Sound waves in even dimensions

[autodidude]

I read on the 'Ask A Mathematician/Ask A Physicist' blog that sound waves behave differently in an even-number of dimensions than they do in odd-number dimensions - that they 'double-back'. Why is this - and what does 'double-back' mean?

http://www.askamathematician.com/2012/05/q-what-would-life-be-like-in-higher-dimensions/

"Sound waves too. In odd dimensions other than 1 (3, 5, 7, …) waves behave the way we normally see and hear things: a wave is formed, it moves out, and it keeps going. However, in even dimensions, and 1 as well, (1, 2, 4, 6, …) waves “double back” on themselves. You can see this in ripples on the surface of water (2-D waves)."

 

[LightHero]

Double-backing in this case refers to the behavior of sound waves in even dimensions where, instead of continuing on in a straight line, the wave reflects off itself and effectively creates a loop. This occurs because the wave has nowhere else to go - in two dimensions there is no third dimension to carry the wave out into, so it reflects back on itself. In odd dimensions, however, the wave does have somewhere else to go, and so can continue on in an outward trajectory.