Main source: http://www.mathpages.com/home/kmath242/kmath242.htm
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