# What happens to the energy when waves perfectly cancel each other?

What happens to the energy when waves perfectly cancel each other (destructive interference)? It appears that the energy "disappear" but the law of conservation of energy states that it can't be destructed. My guess is that the kinetic energy is transformed into potential energy. Or maybe it depends on the context of the waves where do the energy goes? Can someone elaborate on that or correct me if I'm wrong?

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We treated this a while back at University...

First of all, I assume you mean global cancellation, since otherwise the energy that is missing at the cancelled point simply is what is added to points of constructive interference: Conservation of Energy is only global.

The thing is, if multiple waves globally cancel out, there are actually only two possible explanations:

• One (or more) of the sources is actually a drain and converts wave energy into another form of energy, (e.g. whatever is used to generate the waves in sources, like electricity, and also as Anna said, very often heat)
• You are calculating with parts of an mathematical expansion which are only valid when convoluted with a weight function or distribution. For example, plane waves physically don't exist (But when used in the Fourier Transform they are still very useful) because their total energy is infinite
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Waves always travel. Even standing waves can always be interpreted as two traveling waves that are moving in opposite directions (more on that below).

Keeping the idea that waves must travel in mind, here's what happens whenever you figure out a way to build a region in which the energy of such a moving wave cancels out fully: If you look closely, you will find that you have created a mirror, and that the missing energy has simply bounced off the region you created.

Examples include opals, peacock feathers, and ordinary light mirrors. The first two reflect specific frequencies of light because repeating internal structures create a physical regions in which that frequency of light cannot travel - that is, a region in which near-total energy cancellation occurs. An optical mirror uses electrons at the top of their Fermi seas to cancel out light over a much broader range of frequencies. In all three examples the light bounces off the region, with only a little of its energy being absorbed (converted to heat).

A skip rope (or perhaps a garden hose) provides a more accessible example. First, lay out the rope or hose along its length, then give it quick, sharp clockwise motion. You get a helical wave that travels quickly away from you like a moving corkscrew. No standing wave, that!

You put a friend at the other end, but she does not want your wave hitting her. So what does she do? First she tries sending a clockwise wave at you too, but that seems to backfire. Your wave if anything seems to hit harder and faster. So she tries a counterclockwise motion instead. That seems to work much better. It halts the forward progress of the wave you launched at her, converting it instead to a loop. That loop still has lots of energy, but at least now it stays in one place. It has become a standing wave, in this case a classic skip-rope loop, or maybe two or more loops if you are good at skip rope.

What happened is that she used a canceling motion to keep your wave from hitting her. But curiously, her cancelling motion also created a wave, one that is twisted in the opposite way (counterclockwise) and moving towards you, just as your clockwise wave moved towards her. As it turns out, the motion you are already doing cancels her wave too, sending it right back at her. The wave is now trapped between your two cancelling actions. The sum of the two waves, which now looks sinusoidal instead of helical, has the same energy as your two individual helical waves added together.

I should note that you really only need one person driving the wave, since any sufficiently solid anchor for one end of the rope will also prevent the wave from entering it, and so end up reflecting that wave just as your friend did using a more active approach. Physical media such as peacock features and Fermi sea electrons also use a passive approach to reflection, with the same result: The energy is forbidden by cancellation from entering into some region of space.

So, while this is by no means a complete explanation, I hope it provides some "feel" for what complete energy cancellation really means: It's more about keeping waves out. Thinking of cancellation as the art of building wave mirrors provides a different and less paradoxical-sounding perspective on a wide variety of phenomena that alter, cancel, or redirect waves.

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Maybe the question can simply be answered by the observation that a wave like

$$\Psi(x,t)=A \cos(x)-A \cos(x+\omega\ t),$$

where the two cosines cancel at periodic times $$t_n=\frac{2\pi}{\omega}n\ \ \longrightarrow\ \ \Psi(x,t_n)=0,$$ still has nonvanishing kinetic energy, if it looks something like $$E=\sum_\mu\left(\frac{\partial \Psi}{\partial x^\mu} \right)^2+\ ...$$

You really would have to construct an example.

