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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)”

  • Related: https://physics.stackexchange.com/q/129324/2451 – Qmechanic Nov 11 '20 at 21:47
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    I mean, it all depends on how you define the phrase "the speed of wave propagation", which is ambiguous anyway. Nobody is disagreeing on the actual issue here, which is what the Green's function for the wave equation looks like in even dimension. – knzhou Nov 13 '20 at 23:54
  • Kind of like agreeing that you have a pile of sand that's $183.593$ kilograms, but then asking people to clarify whether that sand pile is "big", or whether it is only "large". Does the specific word you use really matter? – knzhou Nov 13 '20 at 23:55
  • @knzhou You’re not wrong. I know it’s a minor issue but I wanted to check anyway if those comments held any truth. Do you know the reasoning behind them? –  Nov 14 '20 at 00:29
  • @knzhou Why is “the speed of wave propagation” ambiguous? –  Nov 14 '20 at 11:46
  • What kind of wave are we talking about here? A solution to the d'Alembertian equation $\Box f = 0$ ? https://en.wikipedia.org/wiki/D%27Alembert_operator – my2cts Nov 18 '20 at 09:47

2 Answers2

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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.

my2cts
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  • Thanks for your answer. I was referring to this wave mentioned in this comment (is it the same as your answer?): https://www.reddit.com/r/Physics/comments/7ad0ju/why_exactly_its_said_that_huygens_principle/dp99kt0/?utm_source=share&utm_medium=ios_app&utm_name=iossmf&context=3 –  Nov 18 '20 at 10:49
  • Also, could you explain your answer in less mathematical terms? Does the wave propagate with less than v or equal or less than v in even dimensions? I just wanted to understand if the comment I linked above is accurate. –  Nov 18 '20 at 10:53
  • Your question should best be self contained so please specify the wave equation. – my2cts Nov 18 '20 at 10:58
  • The comment is linked in the original question. But I’ll try to make a useful edit. –  Nov 18 '20 at 11:03
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    This is true for a plane wave propagating in free space. But the Green's function has a tail in even numbers of spatial dimensions. I took the original question to be asking how the Green's function behaves near the light cone. – Andrew Nov 18 '20 at 11:17
  • I tried to specify the wave equation by editing the post. I couldn’t properly paste the equation here, so I would appreciate if someone could make an edit. –  Nov 18 '20 at 11:22
  • @Andrew exactly. Do you know how to answer it? –  Nov 18 '20 at 12:51
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    I wrote an answer but @Ruslan pointed out that my answer wasn't correct :) (There is a discontinuity in the Green's function right at the light cone that is very delicate to handle properly). He pointed to his answer on math stack exchange to a similar question that looks very good: https://math.stackexchange.com/questions/3867619/how-to-make-sense-of-the-greens-function-of-the-4d-wave-equation – Andrew Nov 18 '20 at 14:09
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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):

phase and group speeds of cylindrical waves of order 0

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
Ruslan
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