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Light waves from two sources of light meet. They are in phase such that they cancel each other out. Is this an exception to the Conservation of Energy Law? If not, why not?
If the waves canceled everywhere, that would indeed violate energy conservation. However, what happens is that for any actual wave patterns, there are regions of both constructive interference and destructive interference, so that net energy is exactly conserved.
For the slightly mathematicaly inclined, hereís an explanation. Take two wave pulses from two completely separate sources, a and b. Initially, the waves donít overlap, so their dot product (meaning the integral over space of the dot products of their fields) is zero. The linear wave equation keeps that dot product constant. The energy depends on the square of the fields, e.g. (E_a)^2 + (E_b)^2 +2 (E_a*E_b). Although if the waves move so that they overlap the last term is not zero everywhere, its integral over space remains zero.
(republished on 07/28/06)
Follow-Up #1: conservation and interference
I keep reading that destructive interference maintains its conservation of energy because the constructive interference makes up for it. But constructive interference seems to make sense: one wave and another get together to make a higher amplitude wave. It seems like people are saying that since 1 and 1 sometimes makes 2 that cancels out the times when 1 and 1 make 0.
- Sal (age 19)
Santa Cruz, CA, USA
Yep, thatís what weíre saying. The reason is that the energy content of a standard wave goes as the square of the displacements.
Thatís because kinetic energy goes as the square of velocity, and potential energy increases usually go as the square of displacements from the equilibrium point. For electromagnetic waves, the energy is proportional to the square of the amplitude of the wave.
So in the energy balance equation, weíre saying that the times when 1&1 give 0 are balanced by the times when 1 & 1 give 22
(published on 11/10/06)
Follow-Up #2: interference and conservation
But what about pure constructive interference ?
If I split a lser beam and then merge the wo beams together in phase. the will have constructive interference everywhere. but if their fields are summed the power is multiplied by 4. How do we get 4 times the energy from merging two beams ?
It's really the same answer. The beams are coming in from different directions, if they aren't actually the same beam. That inevitably leads to a mixture of constructive and destructive interference, with the overall result obeying conservation of energy.
(published on 10/21/09)
Follow-Up #3: Interference effects between two violins
Why we don't observe the interference effects between the sound waves generated by two violins?
- Eduardo (age 19)
At least I can hear the interference effect if there are two violins with a very slight degree of mis-tuning or fingering. Any vibrato by either one will mask the effect. Fortunately the violinists all tune their instruments very closely to the oboe's A.
(published on 10/02/12)
Follow-up on this answer.