Wave interference: where does the energy go?

Apr 07 2010 Published by under Optics, Physics

Last week was a relatively lousy one for me, but it was made up in part by getting a good question from a student on waves and interference after class.  It's really nice to get a question that indicates a genuine interest in the science (as opposed to just wanting an answer to homework), and I thought I'd discuss the question and its answer as a post.

The situation in question is as follows: suppose you have a harmonic wave on a string traveling to the right such that in a snapshot of time, the string looks as follows:

This wave carries energy, and there is a net flow of energy to the right.  Now suppose we excite the string with an additional wave of the same frequency and amplitude, but completely out of phase.  The sum of the two waves then vanishes:

The two waves cancel each other out, leaving a completely unmoving string due to destructive interference.  My student asked me: what happens to the energy?  As posed, it seems that we started with two waves carrying energy, but they canceled each other out, leaving no energy!  This interpretation cannot possibly be correct, so where is the flaw in our description?

There are actually two aspects to the answer that I want to address, each of which is  rather important in the understanding of wave phenomena.  The first of these is the observation that the answer to the question depends on how the waves were generated in the first place!

Let us formulate the problem as follows: one wave is generated far away on the left of the string, perhaps by a hand shaking it up and down.  The second wave is generated locally at a point on the string by some sort of applied force:

Individually, the first wave is propagating entirely to the right; the second wave spreads out in both directions from the point of application.  What does the situation look like when both waves are applied to the string at the same time?  We have:

There is complete destructive interference to the right of the second excitation.  Looking at the picture, though, the actual physical result of our two excitations is that the first wave, propagating to the right, is reflected back to the left!

The proper interpretation is that the second excitation in the center of the string actually adds no energy to the string at all!  In fact, the second excitation on the string just blocks and reflects the first wave.  There is no "energy cancellation" involved.

Waves in a one-dimensional problem like a vibrating string are a bit of a special case; however, a similar argument can be made for higher-dimensional problems.  In general, we can say that the interference of waves doesn't "cancel" energy, it just moves it somewhere else.

The perfect illustration of this is Young's double slit experiment, which I have discussed before:

Light emanating from the two slits in the screen A produces an interference pattern on the measurement screen B, a simulation of which is shown below:

The presence of interference means that more light shows up in some places and less light shows up in other places.  It can be shown, however, using some straightforward mathematics, that the total light collected on the measurement screen B is equal to the total light that passes through the two holes: interference affects where the light goes, but not how much of it there is in total.

This observation -- that interference redirects light, and doesn't cancel it -- actually plays a role in the operation of compact disc players.  The data on a compact disc is encoded in a series of 'pits' surrounded by 'land', which correspond to a '0' or '1' bit.  The disc is read by focusing light onto the surface, and measuring the amount of light reflected back, as crudely illustrated below:

Light from a laser passes through a beam splitter and is focused by a lens onto the disc.  Part of the back-reflected light gets directed by the splitter to the detector, and the absence or presence of light at the detector registers as a '1' bit or '0' bit.

The modulation of light is caused by interference.  Light illuminating 'land' gets directly reflected and collected at the detector.  When a 'pit' is present, however, the focused beam partly reflects from land and partly from pit.  If the pit is taken to be a quarter-wavelength in depth, then the light reflected from the pit is half-a-wavelength out of phase with the light from the land, resulting in destructive interference and less light at the detector:

But, as noted, the light doesn't get "canceled" completely; it just goes somewhere else.  Where does it go?  It gets diffracted into a direction where the lens cannot detect it (color coded for clarity):

To summarize, these are the two thoughts I had when the student asked me his question: (1) one needs to consider all sources of waves when trying to interpret wave interference phenomena, and (2) in general, wave interference results in a change in where light goes.

Even with this in mind, it is easy to come up with circumstances that can really challenge one's understanding of interference.  For instance, let us return to our waves on a string, and consider interference between waves generated by two local excitations:

What do we make of this situation?  We have two excitations, oscillating in sync but with a half-wavelength separating them.  There is destructive interference of the waves on either side of the two excitations, and the string vibrates up and down between the two excitations.

Such a strange string vibration is now known as a nonpropagating excitation, and the first research on such effects was done by my collaborators and I*!  Nonpropagating excitations are actually one-dimensional analogues of the radiationless "invisible" objects that I have discussed previously (here and here, for instance).

