Understanding Destructive Wave Interference

[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?

 

[Dale]

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.

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.

 

[Drakkith]

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?

 

[Dale]

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.

 

[Drakkith]

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?

 

[Dale]

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.

 

[dauto]

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)