# Understanding Destructive Wave Interference

• aaaa202
In summary, destructive interference can be hard to understand when it comes to energy conservation. In the case of two waves on a string, the energy is stored in the kinetic energy of the string, even when it appears to be at rest. In the case of a plane electromagnetic wave approaching two planes, the energy is dissipated in the material of the planes through absorption and radiation. This is how anti-reflective coatings work, using complete destructive interference to absorb the energy of the reflected wave and increase the amount of transmitted energy. However, there may still be some constructive interference on the other side of the film for the transmitted wave. Overall, energy is conserved in this process.

#### aaaa202

Sometimes I find it hard to understand how energy is conserved in destructive interference. Consider two waves on a string which interfere destructively such that the string is at its rest position. Where is the energy contained?
Edit: the string example is not very good at illustrating what I don't understand. Consider instead the following example.
A plane electromagnetic wave approaches two planes. Now the wave is both reflected and travels through the plane to the next wave is reflected again. If we now choose the spacing of the planes such that the reflection on the 2nd plane differs by exactly a 180degree phase to the first reflected wave a situation is created where we have constant destructive interference in the resulting reflected wave. Where is the energy stored?

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aaaa202 said:
Sometimes I find it hard to understand how energy is conserved in destructive interference. Consider two waves on a string which interfere destructively such that the string is at its rest position. Where is the energy contained?
In the kinetic energy of the string. The string may be at its rest position for a moment, but it is not at rest, it is moving. That motion has KE.

aaaa202 said:
A plane electromagnetic wave approaches two planes. Now the wave is both reflected and travels through the plane to the next wave is reflected again. If we now choose the spacing of the planes such that the reflection on the 2nd plane differs by exactly a 180degree phase to the first reflected wave a situation is created where we have constant destructive interference in the resulting reflected wave. Where is the energy stored?
This is how anti-reflective coatings work. The energy is dissapated in the material of the planes, i.e. the waves do work on the matter.

aaaa202 said:
A plane electromagnetic wave approaches two planes. Now the wave is both reflected and travels through the plane to the next wave is reflected again. If we now choose the spacing of the planes such that the reflection on the 2nd plane differs by exactly a 180degree phase to the first reflected wave a situation is created where we have constant destructive interference in the resulting reflected wave. Where is the energy stored?

This is how anti-reflective coatings work. The energy is dissapated in the material of the planes, i.e. the waves do work on the matter.

Forgive me if I'm wrong, but isn't the energy still in the EM wave?

In free space the energy of the wave is conserved, but in the presence of matter the energy of the wave can change due to absorption or radiation. The case the OP mentioned is one with perfect absorption of the reflected wave. It is used in anti reflective coatings.

DaleSpam said:
In free space the energy of the wave is conserved, but in the presence of matter the energy of the wave can change due to absorption or radiation. The case the OP mentioned is one with perfect absorption of the reflected wave. It is used in anti reflective coatings.

Oh, okay, I see that now in the situation the OP gave. How is this the same for anti-reflective coatings? I mean, I know they use destructive interference to work, but in this case the energy is still in the non-reflected EM wave, right?

Drakkith said:
I know they use destructive interference to work, but in this case the energy is still in the non-reflected EM wave, right?
Not 100% of the energy is transmitted through the film, but 0% (at a specific wavelength) is reflected due to the complete destructive interference. The difference is absorbed.

I am not certain, but I think that antireflective coatings increase the amount of light transmitted, so there may be some constructive interference on the other side of the film for the transmitted wave. But I don't know the details.

The interference reduces the amount of energy being reflected while increasing the amount of energy being transmitted in such a way that (surprise) energy is conserved. No energy needs to be absorbed at the surface for that to work (Though there might be some absorption depending on what the coating is made of, of course)

## 1. What is destructive wave interference?

Destructive wave interference is a phenomenon that occurs when two waves of the same frequency and amplitude meet and cancel each other out, resulting in a decrease in amplitude. This is caused by the waves being out of phase with each other, meaning that their peaks and troughs do not align.

## 2. How is destructive wave interference different from constructive interference?

Unlike destructive interference, constructive interference occurs when two waves of the same frequency and amplitude meet and combine, resulting in an increase in amplitude. This is caused by the waves being in phase with each other, meaning that their peaks and troughs align.

## 3. What are some examples of destructive wave interference?

One example of destructive wave interference is noise-canceling headphones, which use destructive interference to cancel out ambient noise. Another example is when sound waves from a loudspeaker interact with reflections from walls, resulting in certain frequencies being canceled out and creating "dead spots" where the sound is quieter.

## 4. How does the principle of superposition apply to destructive wave interference?

The principle of superposition states that when two or more waves meet, their amplitudes add together. In the case of destructive wave interference, this means that the amplitudes of the two waves cancel each other out, resulting in a decrease in amplitude. This principle applies to all types of waves, including sound waves, light waves, and water waves.

## 5. Can destructive wave interference be predicted?

Yes, destructive wave interference can be predicted by analyzing the properties of the two waves involved, such as their frequencies, amplitudes, and wavelengths. This can be done using mathematical equations, such as the principle of superposition and the wave equation. However, external factors such as reflections and diffraction can also affect the outcome of destructive interference, making it difficult to predict in some cases.