Wave Interference and Conservation
Most recent answer: 10/22/2007
- Steve
Northfield MN
Nice question!
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, here's an explanation. Take two wave pulses from two completely separate sources, a and b. Initially, the waves don't 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.
Mike W.
(published on 10/22/2007)
Follow-Up #1: conservation and interference
- Sal (age 19)
Santa Cruz, CA, USA
Yep, that's what we're saying. The reason is that the energy content of a standard wave goes as the square of the displacements. That's 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, we're saying that the times when 1&1 give 0 are balanced by the times when 1 & 1 give 22=4.
Mike W.
(published on 10/22/2007)
Follow-Up #2: interference and conservation
- Coby
Mike W.
(published on 10/20/2009)
Follow-Up #3: Interference effects between two violins
- Eduardo (age 19)
mexico
LeeH
(published on 10/02/2012)
Follow-Up #4: two-beam interference
- Thomas (age 17)
South Africa
There's a tricky assumption hidden in your phrase "these beams are then superimposed such that they intersect in an infinite ray to form a new light beam ". How would that process work? If you try merging the beams at a small angle, then across the width they will alternate between constructive and destructive interference. The way they can really be merged is with a beam-splitter, where one is partially transmitted and the other partially reflected. Then you get two combined beams, going in different directions. If the phase is adjusted to eliminate one beam the other one gets all the energy
If you take some interference pattern with nodes, it's true that any molecule at a node won't absorb any energy. This sort of thing is done routinely in various experiments.
Mike W.
(published on 03/08/2014)