What is Destructive Interference?

[Quarker]

In an electromagnetic interference pattern with two sources of identically sized waves, the dark areas are where the waves destructively interfere. Being identical in size, the waves should completely cancel one another out, leaving an area empty of any trace of electromagnetism. Yet the area of constructive interference arises right next to it, seemingly out of nothingness, to continue the pattern. It’s as if the EM has momentarily left our universe. Where does it go?

 

[Drakkith]

Being identical in size, the waves should completely cancel one another out, leaving an area empty of any trace of electromagnetism.

This would only be true if two waves were entirely out of phase with each other everywhere. Which would require that they occupy the exact same points in space at the same time, just with opposite phase. Which would mean they were emitted from the exact same point. But this is impossible. You cannot emit two waves from the same point at the same time.

Yet the area of constructive interference arises right next to it, seemingly out of nothingness, to continue the pattern. It’s as if the EM has momentarily left our universe. Where does it go?

The energy lost from the areas of destructive interference is gained in the areas of constructive interference. This is just like how two water waves create ripples on the surface of the water. The areas where they destructively interfere are areas of reduced amplitude, while areas where they constructively interfere are areas with increased wave amplitude.

 

[Quarker]

This would only be true if two waves were entirely out of phase with each other everywhere. Which would require that they occupy the exact same points in space at the same time, just with opposite phase. Which would mean they were emitted from the exact same point. But this is impossible. You cannot emit two waves from the same point at the same time.The energy lost from the areas of destructive interference is gained in the areas of constructive interference. This is just like how two water waves create ripples on the surface of the water. The areas where they destructively interfere are areas of reduced amplitude, while areas where they constructively interfere are areas with increased wave amplitude.

View attachment 294324

So complete annihilation is impossible, there is always some trace of electromagnetism within a dark zone?

 

[Drakkith]

So complete annihilation is impossible, there is always some trace of electromagnetism within a dark zone?

No, there can be points with zero amplitude. It's just that the energy that would have been there had the other wave not been present is not distributed elsewhere. What you can't do is produce two waves that completely cancel each other out everywhere, leading to a violation of conservation of energy.

 

[Quarker]

So light with zero amplitude has zero energy?

 

[vanhees71]

If you consider the superposition of two plane waves of equal amplitude. At one given point you have an electric field and magnetic field $\propto$
$$cos(\omega t-k z)+\cos(\omega t-k z+\varphi)=2 \cos \left (\frac{\varphi}{2} \right) \cos \left (\omega t -k z+\frac{\varphi}{2} \right).$$
This means you have destructive interference at the times and places fullfilling $\omega t-k z+\varphi/2=(2n+1) \pi/2$ and constructive interference when $\omega_t -k z+\varphi/2 = n \pi$ with $n \in \mathbb{Z}$. At a given point in time you always have a point constructive interference at one place when there is distructive interference at another. The total energy in the wave (which of course is never a plane wave everywhere, because a literal plane wave has inifinite total energy) is constant thanks to Poynting's theorem, i.e., field energy flowing out of a fixed volume must stream through the volume's surface and thus occur somewhere else.

 

[Quarker]

Let’s make this simpler. When two light waves of the same amplitude destructively interfere, what happens to the electromagnetic wave at the point of zero amplitude? How can it be said to exist if it has zero energy?

 

[vanhees71]

It has zero energy density (!) at the point of destructive interference. A field is always an "object" extended over all space. There is no paradox or something only because there is destructive interference at one point. The total energy of a closed system is always conserved. The closed system in electrodynamics are the electromagnetic field and the charged matter (described by charge and current densities in Maxwell's equations). The total energy-momentum tensor is always (locally) conserved and thus total energy and momentum.

 

[Drakkith]

Let’s make this simpler. When two light waves of the same amplitude destructively interfere, what happens to the electromagnetic wave at the point of zero amplitude? How can it be said to exist if it has zero energy?

At the point of zero amplitude there simply isn't any changing electric or magnetic field vectors. Just like the water waves where the zero amplitude points are just smooth water that doesn't oscillate up and down.

So light with zero amplitude has zero energy?

Let's be careful with terminology here. 'Light' is an EM wave and that wave occupies a region of space and can have zero amplitude at some points and non-zero at other points. So when we talk of amplitude let's talk of points that the wave occupies and not as a whole, since zero amplitude everywhere isn't a wave and thus can't be called light.

