can't resist.
Wow, I'm nervous to post in this thread, given the crazy ground covered, but I find that I can't help myself, for a couple reasons:
- chaos attracts me, and this thread is nutso
- I strongly agree with crashsite (if I may put some words in his mouth) that there is a dangerous tradition among intellectuals (big and small, but maybe more often among the small ones) to equate mathematical modeling with understanding; i.e. "I understand the manipulation of an abstract grammar, and have a clear picture in my head of that intangible symbol-machine, and I can use it to make accurate predictions about the physical world; the nature of that world in front of my tangible eyes presents no psychic discomfort to me because I have a comfortable internal metaphor for it, even though that metaphor is essentially nothing more than an equation that happens to fit the data set". There must be a classic name for the difference in philosophical school between those people and others who want something more to hold on to. I'm not talking here about spiritualism, that's obviously a different subject, and I'm not implying that crashsite demands that any property of the universe fit inside his brain or else he eyes it suspiciously. I'm suggesting that maybe some of the string theorists and particle physicists get a little caught up in the math and the empirical correlation of their data. Since there is only a handful of people on the planet that understand
that math, I'm obviously speculating, but even my high school physics compatriots fell into the trap... Seems like the world of electronics and acoustics are especially rife with this: "but
why does 3-phase 240V blah blah reduce the current in blah blah inductance blah blah?" Answer: "well, simply apply Ohm's law like so..." It's circular logic: they're saying "phenomenon X happens because of equation Y that was developed based on the fact that phenomenon X happens".
It's taken 12 pages so far and crashsite is still unsatisfied. Far more informed minds than mine have attempted answers, but I had to throw in my hat!
(*Let me say that I in no way accuse intellectual people across the board of that stuff, and obviously in some cases equations not only predict results but function as powerful -- and obviously useful -- metaphors for the physical world.)
Here's my own head-model of sound waves, for whatever it's worth. Some or all of this may be misguided. I'm no expert, I just love an explanation challenge:
With air, I imagine a space filled with particles that are repelled from each other (and by virtue of those repelling forces, effectively "attracted" to voids).
These molecules are "jiggling" very rapidly: they are moving, and the kinetic energy they possess (temperature, I'm told) has them traveling in one direction a very short time only to bounce off the field of a neighbor. This motion, which I'll call "random" at the risk of falling off an existential cliff, doesn't result in displacements or shifts on any kind of large aggregate scale. I think in the "crowded room of people" metaphor (not a great metaphor), this is everyone shuffling around in place, but not walking anywhere. If everyone just so happened to shuffle in the same direction at the exact same time, that'd be a different story, but it's a statistical wash, and no such thing happens.
I hold that the dynamics of the repelling and void-seeking are responsible for the particulars (putting aside who-knows-what myriad other sophisticated forces at play), and that this is a knowable system by non-mathematicians (like me, and crashsite). Now, this seems to be the crucial point for crashsite: clearly, our repellent molecules are going to shift when perturbed, but why would they shift the way they do, at the speed they do, etc.
I liked the "long rod" image brought up before. Let me modify it:
Imagine a series of foot-long rulers laid end-to-end for hundreds of feet. There is a one inch gap between each. Pushing the first ruler at a steady rate means that the "gap closing" will move at a much faster rate along the line of rulers than the original push does. (Move three inches, and the "gap closing" is already happening ~3 feet away, a factor of ~12 "faster"). This shows nicely how the propagation rate doesn't relate in a simple way to the perturbation's rate. Of course, these rulers were massless and frictionless.
Now, abstracting a bit, instead of there being a hard edge to the rulers' interactions, their edges have a magnetic repulsion to each other. The same effect can be seen, but no contact is needed.
However, these rulers also have mass, and therefore inertia, and they take time to move away (ignore friction in this universe). You can see that after pushing the first ruler an inch and then holding it there, time is taken for the subsequent rulers to move to their preferred distance. They are still moving one at a time, but now it's kind of blurred: the first ruler is pushed maybe half an inch forward and just held there. As the first "push" ruler moves, the second starts to move away, and the third starts to move as well, before the second has totally settled into its final position, etc.
The interplay of these dynamic forces is crucial: how strong the "spring" repulsion is, how far out its effect reaches (e.g. can a given ruler "feel" just the neighboring ruler, or two rulers away as well?), how much inertia the ruler has, how fast the initial push was, and so forth.
But it's clear that in this line of rulers, if one was to push the first and watch this "wave" propagating down the ruler chain, and one wanted to then take that wave back, one is lost: that wave is out of reach of the first ruler. It can only generate new waves that will follow the original.
Why does the "wave" propagate? It's simply the system of rulers trying to rebalance itself. Or, more microcosmically, it's the result of localized imbalances of forces: they propagate directionally, in this case, because that first ruler has set a new edge they must react to. The first ruler may move back in the future (like a speaker cone returning to rest or beginning a rarefaction cycle), but the propagation has already happened.
If one pushes the first ruler, waits for the second ruler to start to move, but then releases the first ruler before the second one has finished moving, the first ruler starts to move backwards a bit because of the mutually-repelling force from the second ruler, which has not yet achieved its preferred distance from the first, even though it's on the move. So if one wants to wait until there is no danger of the first ruler moving backwards, how long must one hold the first ruler in place? Even with an infinite line of rulers, one must wait forever, actually, since the backwards force is a limit approaching zero. (Back on Earth, of course, there is a noise floor of jiggling, there is damping, and all kinds of things happening that mean you don't have to wait forever.)
Note also that jiggling a given ruler a tiny bit side-to-side, very fast, (e.g. less than a millimeter, at 5000 Hz), has almost no effect on adjacent rulers, because they are too sluggish to react in an appreciable way (though they do of course react).
