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As can be proven, Maxwell's equations don't work for spherical or cylindrical waves, but these waves satisfy the wave equation. Are there any other examples of this type that satisfy the wave equation but not Maxwell's equations?

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    $\begingroup$ Can you elaborate what you mean by "Maxwell's equations don't work for spherical or cylindrical waves"? Any reference will also do. $\endgroup$ – Jia Yiyang Feb 2 '14 at 13:54
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    $\begingroup$ Jackson chapter 8 section 8.2 discusses cylindrical waveguides and cavities, how does that example act as something that doesn't satisfy Maxwell's equations yet satisfy the wave equation? If you read Jackson ch. 6 sec 6.1-6.2 he derives two wave equations from Maxwell's equations in vacuum that are equivalent to Maxwell's equations in vacuum when you impose the Lorentz condition, so I'd be interested to see what you mean? Maybe you mean Maxwell's equations in the presence of source charges, which do not satisfy the wave equation (Landau vol 2 sec 51)? $\endgroup$ – bolbteppa Feb 2 '14 at 19:03
  • $\begingroup$ Cylindrical waveguides is a good example. Let me have a look at Jackson and Landau. However, this question is headed the other way. Spherical and cylindrical waves do satisfy the differential wave equation, but do not satisfy Maxwell's equations due to the additional constraints of ∇∙E=0, and ∇∙B=0, that are, as you said, the indication of the absence of source charges (If I am not wrong haha). $\endgroup$ – Artemisia Feb 6 '14 at 10:18
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Mmm, the wave equation is a consequence of Maxwell's equations plus a gauge condition, and once you cast the wave equation into spherical, or cylindrical coordinates, you can easily find solutions that satisfy it. Maybe you mean waves that are spherically symmetric? In that case, the problem is more about finding non-trivial spherically symmetric vector fields (you can't) and nothing to do with Maxwell's equations. However, the waves can have a spherical character to them, as is the case with dipole radiation.

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