 

[Quarker]

It has zero energy density (!) at the point of destructive interference. A field is always an "object" extended over all space. There is no paradox or something only because there is destructive interference at one point. The total energy of a closed system is always conserved. The closed system in electrodynamics are the electromagnetic field and the charged matter (described by charge and current densities in Maxwell's equations). The total energy-momentum tensor is always (locally) conserved and thus total energy and momentum.

I know about the conservation of the wave energy across the interference pattern. My question is what happens to the wave at the point of zero amplitude. Any image will show the area as dark, as if the wave had momentarily vanished. Within that zone the wave simply does not exist. Let me do some research on energy density. I’ll probably have more questions. Thanks for everyone’s help.

 

[Drakkith]

My question is what happens to the wave at the point of zero amplitude. Any image will show the area as dark, as if the wave had momentarily vanished. Within that zone the wave simply does not exist.

A detector placed at a region of zero amplitude would record no EM wave, no photons, nothing. Whether you interpret this result as the wave not existing is mostly irrelevant as long as you understand the underlying reason why the amplitude is zero and that nearby regions will likely have non-zero amplitudes.

Note that there is nothing special about the points of zero-amplitude. The exact same things are happening to every other point along the waves. The field vectors sum together and you get some resultant value. At some regions this is zero, at others it is above zero, and at some it is greater than what either of the single waves themselves could produce.

 

[Quarker]

There is no paradox.

This seems the perfect example of a paradox. The interference pattern periodically turns to black as the waves interfere. The wave is demonstrably gone from our universe during those intervals. Yet it continues to exist. Where is it?

 

[Drakkith]

The wave is demonstrably gone from our universe during those intervals. Yet it continues to exist. Where is it?

The wave hasn't gone anywhere. It's an extended 'object' or 'phenomenon' that exists across some spatial and temporal range. Just because part of the wave or wavefront happens to be zero doesn't mean the wave is gone. There is no paradox because this is exactly what we would expect based on what we know about wave physics and is exactly what we observe.

I notice that I don't see you questioning the bright regions of the interference pattern, where the amplitude or intensity is higher than either wave individually. Why would this make sense but the reverse not?

Interference, of any kind, is just the result of waves adding and subtracting from each other. You can't have the addition without the subtraction, and sometimes this subtraction leads to a zero. Here's a two-dimensional example that may be easier to understand. As you can see in the right side example, both waves still exist after destructively interfering. The interference pattern of EM waves is just a more complicated example of this.

 

[vanhees71]

You must see a field as a whole, i.e., spread over all space. As was stressed several times now: If you have destructive interference at a certain point at some time, you have constructive interference on other places at the same time and vice versa. The total energy is conserved.

 

[Quarker]

The wave hasn't gone anywhere. It's an extended 'object' or 'phenomenon' that exists across some spatial and temporal range. Just because part of the wave or wavefront happens to be zero doesn't mean the wave is gone. There is no paradox because this is exactly what we would expect based on what we know about wave physics and is exactly what we observe.

I notice that I don't see you questioning the bright regions of the interference pattern, where the amplitude or intensity is higher than either wave individually. Why would this make sense but the reverse not?

Are you trying to argue that the electromagnetism somehow becomes separated from its wave function during the point of destructive interference? Because if there is no electromagnetism at that point, which any interference pattern will show, I don’t see how the wave function can be said to still exist. Within our universe.

 

[Quarker]

You must see a field as a whole, i.e., spread over all space. As was stressed several times now: If you have destructive interference at a certain point at some time, you have constructive interference on other places at the same time and vice versa. The total energy is conserved.

It’s not the total energy I’m concerned with. I want to know what happens to the EM wave when it disappears. And then comes back.

 

[weirdoguy]

I want to know what happens to the EM wave when it disappears.

It does not disappear. It doesn't make sense to talk about wave at a point, since by definition it is a phenomena spreaded spatially.

 

[weirdoguy]

Because if there is no electromagnetism at that point, which any interference pattern will show, I don’t see how the wave function can be said to still exist.

So are you saing that a function that takes value 0 at one point does not exist because of that? Sorry but that is kind of nonsense...

