# Thread: Destructive interference - where does the energy go?

1. Originally Posted by James R

My question is now: In the centre image, where the pulses exactly overlap and the string is flat, where is the energy that was in the two pulses in the image immediately before that one? And where does the energy come from to reconstruct the two pulses after the time when the string is flat?

In this situation, let us assume that the string is very long and the sources of the original pulses are very far away from where the pulses overlap.
Seriously, James, this whole business is *exactly* what's called "transmission" in electronics. Forgive me if it sounds as if I'm being condensending, that's not my intent in the least.

The string is nothing more than an transmission line - and in physics is identical to a transmission line leading from a radio/TV transmitter to the antenna. Or a coax tube in the older versions of computer networks.

In order to transfer energy OUT of the line, it has to do what's called "terminate in the characteristic impedance of the transmission line." Otherwise, some (or all) of the energy is reflected back to the source and results in standing waves being created in the line. And that's what's happening with your string tied to a tree or whatever. I could go on about SWR (standing wave ratio), etc. but it's really not necessary for this simple example.

In the case of electronics, the standing wave is eventually dissipated as heat as the energy reacts with the electrical resistance of the line. In the case of your string, there are TWO things presenting resistance - the flexing of the very string itself and the resistance presented by the air the string is vibrating in. And your "missing energy" in those "flat spots" - more accurately called "nodes" - has been transferred into the standing waves.

And that's really all there is to this whole story - seriously, that's it, nothing more.

I'm not interested in what happens to the energy eventually due to dissipative processes. I'm interested in what happens to it while the string is flat and there's no visible wave on it. And, as I said before, how does the string "know" how to reconstruct the two pulses from the flat string, and where does the energy come from to do that?

3. Originally Posted by James R
Pete:
I can't see the second image - it gives me a "no permission" message.
Whoops... try again

That makes some kind of sense. So the "missing" energy is perhaps absorbed by the sources.
Maybe. Now I'm confusing myself thinking about a second shaker that is attached to the string at only a single point.

If it moves at all, it has to transmit some energy down the string.

But if it doesn't move (ie if it just applies an equal and opposite force to that applies by the incoming wave), then it will just reflect the wave back to the other shaker, resulting in a standing wave.
In that case, neither shaker is putting energy into the string, because neither is applying a force over any distance.

(Thanks, Pete.)

My question is now: In the centre image, where the pulses exactly overlap and the string is flat, where is the energy that was in the two pulses in the image immediately before that one? And where does the energy come from to reconstruct the two pulses after the time when the string is flat?
Oddly enough, Farsight almost had it, he just had tensile and kinetic energy backward.

The string is only flat instantaneously. It is not motionless, but moving sideways. The wave energy is in the kinetic energy of the string.

Just as when the string is motionless (at the wave peak), the energy is in the string tension.

4. Originally Posted by James R
[b]

Tach:

Once again, I have to doubt that you actually have an answer. You seem quite incapable of giving anything but a vague response to physics questions. Just leave this thread alone if you don't know.
Sure, I do, you need to stop being your offensive self and think.
Since the two effects (destructive and constructive) always happen in pairs depending on the phase difference, it means that the energy that is "missing" at the destructive nodes is being "added" at the constructive nodes. The energy is conserved.
The only exception is the case where the two components are in anti-phase everywhere (as in Pete's counter-propagating stationary waves). If you look carefully, even in that case, the cancellation is only instantaneous and alternates with the doubling of the amplitude, so the energy is conserved in that case as well.

5. Originally Posted by Pete
Whoops... try again
Nice illustration, Pete.
To be absolutely correct, the cases of constructive interference would need to show twice the amplitude of the components. This is not the case in the animation above.

6. Originally Posted by Tach
Nice illustration, Pete.
To be absolutely correct, the cases of constructive interference would need to show twice the amplitude of the components. This is not the case in the animation above.
It's not mine (I pinched it from a physics site), but it looks to me like the combined wave does have twice the amplitude of the components.

7. Originally Posted by Pete
It's not mine (I pinched it from a physics site), but it looks to me like the combined wave does have twice the amplitude of the components.
Yes, on further examination is seems to have.

