Sunday Function

Happy Easter, everyone!

This Sunday Function is going to be short and sweet, since unfortunately I've got a lot to do before Monday. Let's get down to business!

Here's a function I've made up out of thin air. It's pretty arbitrary - in fact, it's discontinuous at x = -1/2 and not differentiable at x = 0:


Here is a huge pile of sine and cosine waves, with one bonus constant line which, if you'd like, you can think of as a cosine wave with infinite wavelength:


And here is what you get if you add all of those waves together and graph the result:


It looks quite a bit like the random crazy function I posted first, right? Sure the resemblance is a little hazy, but it is there. What I've done is to pick the relative sizes of those waves very carefully to reproduce my original function as closely as possible. As it is, this is pretty decent for only including a total of 9 waves. Could I make it more accurate by adding more waves of even smaller wavelength? You bet. I won't plot the waves separately since the graph would be an illegible mess, but here's a plot of a total of 61 waves:


Remarkable, right? By adding more and more waves I could get better and better, and it's mathematically provable that I can make the error smaller than any arbitrarily tiny amount by simply adding enough waves.

The key thing to notice is that the original graph didn't have anything to do with waves. But through knowledge of the mathematics of waves - of Fourier series, formally - I can find a relationship between that wacky arbitrary function and the well-known seemingly mundane trig functions. It gives insight into a relationship that otherwise would have been hidden.

I bring this up because of a feisty discussion that erupted a couple items down in the "What is light?" post. A commenter of unorthodox views is determined to believe that there must be a physical medium for light to "wave", despite the null result of Michelson-Morley and others and the astonishing success of both the classical covariant formulation of Maxwell's wave mechanics and the modern quantum electrodynamic formulation. This doesn't convince him because all other waves in his experience actually displace something in position - water, sound, etc.

And of course this is true, if you define a wave as "objects undergoing positional displacement in a certain pattern". But that's an impoverished and stagnating view. It's akin to insisting that Bill Gates is poor because he doesn't physically carry much cash. The whole point in physics is to recognize and mathematically express connections that reveal different and diverse phenomena to be aspects of the same underlying principle. Sure, positional displacement is one kind of wave. It's not the only kind. You can have waves of heat, pressure, gravitational field strength, you name it. Some of these can be thought of as mediated by a physical medium being displaced in position, and others can't. The underlying mathematics is the same, and we now have the power to explain very different problems with simple and unified principles.

Perhaps this reads more like Sunday Sermon! Well, it is Easter, so I hope that such a thing is forgivable.

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I haven't read the original comments, but here is how I think about (simple, classical) light waves:

Suppose there is an electric field E in some volume V. Pick an arbitrary point in V and measure the electric field. You will get the direction and magnitude of the field. Usually this is visualized as an arrow originating from our point. The crucial thing to remember is that this arrow doesn't really have a physical size, it's just a way to visualize the electric field. In actuality you still only have your (infinitesimal) point that has an "invisible" property called the electric field.

The same argument can be made for the magnetic field. Furthermore, a changing electric field induces a magnetic field, and vice versa. In other words, if the electric field is changing in time in one point, the magnetic field will change in the neighbouring points, which changes the electric field in their neighbouring points etc etc.

The result is that light waves in a way DO have a physical medium to propagate through. It is just not enough to look at our 3 spatial dimensions. The change occurs in the direction and magnitude of the electric and magnetic fields (which don't have a physical "size"), and since the neighbouring points are sensitive to these changes, we get a wave propagating through space.

Of course things get more complicated if you think about quantum or relativistic stuff :)

As I understand it, the "relativistic stuff" means that under a lorentz "boost", the electric field partly gets transformed into a magnetic field.

Out of idle curiosity, here's a mathematical question:

Take the difference between the original function and the n^th term of its fourier series. Now count the number zeroes of this difference function and call that Z(n).

Will you ever find Z(n) has a non-zero limit as n->infinity? What is the asymptotic behavior of Z(n)?

My guess would be you normally find Z(n) \propto n^2, but I don't know.

But can your precious waves explain Extended Bell Curve? I thought not.

Sorry, I now return you to your regularly scheduled erudite discussion already in progress.

By Abby Normal (not verified) on 06 Apr 2010 #permalink

meichenl, the nth Fourier coefficient goes to zero as n->infinity.

So if I understand your question correctly, you should always find all the zeroes of the original function in the limit?