Since non-dissipative waves whose equations of motions can be formulated by a Lagrangian will have an energy associated to them, as you say, you'd have to find a situation/theory without an energy quantity. The energy is related to the wave by its relation to the equation of motion. So if the energy is defined as that which is constant because of time symmetry and you don't have such a thing, then there is no question.

Also don't make the mistake and talk about about two different waves with different energy. If you have a linear problem, the wave will be "one wave" in the energy expression, wherever its parts may wander around.

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Just in case anyone (e.g. student) would be interested in the simple answer for mechanical waves:

CASE 1 (global cancellation): Imagine that you have crest pulse moving right and equally large though pulse moving left. For a moment they "cancel", e.g. there is no net displacement at all, because two opposite displacements cancel out. However, velocities add up and are twice as large, meaning that all the energy in that moment is stored within kinetic energy.

Instructive and opposite situation happens, when crest pulses meet. For a moment, displacements add up and are twice as large, meaning that all the energy in that moment is stored within potential energy, as velocities on the other hand cancel out.

Because wave equation is linear differential equation, you can superpose different waves $\psi_{12} = \psi_1 + \psi_2$. As a consequence, after meeting both crest pulses or pair crest / though pulses keep traveling if nothing has had happened.

It is instructive, that you can add velocities separately of amplitudes, as $\dot{\psi}_{12} = \frac{\partial}{\partial t} (\psi_1 + \psi_2) = \dot{\psi}_1 + \dot{\psi}_2$. So even if amplitudes do cancel out at a given moment ($\psi_1 + \psi_2 = 0$), speeds do not ($\dot{\psi}_1 + \dot{\psi}_2 \ne 0$).

It is just as if you see that oscillator is in a equilibrium position at a given moment. That does not mean that it is not oscillating, as it still might posses velocity.

If we generalize written above: in any wave you have exchange of two types of energy: kinetic vs. potential, magnetic vs. electrical. You can make such two waves that one of the energies cancels, but the other energy will become twice as big.

CASE 2 (local cancellation): In case of spatial interference of two continuous waves there are areas of destructive and areas of constructive interferences. Energy is no longer uniformly distributed in space, but in average it equals added up energies of two waves. E.g. looking at standing waves, there is no energy at nodes of the standing waves, while at crests energy is four times the energy of one wave - giving a space average of twice the energy of one wave.

More engineer-like explanations can be found here: http://van.physics.illinois.edu/qa/listing.php?id=1891

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I had prepared this answer for a question that was made duplicate, so here it comes, because I found an instructive MIT video. (the second link) This answer is for electromagnetic waves mainly

Have a look at this video to get an intuition how interference appears photon by photon in a two slit experiment.

It comes because the probability distribution for the photons, as accumulated on the screen, has destructive and and constructive patterns, ruled by the underlying quantum mechanical solution of "photon + two slits".

The classical electromagnetic wave emerges from a great plethora of photons which have phases and such that they build up the electric and magnetic fields. The nu in the E=h*nu of the photon is the frequency of the electromagnetic wave that emerges from the confluence of the individual photons. In order to get an interference pattern the photons have to react with a screen, or some some matter, as in the laser experiments.

The reason that matter is needed for light interference phenomena is due to the very small electromagnetic coupling constant. Photon photon interactions due to the 1/137 end up having a probability of interaction of order of ~10^-16 . With respect to photon electron interactions, which to first order is ~ 10^-4,( and is the main photon-matter interaction) there are 12 orders of magnitude. To all intents two laser beams crossing will go through each other without any measurable interaction, interferences included. (Keep this in mind when you reach the last question at the end of the next video.)

This MIT video is instructive and a real experiment that shows that in destructive interference set up with interferometers there is a return beam, back to the source, as far as classical electromagnetic waves go. So the energy is balanced by going back to the source.

What is happening at the photon level? If the laser emitted photons one by one as in the two slit video? I will hand wave as there is no corresponding video to show:

The quantum mechanical solution with the complicated boundary values of the interferometer allows the elastic scattering ( not small, that is how we get reflections) of photons also back to the source. You can see in the video that there always exists a beam going back to the source, that beam is carried by individual photons scattering elastically backwards through the system of the optics of the interferometer. In total destructive interference all the energy is reflected back ( minus some due to absorption and scattering in the matter of the optical system).

