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If I understand magnetism correctly, then two magnetic fields with opposing directions would repel each other. Could this same affect happen to electromagnetic waves? And if so, could it be reproduced in an experiment?

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It is not that magnetic fields repeal each other, but the bodies that are origin of these fields repel if they are oriented in such a way that their fields in the space between them are pointing in opposite directions.

Electromagnetic waves in vacuum obey linear wave equation, so one does not change the other; they pass each other as if the other one did not exist.

In matter, on the other hand, the description of EM field is more complicated, the equation does not need to be linear and there could be phenomenon that would manifest as waves "repelling" or "attracting" each other. Unfortunately, I do not know any specific examples.

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  • $\begingroup$ An example as suggested in your last paragraph: reflection of light from a mirror. $\endgroup$ – Asher Jul 22 '15 at 4:00
  • $\begingroup$ @Asher, I do not see how reflection of light from a mirror is an example of waves repelling. The reflection is due to metal in the mirror, not due to some other wave. $\endgroup$ – Ján Lalinský Jul 22 '15 at 16:47
  • $\begingroup$ What is the metal made of? When considering light as a wave-like quantum object, then it makes sense to consider the particles with which it interacts, such as electrons in a metal, as wave-like quantum objects as well - which they are. $\endgroup$ – Asher Jul 22 '15 at 19:56
  • $\begingroup$ @Asher, you introduce too many assumptions. I never mentioned wave-like quantum objects. Your idea would be an example of electron wave interacting with EM wave. The original question and my answer are not about that; they are about interaction of two EM waves. $\endgroup$ – Ján Lalinský Jul 22 '15 at 20:19
  • $\begingroup$ Ah, indeed. I was thinking about an electric or magnetic field reflecting/repelling an EM wave, not two EM waves repelling each other. My mistake. $\endgroup$ – Asher Jul 23 '15 at 1:13

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