Constructive interference and energy conservation

In summary: This is because the energy is preserved in the form of a wave. For waves of finite length, the energy is conserved as the waves travel from one point to another.
  • #1
Wminus
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Hi. Let's say two monochromatic laser beams superimpose in a single point in space in such a way that there's constructive interference. Because there's constructive interference there, the total intensity at that point will be larger than the sum of the separate intensities.

Will this mean that the area illuminated by the lasers in the point of superimposition will shrink such that energy is conserved?

thanks :)
 
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  • #2
Essentially, yes. What happens is that if you have regions of constructive interference then you also have regions of destructive interference also. The energy density is higher where there is constructive interference and lower where there is destructive interference, for overall energy conservation.
 
  • #3
What happens when two {small) wave sources are placed much less than half a wavelength apart? Destructive interference everywhere, presumably. What then happens to the energy? I think I know the answer to this paradox, but I post it here as a brain teaser. [If a mentor considers it an inappropriate posting, I'd be happy for it to be removed,]
 
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  • #4
Why would be destructive interference everywhere?
 
  • #5
Wminus said:
Hi. Let's say two monochromatic laser beams superimpose in a single point in space in such a way that there's constructive interference. Because there's constructive interference there, the total intensity at that point will be larger than the sum of the separate intensities.

Will this mean that the area illuminated by the lasers in the point of superimposition will shrink such that energy is conserved?

thanks :)
The total power over the combined spot will be the sum of the power of the two beams. There will be a fringe pattern, and the bright fringes represent the coherent addition of the two amplitudes, so they have four times the intensity (W/m^2) of a single beam. The additional power to provide these bright fringes comes from the dark ones, where cancellation is occurring. For the special case where the lasers are side by side and the measurement is made at great distance, in the radiation far field, the round spot of a single laser will become narrower, forming an ellipse, with a peak intensity four times that of one laser. The additional intensity comes from the smaller spot area, in the way you describe.
 
  • #6
My mistake, and too late to edit! Should say 'constructive everywhere' (because difference in path distances from sources to any point P must also be much less than half a wavelength). So at every point P amplitude is double what it would be from a single source, so energy of oscillation four times as much. Where has the extra energy come from?
 
  • #7
I think that you are forgetting about diffraction. In the single source situation the light diffracts over a very wide angle. In the two source situation the diffraction is less and the light is much more concentrated. It has a higher energy density in the middle, but lower energy density at large angles. The overall energy is still conserved.
 
  • #8
Thanks for the answers guys! :)

DaleSpam said:
Essentially, yes. What happens is that if you have regions of constructive interference then you also have regions of destructive interference also. The energy density is higher where there is constructive interference and lower where there is destructive interference, for overall energy conservation.

Is it really impossible to construct a case where the interference is solely positive? I mean in my example the two rays cross in a single point of constructive interference, so were will the destructive interference take place?
 
  • #9
In that 'single point' there will be places of constructive interference and destructive interference. The total radiated power must be the same, regardless of interference between the two beams.

edit: to be able to talk about interference, we need to think of the rays as having some wave-like properties, so we can think of the region of interference between the two rays, we get constructive and destructive interference. If we shrink this region of interference, the smaller region will still have constructive and destructive interference. It is not possible to keep only constructive interference by making the region of interference smaller.
 
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  • #10
Wminus said:
Thanks for the answers guys! :)
Is it really impossible to construct a case where the interference is solely positive? I mean in my example the two rays cross in a single point of constructive interference, so were will the destructive interference take place?
Yes, it is impossible. Similarly to Philip, you seem to be forgetting about diffraction. It prevents waves that would cross at a single point. You will always have a finite volume of crossing.
 
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Philip Wood said:
What happens when two {small) wave sources are placed much less than half a wavelength apart? Destructive interference everywhere, presumably. What then happens to the energy? I think I know the answer to this paradox, but I post it here as a brain teaser. [If a mentor considers it an inappropriate posting, I'd be happy for it to be removed,]

For the poynting theorem the energy of the laser beams should be conserved no matter how they interfere. A deeper analysis would show the intensity is distributed.
 

What is constructive interference?

Constructive interference is a phenomenon that occurs when two or more waves overlap and their amplitudes add together, resulting in a wave with a larger amplitude. This happens when the waves are in phase, meaning their crests and troughs align.

How does constructive interference relate to energy conservation?

According to the law of energy conservation, energy cannot be created or destroyed, only transferred from one form to another. In the case of constructive interference, when two waves combine and create a larger amplitude wave, the energy of the individual waves is conserved and transferred to the new wave.

What is the difference between constructive interference and destructive interference?

Constructive interference results in a wave with a larger amplitude, while destructive interference results in a wave with a smaller amplitude. This is because in constructive interference, the two waves are in phase and their amplitudes add together, while in destructive interference, the two waves are out of phase and their amplitudes cancel each other out.

Can constructive interference occur with any type of wave?

Yes, constructive interference can occur with any type of wave, including sound waves, light waves, and water waves. However, the conditions for constructive interference to occur may vary depending on the properties of the wave, such as wavelength and frequency.

How is constructive interference used in real-world applications?

Constructive interference is used in a variety of real-world applications, such as in noise-cancelling headphones, where sound waves are deliberately combined to cancel out unwanted noise. It is also used in the field of optics, where constructive interference is used to enhance the brightness and clarity of images.

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