In essence this experiment is a clear demonstration that the system laser-optical-bench is in a coherent quantum mechanical state, the returning photons joining the ensemble of photons within the laser action , which also includes reflections to be generated.

In this video, the first beam carries the information of the phases such that in space interference patterns will form if a screen or other matter intervenes. The energy of the final beam after it leaves the interferometer system and falls on the screen and is redistributed according to the pattern of the interference. The amount of energy carried by the beam there depends on the proportion of energy that manages to leave the interferometer/laser system, i.e. if all of the energy is returned to the laser (destructive interference) , or a part of it goes out of the lasing system to impinge on the screen.

In the case of waves in matter, as sound waves or water waves:

In the case of two sound waves interfering destructively, the temperature of the medium will go up and energy is conserved because it turns into incoherent kinetic energy of the molecules of the medium.

For two water waves, ditto.

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http://www.opticsinfobase.org/josaa/abstract.cfm?uri=josaa-27-11-2468

Oblique superposition of two elliptically polarized lightwaves using geometric algebra: is energy–momentum conserved? Michelle Wynne C. Sze, Quirino M. Sugon, Jr., and Daniel J. McNamara JOSA A, Vol. 27, Issue 11, pp. 2468-2479 (2010)

We added the two elliptically polarized waves and computed the energy–momentum density of their sum. We showed that energy and momentum are not generally conserved, except when the two waves are moving in opposite directions. We also showed that the momentum of the superposition has an extra component perpendicular to the propagation directions of both waves. But when we took the time-average of the energy and momentum of the superposition, we found that the time-average energy and momentum could also be conserved if both waves are circularly polarized but with opposite handedness, regardless of the directions of the two waves. The non-conservation of energy and momentum of the superposition of two elliptically polarized plane waves is not due to the form of the plane waves themselves, but rather to the accepted definitions of the electromagnetic energy and momentum. Perhaps we may need to modify these definitions in order to preserve the energy–momentum conservation. In our computations, we restricted ourselves to the superposition of two waves with the same frequency.

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thanks, an interesting calculation behind a paywall – anna v Apr 25 '12 at 4:38

I think a good way to approach this question is with a Mach-Zehnder interferometer:

The field landing on detector 1 is the interference between two waves, one from the lower path, and one from the upper path. Let's suppose the field in each arm is a collimated beam of coherent light, well-approximated as a plane wave, and the interferometer is well-aligned, so the two outputs are almost perfectly overlapped. By changing the thickness of the sample, we can change the relative phase between the two waves, changing our interference from destructive (less energy on detector 1) to constructive (more energy on detector 1). Where did this energy come from? If the beams are nicely matched, this interference can even be completely destructive, and detector 1 will register zero signal. Where does the energy go?

The short answer is: detector 2. The total energy hitting the two detectors is constant, as you vary the phase shift caused by the sample. Constructive interference at detector 1 goes hand-in-hand with destructive interference at detector 2.

If you only look at one detector or the other, it might seem like energy is created or destroyed by interference, but as other answers mention, we must consider the whole system.

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This figure shows two common situations.

The top is an example where the waves are coming from different directions--one from "S1", one from "S2". Then there is destructive interference in some areas ("nodes") and constructive interference in others ("hot spots"). The energy has been redistributed but the total amount of energy is the same.

The bottom is an example where the two sources S3 and S4 are highly directional plane-wave emitters, so that they can destructively interfere everywhere they overlap. For that to happen, the source S4 itself has to be sitting in the field of S3. Then actually what is happening is that S4 is absorbing the energy of S3. (You may think that running the laser S4 will drain its battery, but ideally, the battery can even get recharged!)

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What about momentum conservation? Do you have a link for an experiment on this? – anna v Apr 19 '12 at 14:19

In physics, the Poynting vector represents the directional energy flux density (the rate of energy transfer per unit area, in Watts per square metre, W·m−2) of an electromagnetic field.

If the antilaser antilaser experiment is performed in the vacuum there is no thermal dissipation, and the Poynting vectors are opposed, and cancel, for the same field intensity and with the fields out-of-phase. For plane waves (WP, link above):
"The time-dependent and position magnitude of the Poynting vector is" : $\epsilon_0cE_0^2\cos^2(\omega t-\mathrm{k\cdot r})$ and the average is different of zero for a single propagating wave, but, for two opposing plane waves of equal intensity and 100% out-of-phase the instantaneous Poynting vector, that measures the flux of energy, is the vector $\vec{S}(t)=\mathrm{\vec{0}}$.

