, but is that a real physical picture, waves made up of actual little wavelets? Sometimes it looks like it is, even when classical theory is an appropriate description (like in acoustics). Which is fucking amazing.

If it's any consolation at all, (1) I'm not being particularly insightful as far as science goes, and (2) when many of my favorite writers go off on semiformal philosophical or literary tangents and the like, I often curl up in a defensive fetal position.

Schmutzie said...

Is that Christian Huygens you're talking about? Dude got it done in many areas, so it wouldn't surprise me.

Speaking of Feynman's take on waves, have you seen this old video?

http://www.youtube.com/watch?v=AU8PId_6xec

Keifus said...

Same guy. One of the polymath geniuses of his time; gets a bit shadowed by that Newton guy.

(Feynman isn't advocating quantum anything in that interview. The sneaky bastard.)

Schmutzie said...

The first peek I got at Christian Huygens was via Sagan's Cosmos. Man, what a brain! Astronomy, physics, mathematics....fuckin' L. Imagine if people like that, like Sir Isaac, like Galileo, had access to 21st Century tools.

"All these waves, light waves,radio, heat, gamma, and they're ALL THERE!! That's the thing that really gets ya!" So excitable. RPF was a true genius.

Keifus said...

They probably do, Schmutz, and working harder than I'd ever care to with a thousand times the brain. They just haven't written next century's textbooks yet. In fact, we're probably much better off than when science was limited to gentlemen's pasttimes.

Although it's a little harder to be a real polymath in these late days. And I don't think we can expect to see Science reinvented like when it woke up way back when.

LentenStuffe said...

Thanks,

Your comment helped, as did your discussion with Michael (thanks for that Feynman video).

I find I must always go back to basics, then proceed, then back again, because I forget the rudimentary concept upon which the advances are built. If you or Michael have any tips on a book for Dummies I'd appreciate it.

Cheers.

Schmutzie said...

John~

Isn't that a great little insight into the mind of Feynman? One of my favorite vids of one of my favorite people.

One of the true measures of his genius was his ability to explain the very complex to idiots like me. Hell, I didn't understand 90% of what he was sayin', but he sure was fun to listen to. (and read)

Keifus said...

Wish I could help. Wish I found something like that a few years ago myself--waves rarely come up in normal Chem E edjumication. [Although this weekend I surprised myself by reading large sections of an undergrad textbook straight through (Optics, by Eugene Hecht), finding it only as basic as necessary and quite enjoyably written. I may be nuts.]

Point being: you're probably better asking Michael. I haven't personally read Dr. F., but understand that he's great on all levels.

Schmutzie said...

"Surely You're Joking Mr. Feynman" is a must read, and I think you already read it John, right? Also, "What Do You Care What Other People Think?"...both co-written with Ralph Leighton.(CalTech student/best buddy)

"Genius" was a pretty good bio, but there's nothing like reading those first two. Not much in the way of nuts and bolts physics, but loads of insight into how he used his brain as a tool.

Greatest story from Surely You're Joking...is how he won a bet with fellow Princeton students who were involved in a serious discussion about whether or not gravity was necessary for urination. Some of the Princetons thought is was, and RPF proved them wrong by standing on his head and taking a piss.

The Feynman Lectures must be out on audiobook by now.

LentenStuffe said...

Yes, read that. And the "Six Easy Pieces", plus the "Pleasure of Finding Things Out" and the Letters. I just missed out on the Three Volume Lectures on Physics on eBay. I hear that's quite accessible. I did physics for the Leaving Cert in school, and got honours, but that hardly amounts to anything, even though it was a great struggle. I think I have the Gleck bio., but not the Sykes. The man was a sheer force of nature, and had what many scientists of his stature lack, common-sense yoked to a good sense of humour. John Nash [at least going on Nasar's bio.] hadn't much of either, but of course he was ill.

Anyway, Keith, you must read Feynman. Michael's far more au fait than I, in fact he turned me on to him, and his judgment is spot-on ... Besides, he's liking Patrick McGinley.

Lastly, have either of you checked out this fellow -- James Healy? some superb writing, if sometimes a lot right-wing.

Schmutzie said...

Keifus~

John knows all of this, but you might find it interesting to know that Feynman wanted to refuse his Nobel in '65. He hated honors. He also hated people in "fancy uniforms."

Artist, drummer, puzzle freak, Manhattan Project to Challenger explosion, quite the life he lead, outside of the Nobel for developing the Space-Time view of Quantum Eloctrodynamics.

Very much a nonconformist, probably why I admire him so much. (And his pot smoking too, I liked learning that.)

When you have a few minutes to blow...(why can't I post links here dammit!!!)...

http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html

LentenStuffe said...

Michael,

Use this <(no space)a(space) href="followed by the http//:link, then close the lot with bracket>type the word or phrase to be highlighted here, then finish off with<(no space)/(no space)a>

I just type the whole thing

Schmutzie said...

Feynman Test

Schmutzie said...

Oh you are the man!!

Thx brother.

Schmutzie said...

Oh and, since you snuck "au fait" into your compliment, and since Feynman signed all of his art Ofey as a play on that, I call that the clever compliment of the year JG.

Keifus said...

Well, I got Six Not-So-Easy Pieces at the bookstore on my way back from my morning trip. (It had a higher fraction of stuff I wanted to read about than Six Easy Pieces--but I have to say I'd have rather picked and chosen my pieces, dammit, because at $27.90 a dozen I didn't want to buy all of them.)

I've not read Mr. Healey, nor McGinley, but will endeavor to do so at some point or other. There's been limited time this year for all the time-wasting I want to do.

Thanks, guys.

LentenStuffe said...

You're welcome, Michael. You got that other ref.

Happy reading, Keith.

artandsoul said...

Okay so today I went on a tour of EBR-1 in Arco, Idaho.

I got to see where Zinn made electricity from fission for the first time. Saw the four bulbs.

Sort of understood it.

I have a picture of my husband pushing the red "SCRAM" button.

So, does that qualify me to read and post here??? Cuz otherwise I'm doomed.

:)

Keifus said...

Evidently, I've disqualified myself too. (Been busy.) But really, if criteria of understanding exist for posting here, then I'm sure I've failed them all over and over. So have at it if you're so inclined.

(I hope they disconnect those red buttons for tourists. Far too tempting.)