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Drop a stone in the pond...a wave propagates radially from the source. The conservation of energy says the wave must decay proportionally to the radial distance. If I drop a steel I-beam in the pond, the same concept holds, only it can be considered a finite linear array of point sources. If the energy must still decay radially, then how do plane waves ever form?

I know we can generate them in labs and obviously express the math, but in nature, do plane waves really exist?

OK...so let's assume it was an infinitely long I-beam. Then I guess you could argue you get plane waves. But look at the assumptions: 1) infinite I-beams don't exist (nothing is infinite) and 2) you have to assume there are ZERO irregularities in the I-beam. These concepts are purely mathematical.

OK...OK....you're "zooming in" on the radial wave and it appears like a plane wave. But do you account for the fact that the wave must always be decaying in time (assume there is no external dissipation involved)?

Restating the main question...do plane waves really exist in nature? If they don't then why do they show up in theories? (Good example is plane waves hitting an open slit for diffraction studies...you started with something that is conceptual, not physical?)

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    $\begingroup$ Yes, plane waves are a mathematical idealization. They show up in theories because they're mathematically simple, and because we can build up any (physical) wave out of plane waves. $\endgroup$
    – knzhou
    Aug 10 '16 at 1:22
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    $\begingroup$ Consider sunlight at Earth's orbit. If you make two measurements of the separated by 15 meters you can observe an angular difference of up to $2 \times 10^{-10}\,\mathrm{rad}$ or a fractional intensity difference of up to $4\times 10^{-10}$. Can you design an experiment for which one of those would be the leading imprecision? Neither can I, which means that sunlight is effectively plane-waves for a very large class of experiments. $\endgroup$ Aug 10 '16 at 2:04
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No, "real" plain waves do not exist in nature and neither does anything "exist" the way a physical theory describes it. That's about as trivial as it is irrelevant. We are not performing experimental mathematics here. In physics we are merely finding approximate explanations to natural observations. My first theory professor said it this way to the entire classroom:

"Physics is the art of approximation. If any of you are not comfortable with this idea, then you should leave this classroom right now and try your luck at the philosophy department."

Then he made the following joke:

"How does a theoretical physicist describe a cow? Well, first he assumes that she is spherical. If that doesn't work, he covers her homogeneously with milk!"

That's pretty much all there is to say about it. Plane waves are one of our favorite spherical cows. Sometimes she gives us perfectly white milk and sometimes she doesn't... in which case we move on to cow #2, which produces harmonic oscillations.

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  • $\begingroup$ What is the deviation of free electromagnetic modes in space from the ideal plane waviness, though? It can't be that horrible of an approximation or are you simply stating that even those have a finite deviation, no matter how small, from the description of a perfect plane wave? $\endgroup$ Aug 10 '16 at 12:17
  • $\begingroup$ P.S. One of my grad school professors made the same comment about the "art of approximation" but didn't lighten the mood with the spherical cow joke ;P $\endgroup$ Aug 10 '16 at 12:18
  • $\begingroup$ @honeste_vivere: Thankfully there are rational criteria to decide when an approximation is useful. I hope I didn't leave the impression that this is all just some vague guessing game. In that theory class we were always asked to give a rough estimate of higher order effects to check if approximations were appropriate. $\endgroup$
    – CuriousOne
    Aug 10 '16 at 13:01
  • $\begingroup$ I am glad to hear it (i.e., checking the higher order effects) and no, it was not vague. I was merely curious because when I read your answer the first time I think I misread it. $\endgroup$ Aug 10 '16 at 13:07
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Plane waves are useful because we can take any physical function of space, e.g. some field, and Fourier transform it to represent it as a sum (well, integral) of plane waves. This is often a very useful way to approach complicated problems. For example Fourier developed the technique as a way of solving the heat equation, and it's the way quantum field theory is developed.

But a plane wave has obvious unphysical properties:

  • it is of infinite length, so it must have existed for an infinite time in the past and continue to exist for an infinite time into the future

  • the wavefronts have infinite area i.e. if you take an infinite plane normal to the direction of propagation the intensity and phase is constant everywhere in that infinite plane

So no, plane waves don't exist. However in many cases we have waves where:

  • the length is large compared to the wavelength/the time the wave has existed is long compared to the period

  • the area of the wavefront is large compared to the wavelength (squared)

So while no plane wave exists there are plenty of situations where waves exist that are experimentally indistinguishable from plane waves. Many physicists (including me) will quite happily talk about plane waves without having to continuously append the caution that they don't really exist.

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