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# Thread:

1. ## Destructive interference - where does the energy go?

Consider two counter-propagating waves on a string with equal frequencies but 180 degrees out of phase so that there is complete destructive interference.

Each wave carries energy - one to the left and the other to the right, let's say. But when they interfere we get a stationary string and, apparently, no energy.

Where did the energy go?

I'd like to nail down the answer to this question. If you look on the internet, there are all kinds of different explanations. Let's see if we can do better at sciforums.

2. Just as a sky blue guess, I'll venture that it shows up as heat.

Instead of string. lets use a flexible strip of superconducting materiel and measure any change in temperature.

3. The string doesn't move, so there's no mechanism for generating heat.

In any case, let's consider an "ideal" string that doesn't have any friction forces on it if/when it moves.

4. Originally Posted by James R
The string doesn't move, so there's no mechanism for generating heat.

In any case, let's consider an "ideal" string that doesn't have any friction forces on it if/when it moves.
I'll give you a hint: destructive interference always happens in conjunction with constructive interference. So, where does the energy go?

5. I don't want a hint, Tach. I want an answer. I've read other internet answers.

6. Haha, this is a fun question, I remember wondering about it a few years back. I was worrying about quantum interference but I guess it is a concern classically too.

My first question would be how did you manage to create these waves in the first place? It doesn't take any energy to create the situation you describe, it is just a motionless string. You will not be able to create even a thought experiment in which you can achieve this configuration of waves from some sources I think.

7. Originally Posted by James R
I don't want a hint, Tach. I want an answer. I've read other internet answers.

You have your answer, you just need to think about it. It is right in front of your nose.

8. Originally Posted by AlexG
Just as a sky blue guess, I'll venture that it shows up as heat.

Instead of string. lets use a flexible strip of superconducting materiel and measure any change in temperature.
No heat is being generated.

9. Originally Posted by James R
The string doesn't move
Are you sure about that? When you say two waves are propagating in opposite directions, I think of something like cos(x - ct) + cos(x + ct) = 2 cos(x) cos(ct), which does move. Am I missing something?

10. Originally Posted by przyk
Are you sure about that? When you say two waves are propagating in opposite directions, I think of something like cos(x - ct) + cos(x + ct) = 2 cos(x) cos(ct), which does move. Am I missing something?
Yeah I think he would prefer they travel in the same direction:

$cos(kx+\omega t) + cos(kx+\omega t + \pi/2) = 0$

11. przyk:

Originally Posted by przyk
Are you sure about that? When you say two waves are propagating in opposite directions, I think of something like cos(x - ct) + cos(x + ct) = 2 cos(x) cos(ct), which does move. Am I missing something?
Hmm....

Maybe I need to alter the scenario to consider co-propagating waves that are out of phase by 180 degrees.

So, to spell it out, what is your answer for the counterpropagating case you describe? And then, what about the co-propagating case?

kurros:

Originally Posted by kurros
Haha, this is a fun question, I remember wondering about it a few years back. I was worrying about quantum interference but I guess it is a concern classically too.
Yes. You can think about it with waves on a string, water waves, light, sound, matter waves, or whatever.

My first question would be how did you manage to create these waves in the first place? It doesn't take any energy to create the situation you describe, it is just a motionless string. You will not be able to create even a thought experiment in which you can achieve this configuration of waves from some sources I think.
Well, co-propagating case. At the left end of the string, I have a shaker that creates a sinusoidal oscillation on the string that moves to the right (say). A little way to the right of that, there's another shaker that applies a force to the string to create the other wave, which also propagates to the right. Turn on one shaker and leave the other one off and you have a wave moving right along the string. Turn both on and suddenly there's no wave to the right of the second shaker. But the shakers are still shaking away (I assume). So what's happening to the energy they are supplying?

Or is this somehow impossible.

Tach:

Originally Posted by Tach
You have your answer, you just need to think about it. It is right in front of your nose.
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.

12. James I don't know if this is the same but if you get 2 people ( yes I know this is practically impossible, its a thought) in a vacume to push against eachother with the same force where does the energy go. Its not sound because they are in a vacume, its not light and I doubt its heat. If done properly it shouldn't deform.

13. Originally Posted by kurros
Haha, this is a fun question, I remember wondering about it a few years back. I was worrying about quantum interference but I guess it is a concern classically too.

My first question would be how did you manage to create these waves in the first place? It doesn't take any energy to create the situation you describe, it is just a motionless string. You will not be able to create even a thought experiment in which you can achieve this configuration of waves from some sources I think.
I've no idea what you're trying to say. It's an easy experiment to do, it DOES require energy and the string most certainly DOES move. It's not just a thought experiment, any kid of 5 can do the real thing.

The answer is that you will always wind up with standing waves - motionless nodes and sections between those nodes that vibrate back and forth. And the final bit is that YES, the energy in the string is converted into heat.

14. Originally Posted by James R
Consider two counter-propagating waves on a string with equal frequencies but 180 degrees out of phase so that there is complete destructive interference.
I don't quite get the picture.
Do you mean this: Or this:

15. Originally Posted by Read-Only
I've no idea what you're trying to say. It's an easy experiment to do, it DOES require energy and the string most certainly DOES move. It's not just a thought experiment, any kid of 5 can do the real thing.

