(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

[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?