If you have one electromagnetic beam at a time then work can be done. If you have two in the above conditions then no work can be extracted. (Energy is canceled, destroyed, ;)

BUT, things can be more complicated then described by the eqs, because a physical emmiter antenna also behaves as a receiving antenna that absorbs and reradiates etc, ... changing and probably trashing my first oppinion.

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The Poyinting vectors, and the momenta vectors as the E, B fields are symetric. When we do 'field shaping' with antenae aggregates we simply use Maxwell eqs and go with waves everytime. When we got near a null in energy in some region of space we dont get infrared radiation to 'consume' the canceled field. E,B vectors additive: Light+Light=0

Antenae in sattelites (vaccum) work the same way as the ones at Earth surface to shape the intensity of the field.
Because the "Poyinting vectors" add to null there is no doubt, imo, that energy vanish.

See the antilaser experiment.

We dont have theory? Then we must rethink.
IMO energy is not transported. What is propagating is only an excitation of the medium (we call it photons) and energy is already 'in site' (vacuum, or whatever name we call the medium).

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When there is a complete destructive interference of two light beams, Maxwell’s equations predict that the energy becomes zero. Let’s have the case of two coherent collinear beams, out of phase 180 degrees, like the case of the antilaser.

\begin{align} E_1 = E_m \sin (kx - \omega t);\quad E_2 = E_m \sin (kx - \omega t + p) \\ B_1 = B_m \sin (kx - \omega t); \quad B_2 = B_m \sin (kx - \omega t + p) \end{align}

$E = E_1 + E_2$ and $B = B_1 + B_2$ \begin{align} E &= E_m \sin (kx - \omega t) + E_m \sin (kx - \omega t + p) \\ B &=B_m \sin (kx - \omega t) + B_m \sin (kx - \omega t + p) \end{align} But, $\sin (kx - \omega t + p) = - \sin (kx – \omega t)$ , Then, $E = 0$ and $B = 0$ and,

\begin{align} UT &= U_E + U_B \\ &= \frac12ԑ_0E^2 + \frac1{2\mu_0}B^2 \\ &= 0 \end{align} This is the classical interpretation of the waves electromagnetism during total destructive interference, following Maxwell. Maxwell's description of the energy of the light wave is of an undulating energy that predictably reaches a maximum and later becomes zero. The proposed solution to this problem is to calculate the mean of the energy when the fields are maxima.

What is the physical meaning of an energy that have to be averaged in order to have the real magnitude. If the principle of conservation of energy is to be applied to this phenomenon, the energy must be constant, have an unique value for each instant during the movement of the wave. What is the meaning of that situation that has not been recognized for more than a century?

What almost nobody want to admit is that electromagnetism is incomplete, because cannot describe the electromagnetic radiation adequately, and generate a violation of the principle of conservation of energy.

As Helder Velez said: “We don’t have theory?”. NO Then we must rethink.” He has a proposition: EM energy is not transported, only is an excitation of the medium, the quantum vacuum, or the quantum plenum as I prefer to call it. But this is only an idea, an intuition, without support or evidence.

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Lots of copy-pasting from poster's other (flawed) answers. – Kyle Kanos Feb 13 '15 at 21:21
I made one and only one reference: Helder Velez. The fact that his comment have negative punctuation does not mean that he is incorrect; only that they disagree with him. I support him in the violation of the conservation of energy. I support the "EMG theory of the foton" by Diogenes Aybar that can be found at journaloftheoretics.com/Links/Papers/EMG%20III.pdf. He thinks that the photon has another inherent field, the gravitational field, where the energy goes from the two electromagnetic fields to the gravitational field, keeping then, at every moment the energy constant. – luis fondeur Feb 15 '15 at 18:39

the waves will obliterate each other but they will still exist, they just won't be moving they would just change form (energy cannot be destroyed it can only change form) so when the waves meet they will cancel each other so sound will change to potential and kinetic will change to sound or whatever

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## protected by Qmechanic♦Aug 15 '15 at 5:47

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