8. I would suggest an experiment, you don't need the beams to be actually facing each other, you could do a point experiment over several regions to determine the actual results by setting up a region of space where the interference occurs as a result of two beams passing at 90 degrees to each other.

9. Originally Posted by siphra
I would suggest an experiment, you don't need the beams to be actually facing each other, you could do a point experiment over several regions to determine the actual results by setting up a region of space where the interference occurs as a result of two beams passing at 90 degrees to each other.
If you use spherical light wavefronts you would be getting alternating areas of total darkness and areas where the light intensity is doubled.

10. Looking at a single wave segment propagating in one direction (>),
the wave peak (>) complimenting the wave trough (>). The remaining slopes of the wave are degrees of counter direction (^) (V).

A "wave package", consisting of two wave segments propagating in entirely opposing directions, likewise, would display complex levels of counter direction of the package.

11. One way to answer this is to use water waves. We begin two wave generators on the opposite sides of a tank. If you do it right, we can make the left side crests-valleys cancel with the reflected right side valleys-crests, while pumping energy into the system. The way to make the energy appear out of nowhere, is to place a partition in the tank so the two wave fronts are physically separated. Then the energy will rise and sink out of the calm tank.

12. At the point both traveling pulses cancel, you have a superposition of waves in an elastic medium of equal amplitude and opposite phase, so wave mechanics says the energy in both waves results in a 'relaxed' state, momentarily; the central image is of a linear 'solution' to the equations of motion (?)
Both waves maintain velocity and direction, so it's just an additive 'result'.

The string's displacement is because of the 'sound' waves traveling along it? So at the point of superposition, the string isn't 'making a sound' . . ?

Hmm, apparently it's important to remember that the material isn't propagating, the state of motion of the material is. The kinetic energy is 'transmitted' from place to place, as ReadOnly states it's connected to effective impedance of the source and sink. That means you want to look at oscillator impedance in an elastic string (?).

I must have a coffee, back later.

13. The energy is converted to potential energy and then back to kinetic energy. The total energy never changes. even when you let go of the rope, if it's moving the rope snaps.

Code:
U sub e = -⌠ - kx dx = 1/2 kx²
⌡
thats my guess on the "perfect" one dimensional analysis anyways.

can i do latex here? if so how?

14. Put your LaTeX code between  tags.

15. If the waves are moving in opposite directions, you get the resulting standing waves, whereas if you set something up to create a second wave travelling in the same direction but 180 degrees out of phase, then the two waves cancel and you get a static string. In any case I believe the potential energy of a sine wave should be, like its kinetic energy, proportional to the amplitude squared. So you'd want to look at the square amplitude in order to check for energy conservation, not just the amplitude itself.

Anyhow, I know the energy in the string must somehow dissipate when you have some external force creating two mutually-destructive waves travelling in the same direction. It wouldn't surprise me if the total energy isn't conserved in the standing wave case, either. Anyhow, my answer to the OP is that whatever energy is gained or lost by the string must be provided or absorbed by whatever it is that's altering the waves on the string. If I have a wave on a string and I use my hand to create a second wave which precisely cancels the first one, then the lost energy has been transferred to my hand and dissipated from there to other things such as Earth itself. I think a similar explanation explains the energy lost or gained when you start with one wave and then produce another one travelling in the opposite direction.

16. Originally Posted by chris25
can i do latex here? if so how?
What James said.
More detail:

17. Originally Posted by James R
I'm not sure exactly what you're saying here. In what form is the wave energy when the string is flat?
I called it pressure, but maybe I should have said pressure/tension. Remember I said imagine the string to be made up of a series of spheres connected by coil-springs under tension ? That relates to Hooke's law:

Consider a mass on the end of a spring, oscillating. When the spring is at maximum compression there's no kinetic energy, only potential energy, and the spring is under pressure. When the spring is at maximum extension, again there's no kinetic energy, only potential energy, and the spring is under tension. Either way, that potential energy is in the electromagnetic field between the atoms. So in the end the flat-string wave energy is electromagnetic in form.

Originally Posted by James R
My question is now: In the centre image, where the pulses exactly overlap and the string is flat, where is the energy that was in the two pulses in the image immediately before that one?
In the string. I said it was pressure, but mmm, maybe it's a combination of pressure and tension. But it's definitely in the string. There's nothing else there, and energy is fundamental, you can't create it or destroy it, conservation of energy always applies.

