Suppose 2 sinusoidal waves undergo totally destructive interference. This results in wave with zero amplitude (and hence zero intensity). What happens to the energy associated with wave (1) and wave (2)?

Asked by: Achilles

Answer

Oh! I like questions like this one, which lead one to arrive at an absurdity using apparently plausible, but impossible in detail arguments. So, lets analyze how this one works:

The main argument is, any wave has an energy associated with it, which is proportional to its amplitude squared, therefore necessarily positive. Then, you take two such waves, let them interfere with phase difference Pi (1.57 will *almost* work) and destroy the two waves. No wave, means no energy, so what happened to the energy?

On a side note, you don't really need total destructive interference -- even partial interference will seemingly violate conservation of energy. But total destructive interference makes the argument a little more drastic.

To start with, when we say a sinusoidal wave, we are talking about waves of the sort

A sin (kx - wt)

The first thing to note is that, such a wave extends indefinitely in time and space, and exists everywhere and every time. With such waves you either have them, or you don't. So you can't cause any sort of interference. Then we must restrict ourselves to waves which are sinusoidal only in part of space, or in part of time.

Now, to have interference, the waves will have to have the same k and same w. Otherwise they will not interfere in terms of energy -- this can be shown, but is probably better left to another question.

The best way to attack the question is with an example. But, the idea can be generalized to any other wave.

Lets take an electrical circuit, with two AC sources of the same frequency, acting on a purely reactive circuit (one that is made of capacitors and/or inductances only) so that no energy is drained by the circuit itself. Let the two sources be exactly in opposite phase -- a phase difference of Pi.

Now, turn on one source. You get a sinusoidal wave on the circuit. It has an energy. Where did the energy of the wave come from? Of course, from the source. Turn it off, and turn on the other source. Again, same result. Now, turn the second one on, too. Result? No wave on the circuit. (Of course, one may need to adjust things rather carefully to get this in practice -- so don't try this at home plugging a capacitor into the AC outlet). Now comes the big question: What happened to the energy of the wave source #1 had set up before we turned on source #2??? The capacitors had charges oscillating, inductors had currents oscillating, all at the same frequency... Where did all that go? The answer is simple. It was absorbed by the second source you just turned on. (Those who know how to analyze circuits, can figure it with two current sources, one capacitor, and letting the second source come on slowly. Then think of the limit of turning it on in an instant). What would happen if I turned on the two sources at exactly the same time? There would not be a wave to start with!

So, the essence of the answer is:

* All waves have sources (and drains). * One must include those in the analysis. * The source that causes the destruction of the wave absorbs the energy.

After all, there is not much point in talking about waves which don't have sources and drains, since they don't interact with anything, and there is not much point in talking about their energy either.
Answered by: Yasar Safkan, Ph.D. M.I.T., Software Engineer, Istanbul, Turkey

Consider the case of two radio transmitters in space, transmitting to earth on the same frequency. Then there will be some places on earth where the two waves cancel out and the received amplitude is zero. But there will be other places where the two sources reinforce each other. So energy is not lost, just moved around. The exact geometry governs the patterns of peaks and valleys.
Answered by: Phil Freedenberg, E.E.D., Exec. VP, Federal Engineering, Inc., Fairfax, VA