I am about to teach a second year undergraduate class on applied differential equation (first time) and, while I won't have time to go into the details, I wanted to show my students the difference between the propagation in even dimensions and the propagation in odd dimensions. Namely, the fact that in even dimensions one gets ripples but not in odd dimensions. Are there simple solutions in one (or 3) dimension and two dimensions that I can use to illustrate this? Thanks

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    $\begingroup$ You may benefit from the answers to this MSE question: Why do odd dimensions and even dimensions behave differently?. I especially like this response from Noam Elkies: "The Euler characteristic of a sphere of dimension 𝑛 is 1+(−1)𝑛 which is zero iff 𝑛 is odd. So you can comb an even-dimensional coconut (whose surface is an odd-dimensional sphere) but not an odd-dimensional one." $\endgroup$ Oct 12 '19 at 17:09
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    $\begingroup$ See here or, for something that doesn't look as scary, here. A longer but possibly more student-friendly explanation is here. $\endgroup$
    – J.G.
    Oct 12 '19 at 18:13
  • $\begingroup$ Pretty hard to cover >1-dimensional wave equation in an undergrad diffeq course. It's a fascinating fact to mention, but you shouldn't get your hopes up that you'll have time to explore something like that in detail. $\endgroup$
    – Jake Mirra
    Oct 17 '19 at 13:25

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