The answer is that you will always wind up with standing waves - motionless nodes and sections between those nodes that vibrate back and forth. And the final bit is that YES, the energy in the string is converted into heat.
Once the wave is set up, it doesn't need energy to maintain, except to replace the damping of air resistance and internal friction. A standing wave is a (kind of inefficient)! energy storage medium.

16. Originally Posted by James R
co-propagating case. At the left end of the string, I have a shaker that creates a sinusoidal oscillation on the string that moves to the right (say). A little way to the right of that, there's another shaker that applies a force to the string to create the other wave, which also propagates to the right. Turn on one shaker and leave the other one off and you have a wave moving right along the string. Turn both on and suddenly there's no wave to the right of the second shaker. But the shakers are still shaking away (I assume). So what's happening to the energy they are supplying?
It might not work, depending on the way the shakers work.

The second shaker has to be able to absorb the energy of the first, because the force it applies is always in the opposite direction to displacement, so the energy expended is negative.

17. Originally Posted by James R
kurros:

Well, co-propagating case. At the left end of the string, I have a shaker that creates a sinusoidal oscillation on the string that moves to the right (say). A little way to the right of that, there's another shaker that applies a force to the string to create the other wave, which also propagates to the right. Turn on one shaker and leave the other one off and you have a wave moving right along the string. Turn both on and suddenly there's no wave to the right of the second shaker. But the shakers are still shaking away (I assume). So what's happening to the energy they are supplying?

Or is this somehow impossible.
Ok well I have not gone off and solved the 1D wave equation with 2 dirac delta in x but sinusoidal in t source functions, but I expect that the solution will be that yes, the string does nothing outside the two shakers, but inside plenty happens so all the energy is contained there. You can only put the shakers at integer wavelength positions to get the waves to cancel on both sides. Admittedly the amplitude of the interior waves won't increase as the shakers move closer together, which seems bad. Hmm. I might have to simulate this to convince myself. Actually maybe the amplitude of the interior wave WILL increase, we have a standing wave in there with continual energy input, I guess the solution will blow up... yes I think this is what I am going to go with.

Anyway if you place the two sources directly over each other they will cancel themselves out so you will be back to the homogeneous wave equation.
I have mostly thought about 2d or 3d scenarios and disallowed myself infinite sources, which is what the 2d equivalent of your scenario would required. I'll get back to you.

18. Originally Posted by James R
Where did the energy go?
Into the string. See this website and look at the two waves travelling in opposite directions. One is green, the other is blue. When they're exactly out of phase the red superpositional wave is a straight line. At that instant the string is flat. Freeze that instant in your mind. If you imagine the string to be made up of a series of spheres connected by coil-springs under tension, then at the flat instant, the tension in the coil springs is reduced. The dimensionality of energy is pressure x volume, tension is negative pressure, and the energy in the string has reduced its tension. If the string was stretched over an archer's bow, the ends of the bow would be starting to move apart at this point.

19. Originally Posted by Pete
Once the wave is set up, it doesn't need energy to maintain, except to replace the damping of air resistance and internal friction. A standing wave is a (kind of inefficient)! energy storage medium.
Precisely. And it's created by the reflection at the far end of the string - which is not a proper terminator (which would allow the energy to pass out of the string) with even a single source of wave input. The second input, as stated in the OP, simply adds even more energy to the system. But nevertheless, the cancellation of the waves will result in creating standing waves - where the energy will reside (just as you said) until it's eventually converted to friction by the air resistance.

Exactly what I said earlier, just with more detail.

20. Asguard:

Originally Posted by Asguard
James I don't know if this is the same but if you get 2 people ( yes I know this is practically impossible, its a thought) in a vacume to push against eachother with the same force where does the energy go. Its not sound because they are in a vacume, its not light and I doubt its heat. If done properly it shouldn't deform.
This is different. With forces, no energy is ever expended unless the object the force acts on moves in some way. So, if two people push on each other with equal forces, no useful work is done and the people don't move. They will be straining their muscles, and with all the muscle twitching going on in their bodies there will be movement there. That movement ultimately produces heat. So, the net conversion in that situation is from chemical energy to mechanical to heat in the muscles.

Pete:

Originally Posted by Pete
I don't quite get the picture.
Do you mean this: Or this:
I can't see the second image - it gives me a "no permission" message.

Originally Posted by Pete
It might not work, depending on the way the shakers work.

The second shaker has to be able to absorb the energy of the first, because the force it applies is always in the opposite direction to displacement, so the energy expended is negative.
That makes some kind of sense. So the "missing" energy is perhaps absorbed by the sources.

Farsight:

Originally Posted by Farsight
Into the string. See this website and look at the two waves travelling in opposite directions. One is green, the other is blue. When they're exactly out of phase the red superpositional wave is a straight line. At that instant the string is flat. Freeze that instant in your mind. If you imagine the string to be made up of a series of spheres connected by coil-springs under tension, then at the flat instant, the tension in the coil springs is reduced. The dimensionality of energy is pressure x volume, tension is negative pressure, and the energy in the string has reduced its tension. If the string was stretched over an archer's bow, the ends of the bow would be starting to move apart at this point.
I'm not sure exactly what you're saying here. In what form is the wave energy when the string is flat?

----

For those who are worried about infinite waves extending off in all directions, consider the following, perhaps simpler, case:

Two identical pulses propagate along a string, but one is upside-down compared to the other, as here:

(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?

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.

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