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I know that classical wave equation holds for spherical wave fronts. In addition, Huygens' principle states that any wave front is a superposition of many spherical wavelets, so why does the equation NOT hold for any wave front such as the cylindrical?

Qmechanic
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Tofi
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    The wave equation is true for any shape wave front. Spherical waves and plane waves are the most commonly discussed, but by no means the only solutions. – Paul T. Mar 11 '16 at 20:40
  • @ Paul T. The wave equation may allow any shape wave front, but Huygens principle does not hold for any shape wave front. For example cylindrical waves do not propagate 'cleanly' without a wake whereas spherical waves and plane waves do. – user45664 Oct 11 '16 at 22:42

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See my paper https://www.researchgate.net/publication/316994209 or the update https://www.researchgate.net/publication/340085346

This paper shows geometrically why plane and spherical waves propagate without a wake. The same approach could be used to show that some other wave shape would not propagate without a wake.

Basically an impulsive sphere and an infinite impulsive plane are examined. The wave field as a function of time at an external point in space is computed, taking into account 1/r type spherical spreading. Then the time derivative of the wave field is taken to get the wave fronts. The derived wave fronts consist of Dirac Deltas corresponding to the start (and end for the sphere) of the wave fields. (The wave field in between is constant so the derivative there is zero.) So for those sources there are no wakes and the waves propagate cleanly. If the process was repeated for a cylindrical wave the results would probably (or certainly) be different.

user45664
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  • https://physics.stackexchange.com/a/160573/45664 corroborates some of the computation for the sphere using an altogether different approach. – user45664 May 19 '17 at 23:23
  • Why would the principle not hold for cylindrical waves? Doesn’t it hold for any 3-dimensional wave? –  Oct 03 '20 at 16:15
  • I think it holds only for spheres and planes in 3D (in my opinion). – user45664 Oct 03 '20 at 16:26
  • Do you mind explaining why this is so? We usually learn it applies to any shape of wave front. Do you think it’s a subject that needs more studying perhaps? –  Oct 03 '20 at 16:50
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    I think it needs more study. (see my referenced paper) – user45664 Oct 03 '20 at 17:07
  • Do you mind answering a question about your paper? Regarding the wake you mention for cylindrical waves, is your reasoning the same as the one explained in these comments, or is it a different approach? Please reply, I tried but couldn’t understand on my own. Thanks. –  Oct 11 '20 at 11:30
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    It is a different approach--more of a geometrical approach. But I have not analyzed anything but planar and spherical waves. – user45664 Oct 11 '20 at 16:42
  • Thanks for the clarification. Wonderful work. –  Oct 11 '20 at 16:45