 

[Quarker]

So are you saing that a function that takes value 0 at one point does not exist because of that? Sorry but that is kind of nonsense...

At that one point, that’s exactly what I’m saying. A zero amplitude EM wave is a physical impossibility. To say that it somehow exists because the wave exists elsewhere makes no sense.

 

[weirdoguy]

To say that it somehow exists because the wave exists elsewhere makes no sense.

So you simply do not understand what a wave is. And it seems that you really do not want to learn, you just keep repeating incorrect statements.

 

[Quarker]

Then you simply do not understand what a wave is.

And you choose to ignore the evidence of your own eyes. Look at an interference pattern. Exactly what is happening where the image turns dark?

 

[weirdoguy]

Exactly what is happening where the image turns dark?

It's been said multiple times. I'm not going to waste my time repeating that again. Just going to report it to moderators.

 

[berkeman]

Thread closed for Moderation...

Edit: Thread reopened (PeterDonis).

 

[PeterDonis]

Being identical in size, the waves should completely cancel one another out, leaving an area empty of any trace of electromagnetism.

No, that's not the way electromagnetism propagates. The fact that the EM field is zero at one point does not mean "an area empty of any trace of electromagnetism" must start at that point. If your intuition is telling you that should be the case, your intuition is wrong and needs to be retrained. The math of EM is perfectly clear and agrees with what we actually observe.

It’s as if the EM has momentarily left our universe.

The fact that the EM field's value is zero at one particular point does not mean "EM has left our universe" at that point. A field's value being zero at a point is not the same as the field not existing at all at that point.

 

[PeterDonis]

Exactly what is happening where the image turns dark?

See my post #24.

 

[Dale]

My question is what happens to the wave at the point of zero amplitude.

The point of zero amplitude is not very important or meaningful. Remember that a wave is described by the wave equation: $$\frac{\partial^2}{\partial t^2}f=c^2 \frac{\partial^2}{\partial x^2} f$$ So the criteria for the wave “not existing” at a point is not simply that $f=0$. For the wave to not exist at a point you need that the function and all of its first and second partial derivatives should all vanish. Only then is there no wave.

In a typical sinusoidal plane wave the zero crossings are also places where the second derivatives vanish, but not the first derivatives. So the wave still exists at those points. In contrast, something like a square wave does indeed have zero first and second derivatives where the function is zero.

 

[Drakkith]

Are you trying to argue that the electromagnetism somehow becomes separated from its wave function during the point of destructive interference? Because if there is no electromagnetism at that point, which any interference pattern will show, I don’t see how the wave function can be said to still exist. Within our universe.

I don't know what 'the electromagnetism' is supposed to refer to. An EM wave is simply a propagating change of the electromagnetic field and the things that are changing are the electric and magnetic field vectors. The wave function or equation describing these vectors and how they may or may not be changing is just our method of describing and predicting changes of the EM field and has no other meaning. It exists just as much or as little as the function $f(x)=2x+5$ regardless of what the object or phenomenon it is describing is doing.

At no point does the behavior of the wave break from that predicted or described by the wave equation.

 

[Quarker]

It would help if everyone just stayed on topic. Telling me I’m wrong by giving me your definition of a wave is not proving anything. I’m talking about one point in an EM wave. If anyone tries to bring in any additional concepts, I will ignore your post.

 

[Ibix]

I don't really understand what your issue is. As the image of water waves in post #2 shows, the destructive interference forms lines of calm water with propagating waves between. What's the problem? Or are you concerned about waves propagating in opposite directions?

If anyone tries to bring in any additional concepts, I will ignore your post.

That's unlikely to be a helpful attitude. If you don't understand the topic, you may well find that there is no answer in the terms you set. If you go into a car forum asking about a problem with loud bangs from your exhaust and declare all responses about spark plugs and ignition timing off topic because you asked about your exhaust, you will not solve your problem.

 

[Dale]

It would help if everyone just stayed on topic. Telling me I’m wrong by giving me your definition of a wave is not proving anything. I’m talking about one point in an EM wave. If anyone tries to bring in any additional concepts, I will ignore your post.

I am also talking about one point in an EM wave. The definition of a wave is important because from that it is easy to see what the criteria is for saying that there is no wave at that point. The criteria is not simply that the value of the wave is zero.