(no subject)

[gusl]

Last night I introduced myself to my upstairs neighbour at 12:15AM, to friendlyly ask him to turn his music down (especially the bass).

I like thinking about acoustics, especially when I want to insulate myself from noise.

So I've been thinking about noise cancellation. The idea is that you copy the incoming noise (possible, since the signal can travel faster than sound) and reproduce out of phase by half a wavelength (or to be moe precise, in phase but with the amplitude inverted). So you're putting more energy in. But by the principle of energy conservation, either the sound gets louder in some places or the waves will all get turned to heat.

Btw (1), I have an excuse to leave my laptop on: saving gas! When I close my laptop, it stops emitting light outside, so all of its energy expenditure becomes heat. Since everything is regulated by a thermostat, my laptop saves a house radiator from getting warmer (though gas energy is probably cheaper).



Btw (2), I've once used a similar thought to prove that

amplitude is additive, energy is conserved in a closed system, energy is a function of amplitude |= energy is proportional to amplitude^2

(the argument was about a light interference pattern)


though perhaps I should refrain from using "|=" until I have a formal proof, or at least formal models for this stuff.

My physics professors were impressed with my "proof", but I just thought (and still think) that this should be normal science. Unfortunately, such a logical approach is missing from science education (and probably research too). Taking my physics classes as an example:
* premises are almost never explicit
* the structure of arguments is informal (even *more* informal than in mathematics classes)

This lack of formality doesn't bother them, and in fact in most cases.

But Feynman tells the story of the S-shaped sprinkler, where people had to resort to experiment in order to resolve the question. But it's the sort of question that should be decided by theory.
Even though there might be fluid effects that are not covered by the theory, the question was meant as a brainteaser: I believe Feynman was asking about the "ideal" sprinkler.

Comments

[altamira16.livejournal.com]

There were many theories on the S-shaped sprinkler problem. Experiment lets you know which theory holds. You should read the section about this in James Gleick's bio on Feynman.

[gustavolacerda.livejournal.com]

The point is that many of those theories could be refuted without doing the experiment. (I imagine so, anyway)

If physicists used a cleaner formulation of their theories, if they had an organized logical framework in which to do their deductions, they might not disagree as much and thus avoid doing expensive experiments.

[altamira16.livejournal.com]

How do you test the validity of the underlying assumptions of any theory?

[gustavolacerda.livejournal.com]

experiment.

But the point of having a theory is that you can generalize. This way, you don't have to do an experiment for each little phenomenon you want to predict.

I would bet that the experiments establishing the laws that govern the behavior of the sprinkler in the puzzle had been done at least a century before Feynman posed the question (and I think everyone working on the problem agrees this was a problem of analyzing known laws, not a problem of discovering physical laws that might be at play).

I think the source of the "puzzle" is that there are conflicting intuitions. That's why it was considered interesting.

If physicists relied more on a coherent logical theory than on their intuition, they would have quickly reached a consensus on it, without any experiments. But such a logical theory doesn't exist yet.

[altamira16.livejournal.com]

There are some theories that do extend science to a certain point, but the interesting problems are the ones that are not predicted by existing theories. Also the number of theories that are at work in given phenomena make problems interesting. Wasn't mathematics historically very sloppy until someone came along and introduced systematic ways of doing proofs?

[gustavolacerda.livejournal.com]

but the interesting problems are the ones that are not predicted by existing theories.
This experiment is hardly the kind of experiment that would exhibit the existence of new phenomena. Do you agree that *some* experiments are not worth doing?

Also the number of theories that are at work in given phenomena make problems interesting.
I agree here. I find it interesting to see how different theories and explanations can interact coherently.

Btw, this argument reminds me of the debate about the Monty Hall problem... some mathematicians were so blinded by their intuition (that the probability should be 1/3 regardless of whether she switches), that they could only revise their beliefs when faced with experimental results (probabilistic simulations). It makes me wonder why they didn't try to convince them with a formal set-theoretical proof. Mathematics, afterall, has such a "logical framework", that I keep referring to.

My dream is to see a physicist showing a computer proof: "See, if you accept conservation of energy, blah-blah-blah and etc; you *MUST* accept this theorem" and no-one could argue.

