# Q & A: wave interference and conservation

Most recent answer: 03/08/2014
Q:
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?
- Steve
Northfield MN
A:

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

Q:
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
A:

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

Q:
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 ?
- Coby
A:
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.

Mike W.

(published on 10/20/2009)

## Follow-Up #3: Interference effects between two violins

Q:
Why we don't observe the interference effects between the sound waves generated by two violins?
- Eduardo (age 19)
mexico
A:
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.

LeeH

(published on 10/02/2012)

## Follow-Up #4: two-beam interference

Q:
Suppose a laser is split into two coherent beams and these beams are then superimposed such that they intersect in an infinite ray to form a new light beam which is the combination of the two. Now suppose one of the two beams is made to be perfectly out of phase with the other; should I expect the beam which is the combination of the two to disappear since the two light beams that make it up will always destructively interfere? If so, how does the conservation of energy work in this case? Moreover, what does this "beam" interact with? For example, suppose there are two beams of light that perfectly destructively interfere for a distance of the diameter of an atom, and one places an atom at such a destructive point -- will the light beam pass straight through?
- Thomas (age 17)
South Africa
A:

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)