# How to best describe wave behavior on even dimensions?

On this article, as far as I understand, the author claims that wave behavior on even dimensions would give rise to many waves with infinitely different velocities, all spreading outwards. In the context of even dimensions, it wouldn’t spread cleanly, but diffusely.

See quote: “For the case of two dimensional space this doesn't work (...) We can still solve the wave equation, but the solution is not just a simple spherical wave propagating with unit velocity. Instead, we find that there are effectively infinitely many velocities, in the sense that a single pulse disturbance at the origin will propagate outward on infinitely many "light cones" (and sub-cones) with speeds ranging from the maximum down to zero.”

However, I was informed by another source (a fellow forum user here) that the best way to describe wave behavior on even dimensions is that it, while going forward, also keeps reflecting back, generating backward waves. Now, which is the right interpretation of the refered article? Because as long as I understand there’s no mention of backward wave canceling only on odd dimensions and not on even dimensions (only that it doesn’t propagate sharply on the latter).

The refered dialogue with fellow user can be found here on comments: Open-open pipe standing waves

• Actually both quotes are just different ways of phrasing the same thing, though I have a minor quibble with the second one. – knzhou Aug 8 '18 at 19:54
• As I see, one point source emanating “a series of waves” of multiple velocities, all outwards, is an altogether different thing than waves going forward and backwards at the same time, hence my confusion with the colleague’s interpretation – Roberto Valente Aug 8 '18 at 20:04
• I’d be very pleased if you could elaborate more on your interpretations, but thanks in advance – Roberto Valente Aug 8 '18 at 20:06
• For instance, see this quote from my second source: “A few interesting examples of how this works. In 3 dimensions the retarded wave is canceled. As a result, we don't hear the echo of our voice reflected from the air (but only as reflections from distant objects). Similarly, a flashlight doesn't blind us, because light does not reflect back from the empty space. However, it would do so in 2 or 4 dimensions.” I might be wrong but it doesn’t seem entirely accurate. From this question: physics.stackexchange.com/a/365491/137288 – Roberto Valente Aug 8 '18 at 20:13
• Sorry, I was getting to it but have been busy! I'll type up a long-ish answer by the end of the weekend. – knzhou Aug 9 '18 at 20:44