[spoonless.livejournal.com]

If physicists relied more on a coherent logical theory than on their intuition, they would have quickly reached a consensus on it, without any experiments. But such a logical theory doesn't exist yet.

Many times, though, we run into several different theories which are all logically consistant and the only way to sort them out is by doing experiments. It sort of works in a cycle... spend a while making the theory make sense and be consistant, then spend a while figuring out what other parts of reality we haven't discovered yet which need to be added to the theory.

At present, there is a lot of focus on trying to figure things out sheerly from logical consistancy (primarily, in constructing a theory of quantum gravity which is the next step). But the way it's seen by most physicists, I think, is that this is the slow way to figure things out and we're only doing it this way now because the experiments we want to do which would give us the clues to speed things up can't be done yet due to lack of technology and resources (like a huge amount of energy).

[gustavolacerda.livejournal.com]

Many times, though, we run into several different theories which are all logically consistant and the only way to sort them out is by doing experiments.
I don't think this is relevant to my point. The problem isn't about which theory is "correct", but about predicting a specific phenomenon where all the underlying principles are understood.


Doesn't it seem odd to you that scientists sometimes find it easier to perform huge experiments than to work out the theory? Maybe I expect logical omnipotence... my spiel is that while computers are widely used in simulations, etc, their potential uses in theoretical work are ignored...

[spoonless.livejournal.com]

I don't think this is relevant to my point. The problem isn't about which theory is "correct", but about predicting a specific phenomenon where all the underlying principles are understood.

Well for the most part, I think, that's how it works. If we already understand the underlying principles for something then there's no reason to do the experiment. Experiments are usually chosen to test the limits of a theory, or to find new things which were never predicted. Even if it's an intractible calculation by hand, simulations are usually done before experiments if the physics is already well-understood.

But there are situations when we still need to do experiments even though it could in principle be calculated... sometimes it's because there are so many different factors going into it that we're not even sure which ones are dominant. Other times it's because the calculations would just be too heinous to work out, even with the aid of a supercomputer. (Yes, this does occur sometimes.) For instance, in theory we should be able to calculate what the properties of each of the 92 natural elements on the periodic table are just based on how the electrons move around the nucleus. But in practice, this is only done for hydrogen and helium, and maybe in very rare special cases for other elements where we can make a lot of approximations and ignore most of the behavior. For the rest of the elements, it just gets so complicated that it's easier to ask a chemist to test the element and see how it behaves rather than try to work it all out from first principles. It's not a matter of laziness, it's just that the task is very very difficult and complex. There are simple physical processes which would take millions of years to calculate on an extremely fast supercomputer. This is one of the reasons quantum computers might be important for the next generation--because classical computers simply can't handle the complexity of modern physics.

I don't know exactly what this "sprinkler" problem is, it sounds vaguely familiar. But I would naively think the same as you, that in that case it should be simple enough to figure out doing the experiment.

my spiel is that while computers are widely used in simulations, etc, their potential uses in theoretical work are ignored...

They're actually starting to be used quite a bit. But I agree, they should be used more. The research I did this summer was all computer simulations... of statistical mechanics that's too complex to calculate by hand but works nicely on a fast computer and is a lot easier than going and doing the experiments. We just got the first draft of the paper written, so it'll be submitted soon for publication.

[spoonless.livejournal.com]

But I would naively think the same as you, that in that case it should be simple enough to figure out doing the experiment.

I meant to type "to figure out without doing the experiment."

[kvschwartz.livejournal.com]

What turned out to be the answer to the S-sprinkler problem?

[gustavolacerda.livejournal.com]

well, it seems people still don't agree.

But if you ignore fluid effects, then the sprinkler shouldn't move at all.

Anyway, this is the most reasonable answer I've found:

A friend and I discussed this one a long time ago and I think we decided that the steady state answer is that it wouldn't spin, but that in turning it on you'd get a small kick spinning it backwards which would die out due to the viscosity of the water. This was for a model which used pipes that went out radially and then bent. If you allowed the bent part to start at the center then its possible you'd get some steady state motion. In that case the hose could carry away water which still had angular momentum whereas in a long radial section the angular momentum would bleed away against the sides of the pipe.

and this is the coolest argument:

Mechanics is supposedly "time reversible". In other words, it does not depend on the direction of time.
So, lets say that the sprinker is always submerged in order to make the two different situations (water in vs water out) look time reversible.
Then clearly under time reversibility, if the sprinkler goes in one direction in one instance, it must go in the other direction when we reverse time.

