## 12 October 2007

### Huygens' principle

A weird fact I came across yesterday: Huygens' principle for the wave equation.

When you throw a pebble on the surface of some water, circular waves propagate outward from that point. However, inside the circular wavefront there will still be some sort of disturbance.

Light, in three dimensions, doesn't work the same way. If I have a pulse of light that lasts one second, then one second after that pulse stops, then the wave resulting from that pulse will be supported on the annulus from one light-second to two light-seconds from me. There will not be any sort of wave going on closer than one light-second from me.

I doubt this was known to Huygens, simply because he lived in the seventeenth century when they didn't have partial differential equations. They had waves, though, and the page I linked to above implies that Huygens knew that the leading edge of a wave travels at a constant speed, usually denoted by c; the most important example of waves is light waves, so c is now reserved for the speed of light. (I suspect he was aware of ripples in water; it's less obvious that there aren't analogous ripples in three dimensions, though, simply because there don't seem to be three-dimensional phenomena where the wave travels at a reasonable speed that one could observe. So observation is not particularly useful here, and one simply has to trust the symbols.)

Okay, this seems reasonable... things in different numbers of dimensions behave differently. But in four, six, eight, ... dimensions, solutions to the wave equation behave like the two-dimensional case (the surface of water), in five, seven, nine, ... dimensions they behave like the three-dimensional case (like light). I wouldn't have been surprised to learn that the behavior is different in low-dimensional space than in high-dimensional space. Or even that the behavior was different in "medium"-dimensional space -- it's similar to how things work in the theory of manifolds, where one- and two-dimensional manifolds are basically trivial to study, five-dimensional and higher manifolds aren't that hard, but three and four dimensions are difficult. (I'm vastly oversummarizing here; this isn't my area, and I'm just going on things I hear in the halls.) But who would think it would depend on the parity?

It seems that the dependence on the parity falls out of the series solutions to the spherically symmetrized wave equation, and this equation includes the dimension as a coefficient. I'm having a bit of trouble parsing the page I linked to (although I'm not working too hard on it). The author comments:
It would be interesting to work out the connections between Huygens' Principle and the zeta function (whose value can only be given in simple closed form for even arguments) and the Bernoulli numbers (which are non-zero only for even indices).

This isn't something I would have thought of.

John Armstrong said...

Well, remember that waves are all really about Fourier transforms -- unitary representation theory.

So what sort of theory talks about spherically symmetric propagation in \$n\$-dimensional space? The Fourier theory of \$SO(n)\$! And we know that \$SO(2k)\$ and \$SO(2k+1)\$ behave very differently.

Charles said...

Well, there is actually one big three dimensional wave phenomenon that Huygen's could have (and did) study: sound waves. He also believed that light was a wave (an unfashionable belief at the time, overshadowed by Newton's opposition), and so he had two examples of waves in three dimensions that didn't ripple like waves in two dimensions do.

Isabel said...

Charles,

I thought of that, but it's a lot harder to see the propagation of sound waves than of water waves. Can you really be sure, purely from low-technology observation, that sound waves don't ripple?

Charles said...

The fact that the universe is relatively quiet does it. If they rippled, you'd hear the same sound over and over again several times, which we don't. Instead, we hear something once and then it is done. (this is ignoring the case where there is something for the wave to reflect off of and give an echo)

Mohan K.V said...

Nice Post :-) ! You should definitely take a look at this paper: