Regarding the velocity of waves in even dimensions

Question

A few years ago I asked on Reddit about the behavior of wave propagation in even and odd dimensions. I received this answer:

"The answer lies in the solutions to the wave equations. Essentially, in odd dimensions a wave will propagate at a single characteristic velocity $v$, while in even dimensions it propagates with all velocities $<v$."

Another user added: "If you interpret the mathematics strictly, the speeds are all strictly less than $v$."

This article, however, says in the second paragraph: "Of course, the leading edge of a wave always propagates at the characteristic speed $c$."

For that reason, I was wondering, is that information on Reddit correct? Does the wave, in even dimensions, propagate with all speeds less than $v$, or does it propagate with all speeds equal or less than $v$?

Edit: The original comment (which is linked above) refers to the wave equation in this manner (direct quote):

“(I think the wave equation can approximately be written as v2 d2 /dx2 - d2 /dt2 = 0 in terms of v, at least up to some dimensionless constant)”

Answer 1

I assume a wave equation $(\Box +m^2) f = 0$. This is simply the wave version of Einstein's famous relation $E^2 = m^2c^4 + p^2c^2$. Thus $v \in [0,c)$ for $m\neq0$ and $v=c$ for $m=0$ for any positive number of space dimensions.

Answer 2

It doesn't make sense to critique statements about wave speed when they don't even specify what speeds they are talking about.

In $n$-spherical waves there are at least three speeds, all of which are generally different: phase speed, group speed and leading edge speed. The latter is obviously always $c_0$, the speed of wave in the medium. The former two are generally different—both from $c_0$ and from each other. Moreover, these speeds also depend on distance from origin, approaching $c_0$ as the distance increases (this is because the wavefronts flatten, become closer to those of plane waves, which always travel at $c_0$).

In ref. 1 we can find the expressions for phase and group speeds of cylindrical and spherical monochromatic waves (expressible in cylindrical and spherical Hankel functions, respectively). The graphs for order $0$ functions can also be found there (figure 6):

As you can see, group speed is, at small distances $r$ from origin, lower than $c_0$, but at some point it overtakes $c_0$, then reaches a maximum and starts to asymptotically approach $c_0$ from above as $r\to\infty$. Phase speed, on the other hand, is always less than $c_0$. This one might be what the Reddit posters were referring to.

In spherical wave case there's not much interesting: all three speeds are equal to $c_0$ and don't depend on $r$ (which is to be expected, since it's an odd-dimensional space).

References

1. Dash, Ian & Fricke, Fergus. (2009). Phase Velocity and Group Velocity in Cylindrical and Spherical Waves. 10.13140/2.1.1054.4968