[kvschwartz.livejournal.com]

So no one has actually bothered to carry out this experiment?

[spoonless.livejournal.com]

So I've been thinking about noise cancellation. The idea is that you copy the incoming noise (possible, since the signal can travel faster than sound) and reproduce out of phase by half a wavelength (or to be moe precise, in phase but with the amplitude inverted). So you're putting more energy in. But by the principle of energy conservation, either the sound gets louder in some places or the waves will all get turned to heat.

I don't think heat has much to do with it; it's just that being "in-phase" or "out-of-phase" is a function of where in the room you're standing. You can only make them out-of-phase for particular points (or perhaps a central line if you set it up right). Near those points, the sound will be diminished. Near other points, it will be greatly enhanced. So overall, energy is conserved. (I think you were mostly implying this, I just wanted to expound on it in case it was unclear).

amplitude is additive, energy is conserved in a closed system, energy is a function of amplitude |= energy is proportional to amplitude^2

hmmm... are you leaving out the logic here? How does being additive imply energy is proportional to the square of the amplitude? I'd be curious to hear that, if you have a connection between the two.

[gustavolacerda.livejournal.com]

I don't think heat has much to do with it; it's just that being "in-phase" or "out-of-phase" is a function of where in the room you're standing. You can only make them out-of-phase for particular points (or perhaps a central line if you set it up right).
hm... ok. That's what I thought also, but when I saw a noise cancellation device in the Nemo museum here in Amsterdam (two speakers), I didn't manage to find a point of constructive interference. Could they block the sound from those points? (if so, it would turn to heat anyway)


How does being additive imply energy is proportional to the square of the amplitude?

Construct a model (in the logic sense), i.e. imagine a situation where you have two light sources close to each other: L1 and L2, of amplitude S1 and S2 respectively.

If energy is a function of amplitude, we can give it a name: f.
So the energy of L1 is f(S1) and likewise for L2.

Look at a wall (perhaps a circular wall around the lights). You are going to get an interference pattern with the highs at S1 + S2 and the lows at S1 - S2 (WOLOG let S1 be the stronger one), since amplitude is additive.

At the highs, the energy density (or power density!) of the point will be f(S1 + S2), and at the lows it will be f(S1 - S2)

By conservation of energy, f(S1) + f(S2) = the average of f(S1 - S2) and f(S1 + S2)... of course, the exact curve matters, but in the end I think we can say this is (f(S1 - S2) + f(S1 + S2)) / 2 .

f(S1) + f(S2) = (f(S1 - S2) + f(S1 + S2)) / 2

After some non-trivial math, you arrive that either f(x) = 0 or c * x^2 for some c.

If you work out the details, please show it to me.

[spoonless.livejournal.com]

Hmm, I'm not sure about some of that, but it looks very clever. Assuming all your assumptions are right up to the last line... I'm not sure those would be the only two functions. Maybe if you stipulated they were analytic functions (smooth in the sense of being infinitely differentiable) then you could use a Taylor series expansion to show it had to be one of the two. If all this is right, then it's probably indicative of a deeper principle going on here. But I'm not sure just what. It could be related to Parseval's Theorem, which says that integrating the square of a function over time is the same as integrating the square of its Fourier transform (very useful for proving things about the energy of waves)... or it could be related to something called the Mean Value Theorem in electrostatics; the theorem says that the potential energy of an electrical field at any point is the average of everything around it. Quite an interesting observation, though.

[gustavolacerda.livejournal.com]

I had someone prove this to me once.... we started with the integers, and then extended it to the rationals... it wasn't that much work.

Anyway, if you know any physicists who would like to help me write this up nicely, please let me know! If I ever use it, it will certainly be acknowledged.

[spoonless.livejournal.com]

hm... ok. That's what I thought also, but when I saw a noise cancellation device in the Nemo museum here in Amsterdam (two speakers), I didn't manage to find a point of constructive interference. Could they block the sound from those points? (if so, it would turn to heat anyway)

I don't know of any way they could have blocked sound from certain points. Were the speakers set up to cancel each other out, or was there a microphone set up to pick up some noise they were supposed to both help cancel out? How far away from each other were the speakers?