I still haven't come up with a great way to explain what is going on here.  Let us imagine that we start applying our excitations with the string at rest.  Energy goes into generating the nonpropagating excitation and some transient waves that propagate away after a short period of time.  When the transients are gone, however, no more energy is added to the system.  Just like the case discussed at the beginning of the post, excitation 2 serves only to reflect waves traveling from the left, and excitation 1 serves only to reflect waves traveling from the right.  The two sources are playing a sort of "wave ping-pong", knocking the waves back and forth between them!  Beyond that odd statement, it is difficult to describe the exact origin of the waves without specifying the physical mechanism used to excite the string.

So, there's no disappearance of energy in wave interference, but that doesn't mean that interference isn't a really weird phenomenon!

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* My first paper on the subject is M. Berry, J.T. Foley, G. Gbur and E. Wolf, "Nonpropagating string excitations", Am. J. Phys. 66(2) (1998), 121.  It is one of my favorite papers because of the weirdness and elegance of the subject, and also because it is probably the only time I'll have two world-renowned brilliant physicists -- Emil Wolf and Michael Berry -- as coauthors on the same paper.  The nonpropagating excitation described in the paper is more complicated than the two point source case described here; that example was suggested by B. Denardo, "A simple explanation of simple nonradiating sources in one dimension," Am. J. Phys. 66 (11) (1998), 1020.

8 responses so far

  • meichenl says:

    Great post! I remember wondering about a similar question during my waves class. I wanted to consider something like your first picture, where two waves, each with energy, add up to nothing. Instead of thinking about sources, I imagined the waves were simply traveling down a dispersionless string from opposite directions. In that case when the waves destructively interfere the amplitude is zero everywhere, but the string is in motion. The energy is entirely kinetic.

    We could also imagine two electromagnetic waves with the same frequency propagating towards each other from opposite directions in a vacuum. In that case, the electromagnetic fields could momentarily cancel, and there would be no kinetic energy around. But if the electromagnetic fields were canceling, the magnetic fields should constructively interfere.

    I am going to have to dig out my EM textbook to see what we can say about energy conservation working directly from Maxwell's equation.

    I normally enjoy Am.J.Phys articles, but I'm no longer at a university. Is there a place I can read a copy of your paper?

    • Glad you like the post!

      I normally enjoy Am.J.Phys articles, but I’m no longer at a university. Is there a place I can read a copy of your paper?

      Send me an email, and I'll send you the relevant pdfs directly!

    • Glad you like the post!

      I normally enjoy Am.J.Phys articles, but I’m no longer at a university. Is there a place I can read a copy of your paper?

      Send me an email, and I'll send you the relevant pdfs directly!

  • Wade Walker says:

    Would it help to think about these interference questions the same way we think about real waves in physical media?

    The ideal string that we derive these textbook wave examples from has internal "strain energy" due to tension and compression, similar to pressure energy in air. If you looked at the strain energy before wave cancellation, you'd see wave patterns travelling along the string, just like pressure waves travelling in air. I assume that after wave cancellation, the strain energy would be constant everywhere in the string, but at a higher value that conserves overall energy.

    So with a change of wave medium from string to air, the nonpropagating excitation case could be thought of as two in-phase loudspeakers aimed at each other, but positioned a half-wavelength apart. They'd create a time-varying pressure between them, and silence (albeit with increased static pressure) everywhere else.

    If you increased the distance between the speakers to a whole wavelength, I guess you'd see the opposite -- silence with higher pressure between the speakers, and sound everywhere else.

    But in a physical system like this, it seems like there is still energy flowing out of the sources over time, driving the increase in static pressure which is silently propagating out into the environment.

  • Alex Haig says:

    I'm currently researching ultrasonic guided waves for the non-destructive testing of engineering structures with particular interest in array design for wave mode isolation and directional transduction. I gave a workshop on manipulating superposition last week and I really wish I'd seen this post in time to bring it up in the class. Unfortunately, I've only just found it now, but nevertheless I will be forwarding it on to all of the attendees for them to consider.

    Thanks for producing it!

  • navid says:

    thank you for sharing your thinking time with us. this post is SO helpful.

  • Salem says:

    thank you that was helpful, i found this post when i was thinking about this:
    2 electromagnetic radiations (plane but not infinitely) long in opposite directions and at one instant they completely destroy each other, the same thing on a standing wave when the field is all zero at one instant, where did it go,, for example on a string there is still kinetic energy without potential energy but for EM-radiation where is the energy?