Note also that after pushing the first ruler and holding it, the second ruler may accelerate on it's way to the third ruler, and the gathered speed may actually result in the second ruler "bouncing" back from the third ruler, depending on the springs and how they function over distance, etc. As a result, the character of the "push" of the first ruler (sharp? slow? happening in gradual steps?) may not be perfectly translated into the subsequent rulers, depending on all these factors, and the balance of masses and springs and so forth may result in "bouncy" behavior amongst the rulers instead of a clean "wave" going down the line. It all depends on the parameters of the rulers and the nature of the perturbation.
If these rulers also have an attracting ability when they get too far from each other, you can see the analogous effects when creating a rarefaction with the first ruler.
The continual interplay of these forces is obviously "complex", and hopefully it's clear that it can result in behavior that is difficult to wrap one's head around, or at least to predict casually. For example: rulers at opposite ends of the chain are moved, and two waves come towards each other. Visualizing how they pass through each other without affecting each other is kind of a head-trip, but it works out if you play it out.
Thinking of this cascading effect moving down the chain is a human-applied metaphor: is it a "wave", "moving through a medium"? Sure, whatever, Human! It's stuff reacting to stuff. So when those two waves in the ruler chain come at each other, are they "passing through each other without affecting each other" or are they "bouncing off each other" or are they trading some energy, dancing the Charleston, etc? We can look at it how we like, but it's stuff reacting to stuff.
The first ruler may move at a certain rate, but the second ruler moves at whatever rate the spring/inertia system dictates as a result of the first ruler's movement (may be slower, may start out slower but then actually get faster than the original movement as momentum gathers, etc). The third ruler is again once-removed from the nature of the original push. Conceptually, the energy of the initial push lasts forever, but as it travels down an infinite ruler chain, it gradually fuzzes out (the area of compression or rarefaction becomes wider and shallower as it travels) until it gets lost in the jiggling. There is still a real "push" happening through the system, but it's so small as to be unmeasurable. (Of course in the real world of energy bouncing around there's a lot to get lost in.) This maybe also explains why lower frequencies travel further, since the lower the frequency the less it depends on the next ruler to "catch up" to the movement in time to represent.
The crucial point is: the interplay of forces and masses involved (the nature of the attraction/repulsion springs, the mass of the rulers and their concomitant inertia, the average gap between them) results in the "speed" of the "wave" down the chain, results in what energies and frequencies we consider "jiggling" and what we consider "displacement", and results in the rate at which the character of the initial push is lost as it "fuzzes out", and so on. If you tweak these variables you can see that the system is going to act wildly different: imagine mile-long rulers, or millimeter-long rulers; imagine springs that keep the rulers a mile apart, or just a millimeter, imagine springs that don't have a linear relationship of force-to-compression, imagine rulers spinning around, and so on. This tweaking results in very different pictures of how different initial pushes are treated by the system, whether or not they propagate, and how.
I think this all extrapolates pretty clearly from one to three dimensions. Things certainly get more complicated, but it's the same ideas. An air molecule's graduated "sphere of influence" is the volumetric extent of its repellent field (which has a measurable size, just as the ruler and the ruler's magnetic repulsion has a measurable width). "Air pressure" is the degree to which all those molecules are pressed together, compressing the springs. Temperature is the average jiggling kinetic energy the system already has as they bounce off each other (or at least shake around a bit) in no particular coordinated direction (I'm sure there's something wrong with my definition of temperature, there, but suffice to say...) The idealized compression "wave" is a disturbance of a sufficient number of those jiggly molecules in aggregate that radiates out from its source as a growing sphere. It's all springs and masses bouncing around. (Why does sound travel faster or slower in X, how does temperature affect it, etc. Given all the above, these questions are clearly going to be non-trivial to answer, but perhaps educated guesses could be made just from the above descriptions.)
So the essential question, of "how and why does sound propagate through air" perhaps could be answered thus: The balance of the forces of repulsion of molecules from each other, the character of that repulsion, the inertia of the molecules, and the average spacing they have from each other, results in a dynamic system of particles amongst which disturbances of appropriate magnitude and speed (if we're speaking cyclically, frequency results in speed) will result in perceptible propagation of a "wave" through that system. Disturbances of too-little magnitude, or too-high frequency at a given magnitude, or disturbances that for other fluid-dynamic reasons result in non-directional turbulence instead of coordinated compression/rarefaction, won't."
As to "how is the perceived integrity of the sound maintained through space", that answer also seems to follow... if you grok how one "wave" through the rulers model could "pass through" a second wave coming the other direction, then it should be clear that whatever pattern of pushing/pulling happens on the first ruler is rolling down the line of rulers (again, crucially, tweak the springs and masses in your mind until this becomes plausible). The waves bounce off the theoretical wall at the end (the last ruler is repelled by the wall, of course), and the "reflected" "waves" "pass through" each other on the way back to the first ruler where, if they haven't been overly damped or otherwise degraded, they reasonably replicate the motion of the original ruler (actually, the opposite of the motion, but it's the same difference with sound).
I think the crux of your confusion is your internal tweaking of the springs, inertia, etc, of air molecules. Your internal system's variables are not dialed-in in a way that lends itself to seeing this phenomena happening in air. Try being playful with the variables and see if it doesn't make more sense.
Nervously awaiting your response,
-C
P.S. It's funny that I found this thread, because I was recently running through some calculations about speaker cones: I was wondering how close the front of a speaker comes to the speed of sound when moving. Turns out, not very close (at sane sound pressure levels, anyway). Phew. Sound is definitely a trip, for all kinds of reasons. The aspects that confuse me are the frequency-domain type analyses: talk about people using equations as bogus explanations, geese.