18. When the waves cancel, the medium is 'moving' in both vertical directions simultaneously and the 'forces' cancel each other. The medium isn't displaced horizontally, but pressure must be transmitted in both horizontal directions.

The applied forces are in phase with the velocity of the medium's vertical displacement, so that's in a state of equilibrium but the horizontal displacement of each wave pulse is not in phase. So the horizontal displacement is not 'attenuated' at all by the resistance and the medium preserves the displacement wave(s), because of reactance, and the waves appear to move 'through' each other although they are only traveling along the string 'independently' (??). It really is just the addition of vectors in a rotating phase diagram?

It's a little more complicated; the applied force vectors that generate each pulse are out-of-phase with the displacement (amplitude), the velocity, and the acceleration of each pulse along the string. When the two meet, the vectors cancel in one dimension. Reactance is perhaps the door that leadeth to understanding? It includes the elastic constant, k, and the angular frequency (or in the case of the example, the inverse wavelength), yeah?

Because there is no visible reactance--the string is not vertically displaced by the applied force in the middle diagram--then $m \omega_f = \frac {k} {\omega_f}$, and reactance is "effectively" zero, the string is in a relaxed state of motion. But there are two opposing waves (I suppose they are pressure waves) moving in opposite horizontal directions continuously.

phew.

19. Keep in mind that the string exists in two dimensions.

Also keep in mind that you may be asking the wrong question - do the waves propogate past each other, or do they reflect off each other.

I also suspect that only half of the story is being considered. I don't think I've seen anybody mention the angular momentum of the string - and I suspect that's because people are looking at the wrong moment. The important moment is when the two waves are just starting to overlap, that instant when they form an 's' bend. My instinct is that the answer lies in that instant.

Consider this:
The point where the two waves precisely overlap, and have zero amplitude is a node. You could stick a pin in that point and it would have zero influence on the outcome of the experiment, however, I'm fairly sure it's pretty obvous how it would affect a single pulse travelling along the string. The pulse would reflect off the pin, back to the source, and a smaller wave would probably be propagated in the opposite direction.

Consider what happens when the waves just start overlapping.
The amplitude starts to decrease, but what does that actually mean.
If we draw a line, from peak to peak, and take a series of snapshots, as well as getting shorter, the line begins to rotate. The elastic force of the string does work, and converts the linear kinetic energy to rotational kinetic energy, and the points on the surface of the string begin to rotate about the nodal point. The instant when the string is perfectly flat, all of the linear velocity, and linear energy has been converted to its rotational form, and the stretch of string has angular momentum. The individual component 'bits' of the string continue rotating until some critical tension is reached, at which point more work is done by the elastic tension in the string, and everything is converted back into it's linear form, only the vectors have been rotated by 180°.

I hope that makes sense to somebody other than me, but it seems to me to be the most intuitive way of explaining the observations. I'm also fairly sure that if I sit down and have a think about it, I can probably derive some math to quantify my qualitative explanation.

(and it seems that while I have been formulating this reply, and trying to envisage the scenario mentally, Arfa has come to strikingly similar conclusions).

20. Originally Posted by arfa brane
At the point both traveling pulses cancel, you have a superposition of waves in an elastic medium of equal amplitude and opposite phase, so wave mechanics says the energy in both waves results in a 'relaxed' state, momentarily
That's the reduced tension I was on about. Nice to hear your mention of impedance along with read-only talking about a transmission line. I wonder if we'll get on to vacuum impedance $Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}}$ along with electromagnetic waves, displacement current, and vacuum energy? The photon is a pressure pulse, space acts like a lossless line wherein $Z_0 = \sqrt{\frac{L}{C}}$, that kind of thing.

Originally Posted by Pete
Oddly enough, Farsight almost had it, he just had tensile and kinetic energy backward.
I don't rate kinetic energy, it isn't fundamental enough. Show me a moving mass and I'm boiling it down to a single hydrogen atom then annihilating the electron with a positron and annihilating the proton with an antiproton. Then I'm back to electromagnetism, and telling James that the energy is in his flat string is still there in the electromagnetic field between the atoms, like in the compressed then stretched spring in the harmonic oscillator.